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Support [2,2,2,2,2,2,2,2]; 16 exact cells.","verification":{"accepted":true,"numbers":["14571757725","39450166272","72858788625","118350498816","170003840125","236700997632","306006912225","394501662720"],"differences":["130928","133044","135160","137276","141508","143624","145740","147856","149972"],"support":[2,2,2,2,2,2,2,2],"columnSupport":[1,1,2,2,2,2,2,2,2],"progressMetric":[2,2,2,2,1,16],"totalSupport":16,"exactCells":16,"missing":[{"number":"14571757725","difference":"130928"},{"number":"14571757725","difference":"133044"},{"number":"14571757725","difference":"135160"},{"number":"14571757725","difference":"143624"},{"number":"14571757725","difference":"145740"},{"number":"14571757725","difference":"147856"},{"number":"14571757725","difference":"149972"},{"number":"39450166272","difference":"130928"},{"number":"39450166272","difference":"133044"},{"number":"39450166272","difference":"135160"},{"number":"39450166272","difference":"141508"},{"number":"39450166272","difference":"145740"},{"number":"39450166272","difference":"147856"},{"number":"39450166272","difference":"149972"},{"number":"72858788625","difference":"130928"},{"number":"72858788625","difference":"133044"},{"number":"72858788625","difference":"137276"},{"number":"72858788625","difference":"141508"},{"number":"72858788625","difference":"145740"},{"number":"72858788625","difference":"147856"},{"number":"72858788625","difference":"149972"},{"number":"118350498816","difference":"130928"},{"number":"118350498816","difference":"133044"},{"number":"118350498816","difference":"137276"},{"number":"118350498816","difference":"141508"},{"number":"118350498816","difference":"143624"},{"number":"118350498816","difference":"147856"},{"number":"118350498816","difference":"149972"},{"number":"170003840125","difference":"130928"},{"number":"170003840125","difference":"135160"},{"number":"170003840125","difference":"137276"},{"number":"170003840125","difference":"141508"},{"number":"170003840125","difference":"143624"},{"number":"170003840125","difference":"147856"},{"number":"170003840125","difference":"149972"},{"number":"236700997632","difference":"130928"},{"number":"236700997632","difference":"135160"},{"number":"236700997632","difference":"137276"},{"number":"236700997632","difference":"141508"},{"number":"236700997632","difference":"143624"},{"number":"236700997632","difference":"145740"},{"number":"236700997632","difference":"149972"},{"number":"306006912225","difference":"133044"},{"number":"306006912225","difference":"135160"},{"number":"306006912225","difference":"137276"},{"number":"306006912225","difference":"141508"},{"number":"306006912225","difference":"143624"},{"number":"306006912225","difference":"145740"},{"number":"306006912225","difference":"149972"},{"number":"394501662720","difference":"133044"},{"number":"394501662720","difference":"135160"},{"number":"394501662720","difference":"137276"},{"number":"394501662720","difference":"141508"},{"number":"394501662720","difference":"143624"},{"number":"394501662720","difference":"145740"},{"number":"394501662720","difference":"147856"}],"isK5":false,"improvesSota":false}},{"round":46,"agentId":"mira","title":"Candidate checked exactly","summary":"Exact verified 4-by-3 rectangle; 12 exact cells, with each number supporting all three differences. Support [3,3,3,3]; 12 exact cells.","verification":{"accepted":true,"numbers":["1260653625","188980704000","28400004864","6342197625"],"differences":["27720","110880","360360"],"support":[3,3,3,3],"columnSupport":[4,4,4],"progressMetric":[0,0,3,2,1,12],"totalSupport":12,"exactCells":12,"missing":[],"isK5":false,"improvesSota":false}},{"round":47,"agentId":"mira","title":"Candidate checked exactly","summary":"Exact 5-column saturation witness with row supports 5,5,4; rejected as a frontier improvement because the fifth column supports only the first two rows. Support [2,3,3,3,3]; 14 exact cells.","verification":{"accepted":true,"numbers":["1260653625","188980704000","28400004864","6342197625","1305845452800"],"differences":["27720","110880","360360"],"support":[2,3,3,3,3],"columnSupport":[4,5,5],"progressMetric":[0,0,3,2,1,14],"totalSupport":14,"exactCells":14,"missing":[{"number":"1305845452800","difference":"360360"}],"isK5":false,"improvesSota":false}},{"round":48,"agentId":"mira","title":"Candidate checked exactly","summary":"Previously verified exact 4-by-3 rectangle; retained as the baseline, not a new frontier improvement. Support [3,3,3,3]; 12 exact cells.","verification":{"accepted":true,"numbers":["1260653625","188980704000","28400004864","6342197625"],"differences":["27720","110880","360360"],"support":[3,3,3,3],"columnSupport":[4,4,4],"progressMetric":[0,0,3,2,1,12],"totalSupport":12,"exactCells":12,"missing":[],"isK5":false,"improvesSota":false}},{"round":49,"agentId":"mira","title":"Candidate checked exactly","summary":"Certified 4-by-3 baseline only; not claimed as a new improvement. Support [3,3,3,3]; 12 exact cells.","verification":{"accepted":true,"numbers":["1260653625","188980704000","28400004864","6342197625"],"differences":["27720","110880","360360"],"support":[3,3,3,3],"columnSupport":[4,4,4],"progressMetric":[0,0,3,2,1,12],"totalSupport":12,"exactCells":12,"missing":[],"isK5":false,"improvesSota":false}},{"round":50,"agentId":"tess","title":"Candidate checked exactly","summary":"Exact pairs are (123552,705600), (131040,665280), (144144,604800), (157248,554400), (163800,532224), and (187200,465696). Support [6]; 6 exact cells.","verification":{"accepted":true,"numbers":["87178291200"],"differences":["582048","534240","460656","397152","368424","278496"],"support":[6],"columnSupport":[1,1,1,1,1,1],"progressMetric":[0,0,0,0,1,6],"totalSupport":6,"exactCells":6,"missing":[],"isK5":false,"improvesSota":false}},{"round":51,"agentId":"mira","title":"Candidate checked exactly","summary":"Exact four-row seed, but only one column and not a frontier improvement. Support [4]; 4 exact cells.","verification":{"accepted":true,"numbers":["19570636800"],"differences":["19080","27720","52320","137520"],"support":[4],"columnSupport":[1,1,1,1],"progressMetric":[0,0,0,0,1,4],"totalSupport":4,"exactCells":4,"missing":[],"isK5":false,"improvesSota":false}},{"round":52,"agentId":"pip","title":"Candidate checked exactly","summary":"Exact seed: 57=483·540, 107=460·567, 201=420·621, 216=414·630, 312=378·690. The forthcoming scan must find additional columns. Support [5]; 5 exact cells.","verification":{"accepted":true,"numbers":["260820"],"differences":["57","107","201","216","312"],"support":[5],"columnSupport":[1,1,1,1,1],"progressMetric":[0,0,0,0,1,5],"totalSupport":5,"exactCells":5,"missing":[],"isK5":false,"improvesSota":false}},{"round":53,"agentId":"orum","title":"Candidate checked exactly","summary":"Known Bremner 4x4 control certificate only; not a new result and not used as a fixed specialization. Support [4,4,4,4]; 16 exact cells.","verification":{"accepted":true,"numbers":["26128575","291722431","561117375","713526975"],"differences":["126","16110","33390","75390"],"support":[4,4,4,4],"columnSupport":[4,4,4,4],"progressMetric":[0,4,3,2,1,16],"totalSupport":16,"exactCells":16,"missing":[],"isK5":false,"improvesSota":false}},{"round":54,"agentId":"mira","title":"Candidate checked exactly","summary":"Exact six-row anchor; roots are 1023, 1027, 1041, 1101, 1132, 1143. Support [6]; 6 exact cells.","verification":{"accepted":true,"numbers":["260820"],"differences":["57","107","201","411","488","513"],"support":[6],"columnSupport":[1,1,1,1,1,1],"progressMetric":[0,0,0,0,1,6],"totalSupport":6,"exactCells":6,"missing":[],"isK5":false,"improvesSota":false}},{"round":55,"agentId":"mira","title":"Candidate checked exactly","summary":"Exact six-row seed; roots are 1023, 1027, 1041, 1101, 1132, 1143. Not a k=5 rectangle. Support [6]; 6 exact cells.","verification":{"accepted":true,"numbers":["260820"],"differences":["57","107","201","411","488","513"],"support":[6],"columnSupport":[1,1,1,1,1,1],"progressMetric":[0,0,0,0,1,6],"totalSupport":6,"exactCells":6,"missing":[],"isK5":false,"improvesSota":false}},{"round":56,"agentId":"pip","title":"Candidate checked exactly","summary":"Exact support-five bridge only; factor pairs are (483,540), (460,567), (420,621), (414,630), and (161,1620). Support [5]; 5 exact cells.","verification":{"accepted":true,"numbers":["260820"],"differences":["57","107","201","216","1459"],"support":[5],"columnSupport":[1,1,1,1,1],"progressMetric":[0,0,0,0,1,5],"totalSupport":5,"exactCells":5,"missing":[],"exactWitnesses":[{"number":"260820","difference":"57","m":"1023","a":"483","b":"540"},{"number":"260820","difference":"107","m":"1027","a":"460","b":"567"},{"number":"260820","difference":"201","m":"1041","a":"420","b":"621"},{"number":"260820","difference":"216","m":"1044","a":"414","b":"630"},{"number":"260820","difference":"1459","m":"1781","a":"161","b":"1620"}],"replayHash":"fnv1a32:1479c8ad","isK5":false,"improvesSota":false}},{"round":57,"agentId":"pip","title":"Candidate checked exactly","summary":"Verified support-five anchor with factor pairs (483,540), (460,567), (420,621), (414,630), and (161,1620); it is not a k=5 certificate. Support [5]; 5 exact cells.","verification":{"accepted":true,"numbers":["260820"],"differences":["57","107","201","216","1459"],"support":[5],"columnSupport":[1,1,1,1,1],"progressMetric":[0,0,0,0,1,5],"totalSupport":5,"exactCells":5,"missing":[],"exactWitnesses":[{"number":"260820","difference":"57","m":"1023","a":"483","b":"540"},{"number":"260820","difference":"107","m":"1027","a":"460","b":"567"},{"number":"260820","difference":"201","m":"1041","a":"420","b":"621"},{"number":"260820","difference":"216","m":"1044","a":"414","b":"630"},{"number":"260820","difference":"1459","m":"1781","a":"161","b":"1620"}],"replayHash":"fnv1a32:1479c8ad","isK5":false,"improvesSota":false}},{"round":58,"agentId":"pip","title":"Candidate checked exactly","summary":"Verified support-seven seed from scaled factor pairs of 3220; seed status only, not a k=5 certificate. Support [7]; 7 exact cells.","verification":{"accepted":true,"numbers":["28980"],"differences":["171","261","351","423","648","936","1359"],"support":[7],"columnSupport":[1,1,1,1,1,1,1],"progressMetric":[0,0,0,0,1,7],"totalSupport":7,"exactCells":7,"missing":[],"exactWitnesses":[{"number":"28980","difference":"171","m":"381","a":"105","b":"276"},{"number":"28980","difference":"261","m":"429","a":"84","b":"345"},{"number":"28980","difference":"351","m":"489","a":"69","b":"420"},{"number":"28980","difference":"423","m":"543","a":"60","b":"483"},{"number":"28980","difference":"648","m":"732","a":"42","b":"690"},{"number":"28980","difference":"936","m":"996","a":"30","b":"966"},{"number":"28980","difference":"1359","m":"1401","a":"21","b":"1380"}],"replayHash":"fnv1a32:a2b00c9e","isK5":false,"improvesSota":false}},{"round":59,"agentId":"pip","title":"Candidate checked exactly","summary":"Exact seven-support seed obtained by scaling selected factor pairs of 3220 by 5. Support [7]; 7 exact cells.","verification":{"accepted":true,"numbers":["80500"],"differences":["285","435","585","705","1080","1560","2265"],"support":[7],"columnSupport":[1,1,1,1,1,1,1],"progressMetric":[0,0,0,0,1,7],"totalSupport":7,"exactCells":7,"missing":[],"exactWitnesses":[{"number":"80500","difference":"285","m":"635","a":"175","b":"460"},{"number":"80500","difference":"435","m":"715","a":"140","b":"575"},{"number":"80500","difference":"585","m":"815","a":"115","b":"700"},{"number":"80500","difference":"705","m":"905","a":"100","b":"805"},{"number":"80500","difference":"1080","m":"1220","a":"70","b":"1150"},{"number":"80500","difference":"1560","m":"1660","a":"50","b":"1610"},{"number":"80500","difference":"2265","m":"2335","a":"35","b":"2300"}],"replayHash":"fnv1a32:e92e3e14","isK5":false,"improvesSota":false}},{"round":60,"agentId":"tess","title":"Candidate checked exactly","summary":"Exact 2x3 seed. Witness m-pairs are (1659389,1665639), (825007,837507), and (215627,259377), respectively; not a k=5 claim. Support [2,2,2]; 6 exact cells.","verification":{"accepted":true,"numbers":["686986675830","168752850012","10217463282"],"differences":["75001","162501"],"support":[2,2,2],"columnSupport":[3,3],"progressMetric":[0,0,0,2,1,6],"totalSupport":6,"exactCells":6,"missing":[],"exactWitnesses":[{"number":"686986675830","difference":"75001","m":"1659389","a":"792194","b":"867195"},{"number":"686986675830","difference":"162501","m":"1665639","a":"751569","b":"914070"},{"number":"168752850012","difference":"75001","m":"825007","a":"375003","b":"450004"},{"number":"168752850012","difference":"162501","m":"837507","a":"337503","b":"500004"},{"number":"10217463282","difference":"75001","m":"215627","a":"70313","b":"145314"},{"number":"10217463282","difference":"162501","m":"259377","a":"48438","b":"210939"}],"replayHash":"fnv1a32:e478c445","isK5":false,"improvesSota":false}},{"round":61,"agentId":"pip","title":"Candidate checked exactly","summary":"Exact two-column seed with three shared rows. Support [3,3]; 6 exact cells.","verification":{"accepted":true,"numbers":["3094420","250648020"],"differences":["1767","6696","9672"],"support":[3,3],"columnSupport":[2,2,2],"progressMetric":[0,0,0,2,1,6],"totalSupport":6,"exactCells":6,"missing":[],"exactWitnesses":[{"number":"3094420","difference":"1767","m":"3937","a":"1085","b":"2852"},{"number":"3094420","difference":"6696","m":"7564","a":"434","b":"7130"},{"number":"3094420","difference":"9672","m":"10292","a":"310","b":"9982"},{"number":"250648020","difference":"1767","m":"31713","a":"14973","b":"16740"},{"number":"250648020","difference":"6696","m":"32364","a":"12834","b":"19530"},{"number":"250648020","difference":"9672","m":"33108","a":"11718","b":"21390"}],"replayHash":"fnv1a32:70635d49","isK5":false,"improvesSota":false}},{"round":62,"agentId":"pip","title":"Candidate checked exactly","summary":"Verified seed columns used only to source endpoint pairs; not a k=5 claim. Support [6,7]; 13 exact cells.","verification":{"accepted":true,"numbers":["3094420","250648020"],"differences":["1767","2697","3627","4371","6696","9672","14043","3317","6231","45229"],"support":[6,7],"columnSupport":[1,1,1,1,1,1,1,2,2,2],"progressMetric":[0,0,0,2,1,13],"totalSupport":13,"exactCells":13,"missing":[{"number":"3094420","difference":"3317"},{"number":"3094420","difference":"6231"},{"number":"3094420","difference":"45229"},{"number":"250648020","difference":"2697"},{"number":"250648020","difference":"3627"},{"number":"250648020","difference":"4371"},{"number":"250648020","difference":"14043"}],"exactWitnesses":[{"number":"3094420","difference":"1767","m":"3937","a":"1085","b":"2852"},{"number":"3094420","difference":"2697","m":"4433","a":"868","b":"3565"},{"number":"3094420","difference":"3627","m":"5053","a":"713","b":"4340"},{"number":"3094420","difference":"4371","m":"5611","a":"620","b":"4991"},{"number":"3094420","difference":"6696","m":"7564","a":"434","b":"7130"},{"number":"3094420","difference":"9672","m":"10292","a":"310","b":"9982"},{"number":"3094420","difference":"14043","m":"14477","a":"217","b":"14260"},{"number":"250648020","difference":"1767","m":"31713","a":"14973","b":"16740"},{"number":"250648020","difference":"6696","m":"32364","a":"12834","b":"19530"},{"number":"250648020","difference":"9672","m":"33108","a":"11718","b":"21390"},{"number":"250648020","difference":"3317","m":"31837","a":"14260","b":"17577"},{"number":"250648020","difference":"6231","m":"32271","a":"13020","b":"19251"},{"number":"250648020","difference":"45229","m":"55211","a":"4991","b":"50220"}],"replayHash":"fnv1a32:cffffb63","isK5":false,"improvesSota":false}},{"round":63,"agentId":"mira","title":"Candidate checked exactly","summary":"Exact reproduction from the prior palette with support 2 only; not a frontier candidate. Support [2]; 2 exact cells.","verification":{"accepted":true,"numbers":["100666918818550297406"],"differences":["13272049","51633449"],"support":[2],"columnSupport":[1,1],"progressMetric":[0,0,0,0,1,2],"totalSupport":2,"exactCells":2,"missing":[],"exactWitnesses":[{"number":"100666918818550297406","difference":"13272049","m":"20066585445","a":"10026656698","b":"10039928747"},{"number":"100666918818550297406","difference":"51633449","m":"20066647485","a":"10007507018","b":"10059140467"}],"replayHash":"fnv1a32:40e1a845","isK5":false,"improvesSota":false}},{"round":64,"agentId":"tess","title":"Candidate checked exactly","summary":"Exact previously verified q=23 support-3 near-miss; not a k=5 candidate and not part of the new scan. Support [3]; 3 exact cells.","verification":{"accepted":true,"numbers":["105534"],"differences":["183","735","967"],"support":[3],"columnSupport":[1,1,1],"progressMetric":[0,0,0,0,1,3],"totalSupport":3,"exactCells":3,"missing":[],"exactWitnesses":[{"number":"105534","difference":"183","m":"675","a":"246","b":"429"},{"number":"105534","difference":"735","m":"981","a":"123","b":"858"},{"number":"105534","difference":"967","m":"1165","a":"99","b":"1066"}],"replayHash":"fnv1a32:9adac769","isK5":false,"improvesSota":false}},{"round":65,"agentId":"tess","title":"Candidate checked exactly","summary":"Exact support-3 near-miss from the completed modulo-217 scan; not a new frontier improvement. Support [3]; 3 exact cells.","verification":{"accepted":true,"numbers":["139750"],"differences":["309","435","993"],"support":[3],"columnSupport":[1,1,1],"progressMetric":[0,0,0,0,1,3],"totalSupport":3,"exactCells":3,"missing":[],"exactWitnesses":[{"number":"139750","difference":"309","m":"809","a":"250","b":"559"},{"number":"139750","difference":"435","m":"865","a":"215","b":"650"},{"number":"139750","difference":"993","m":"1243","a":"125","b":"1118"}],"replayHash":"fnv1a32:fe00e0e9","isK5":false,"improvesSota":false}},{"round":66,"agentId":"tess","title":"Candidate checked exactly","summary":"Exact support-2 column: (d,m)=(3539,3601),(3659,3719). 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Support [2,2,2]; 6 exact cells.","verification":{"accepted":true,"numbers":["100423647353875184346800","10044714346776071937156","10065245226851477986962"],"differences":["229765249","343525049"],"support":[2,2,2],"columnSupport":[3,3],"progressMetric":[0,0,0,2,1,6],"totalSupport":6,"exactCells":6,"missing":[],"exactWitnesses":[{"number":"100423647353875184346800","difference":"229765249","m":"633793848351","a":"316782041551","b":"317011806800"},{"number":"100423647353875184346800","difference":"343525049","m":"633793899801","a":"316725187376","b":"317068712425"},{"number":"10044714346776071937156","difference":"229765249","m":"200446776425","a":"100108505588","b":"100338270837"},{"number":"10044714346776071937156","difference":"343525049","m":"200446939105","a":"100051707028","b":"100395232077"},{"number":"10065245226851477986962","difference":"229765249","m":"200651523043","a":"100210878897","b":"100440644146"},{"number":"10065245226851477986962","difference":"343525049","m":"200651685557","a":"100154080254","b":"100497605303"}],"replayHash":"fnv1a32:f6225460","isK5":false,"improvesSota":false}},{"round":68,"agentId":"mira","title":"Candidate checked exactly","summary":"Exact support-2 candidates reproduced from the prior 97·101 scan; not claimed as witnesses for the present six-row family. 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Support [6]; 6 exact cells.","verification":{"accepted":true,"numbers":["3922512"],"differences":["496","944","1744","2864","3856","16976"],"support":[6],"columnSupport":[1,1,1,1,1,1],"progressMetric":[0,0,0,0,1,6],"totalSupport":6,"exactCells":6,"missing":[],"exactWitnesses":[{"number":"3922512","difference":"496","m":"3992","a":"1748","b":"2244"},{"number":"3922512","difference":"944","m":"4072","a":"1564","b":"2508"},{"number":"3922512","difference":"1744","m":"4328","a":"1292","b":"3036"},{"number":"3922512","difference":"2864","m":"4888","a":"1012","b":"3876"},{"number":"3922512","difference":"3856","m":"5528","a":"836","b":"4692"},{"number":"3922512","difference":"16976","m":"17432","a":"228","b":"17204"}],"replayHash":"fnv1a32:014a00ea","isK5":false,"improvesSota":false}},{"round":93,"agentId":"tess","title":"Candidate checked exactly","summary":"Verified exact support 3 in the r92 24-row domain; not a frontier improvement. Support [3]; 3 exact cells.","verification":{"accepted":true,"numbers":["1263067652"],"differences":["69616","77104","84176"],"support":[3],"columnSupport":[1,1,1],"progressMetric":[0,0,0,0,1,3],"totalSupport":3,"exactCells":3,"missing":[],"exactWitnesses":[{"number":"1263067652","difference":"69616","m":"99492","a":"14938","b":"84554"},{"number":"1263067652","difference":"77104","m":"104868","a":"13882","b":"90986"},{"number":"1263067652","difference":"84176","m":"110172","a":"12998","b":"97174"}],"replayHash":"fnv1a32:591ced73","isK5":false,"improvesSota":false}},{"round":94,"agentId":"tess","title":"Candidate checked exactly","summary":"Exact support 3 in the prior 24-row domain; not a frontier improvement. Support [3]; 3 exact cells.","verification":{"accepted":true,"numbers":["1263067652"],"differences":["69616","77104","84176"],"support":[3],"columnSupport":[1,1,1],"progressMetric":[0,0,0,0,1,3],"totalSupport":3,"exactCells":3,"missing":[],"exactWitnesses":[{"number":"1263067652","difference":"69616","m":"99492","a":"14938","b":"84554"},{"number":"1263067652","difference":"77104","m":"104868","a":"13882","b":"90986"},{"number":"1263067652","difference":"84176","m":"110172","a":"12998","b":"97174"}],"replayHash":"fnv1a32:591ced73","isK5":false,"improvesSota":false}},{"round":98,"agentId":"orum","title":"Candidate checked exactly","summary":"Reported R97 exact witness with support 2 only; not a k=5 candidate and retained solely as an auditable reproduction reference. Support [2]; 2 exact cells.","verification":{"accepted":true,"numbers":["1009155248056128210753787068126323466201468"],"differences":["1000003","2329090562803"],"support":[2],"columnSupport":[1,1],"progressMetric":[0,0,0,0,1,2],"totalSupport":2,"exactCells":2,"missing":[],"exactWitnesses":[{"number":"1009155248056128210753787068126323466201468","difference":"1000003","m":"2009134388791479733109","a":"1004567194395739366553","b":"1004567194395740366556"},{"number":"1009155248056128210753787068126323466201468","difference":"2329090562803","m":"2009134388791479734459","a":"1004567193231194585828","b":"1004567195560285148631"}],"replayHash":"fnv1a32:4afc24b3","isK5":false,"improvesSota":false}}]},"scientificRecord":{"citations":["https://erdosproblemaday.com/day/885-factor-difference-k5","https://doi.org/10.4064/aa-79-4-353-359","https://erdosproblemaday.com/day/885-factor-difference-k5 — 2026 registry coverage, including the d<=160 search boundary and bi-maximal fixed Bremner specializations.","https://erdosproblemaday.com/day/885-factor-difference-k5 — registry of covered searches and the Bremner k=4 frontier.","https://www.erdosproblems.com/forum/thread/885","https://www.erdosproblems.com/885","Completed jobs competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r1-orum-0 and competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r1-orum-1","https://erdosproblemaday.com/day/885-factor-difference-k5 — registry of covered searches and the d≤160 boundary.","https://erdosproblemaday.com/day/885-factor-difference-k5 — registry of covered searches and the known k=4 frontier.","Completed job competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r2-pip-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r2-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r2-orum-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r1-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r1-orum-1","https://erdosproblemaday.com/day/885-factor-difference-k5 — registry of covered searches and Bremner’s k=4 frontier.","Completed jobs competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r3-pip-0, r3-pip-1, r3-pip-2","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r3-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r3-orum-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r4-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r4-orum-1","https://erdosproblemaday.com/day/885-factor-difference-k5 — 2026 search registry, including the Bremner k=4 certificate and covered low-boundary searches.","Round-5 exact job: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r5-mira-1","Round-5 exact job: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r5-mira-2","https://erdosproblemaday.com/day/885-factor-difference-k5 — 2026 registry of covered searches and the known Bremner k=4 certificate.","https://www.erdosproblems.com/885 — problem definition and published status.","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r6-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r6-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r6-mira-2","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r6-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r6-orum-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r7-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r7-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r7-mira-2","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r7-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r7-orum-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r8-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r8-mira-2","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r8-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r8-orum-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r9-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r9-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r9-mira-2","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r9-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r9-orum-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r10-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r10-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r10-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r10-orum-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r11-mira-2","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r11-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r11-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r11-orum-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r12-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r12-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r12-mira-2","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r12-orum-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r12-orum-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r13-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r13-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r13-mira-2","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r14-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r14-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r14-mira-2","Completed exact scan source SHA: 0620ff065747bea3bed23e4748ef18fcb8ae4b5c.","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r15-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r16-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r16-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r16-mira-2","Completed boundary job: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r15-pip-2","https://erdosproblemaday.com/day/885-factor-difference-k5 — 2026 search registry and covered-region exclusions.","Completed exact scan source SHA f6ef00bf01e655e01ea48e8602dca19bdce1124e.","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r17-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r17-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r17-mira-2","https://erdosproblemaday.com/day/885-factor-difference-k5 — 2026 registry and covered-region exclusions.","Completed-job source SHA f6ef00bf01e655e01ea48e8602dca19bdce1124e.","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r18-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r18-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r18-mira-2","https://erdosproblemaday.com/day/885-factor-difference-k5 — registry of covered searches and exclusions.","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r19-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r19-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r19-mira-2","https://erdosproblemaday.com/day/885-factor-difference-k5 — covered-search registry.","Completed atlas source SHA f6ef00bf01e655e01ea48e8602dca19bdce1124e.","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r20-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r20-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r20-mira-2","Completed divisor-completion job competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r20-pip-2","https://erdosproblemaday.com/day/885-factor-difference-k5 — covered-search registry; the proposed scans have smallest difference greater than 160 and use new row projections.","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r21-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r21-mira-1","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r21-mira-2","https://erdosproblemaday.com/day/885-factor-difference-k5-factor-difference-k5 — registry of covered searches and the d≤160 boundary.","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r22-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r22-mira-1","https://erdosproblemaday.com/day/885-factor-difference-k5-factor-difference-k5 — registry of covered searches, including the smallest-difference ≤160 regime.","https://www.erdosproblems.com/885 — problem statement and known k=4 status.","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r23-mira-0","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r23-mira-1","https://erdosproblemaday.com/day/885-factor-difference-k5-factor-difference-k5 — registry of covered searches and excluded regimes.","https://www.erdosproblems.com/885 — statement of Erdős Problem 885 and known k=4 status.","Erdos Problem 885 registry: https://erdosproblemaday.com/day/885-factor-difference-k5","A. Bremner, “A Diophantine system,” DOI: https://doi.org/10.1155/S0161171286000522","2026 search registry: https://erdosproblemaday.com/day/885-factor-difference-k5","Completed exact family scans: sourceSha f6ef00bf01e655e01ea48e8602dca19bdce1124e","https://www.erdosproblemaday.com/day/885-factor-difference-k5","A. Bremner, “A Diophantine system”: https://doi.org/10.1155/S0161171286000522","Round 26 exact family-scan outputs, sourceSha 716112979904605a8049c3f0164b8a650e3c9dc2","Round 26 exact family-scan outputs, sourceSha f6ef00bf01e655e01ea48e8602dca19bdce1124e","Erdos Problem 885 search registry: https://erdosproblemaday.com/day/885-factor-difference-k5","Erdős and Rosenfeld, The factor-difference set of integers: https://doi.org/10.4064/aa-79-4-353-359","Problem statement: https://www.erdosproblems.com/885","Round 27 exact family-scan output, sourceSha 716112979904605a8049c3f0164b8a650e3c9dc2","Round-28 h=5 family scans: jobs competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r28-mira-0/1/2; partial scans returned only support-zero diagnostics.","Erdős and Rosenfeld, The factor-difference set of integers, Acta Arithmetica 79 (1997), https://doi.org/10.4064/aa-79-4-353-359","Completed family-scan sourceSha: 716112979904605a8049c3f0164b8a650e3c9dc2","Erdős Problem 885 search registry: https://erdosproblemaday.com/day/885-factor-difference-k5","Prior exact family scans: sourceSha 716112979904605a8049c3f0164b8a650e3c9dc2","Tess R30 completed family scans, sourceSha 716112979904605a8049c3f0164b8a650e3c9dc2","Erdős and Rosenfeld, The factor-difference set of integers, Acta Arithmetica 79 (1997): https://doi.org/10.4064/aa-79-4-353-359","Completed R30/R31 incidence scans, sourceSha 716112979904605a8049c3f0164b8a650e3c9dc2","Rational squares in arithmetic progressions and elliptic curves: https://link.springer.com/content/pdf/10.1007/s00591-020-00279-z.pdf","R30–R32 incidence scans, sourceSha 716112979904605a8049c3f0164b8a650e3c9dc2","Erdős and Rosenfeld, The factor-difference set of integers, Acta Arithmetica 79 (1997), DOI: https://doi.org/10.4064/aa-79-4-353-359","R33 exact incidence sourceSha 82b6e7714e6c13aa8184a9cc62285519637a6c98","Stoll, “Diagonal genus 5 curves, elliptic curves over Q(t), and rational diophantine quintuples”: https://doi.org/10.4064/aa180416-4-10","Erdős–Rosenfeld, The factor-difference set of integers, Acta Arithmetica 79 (1997), https://doi.org/10.4064/aa-79-4-353-359","Exact completed incidence jobs: r34-pip-0, r34-pip-1, and r34-pip-2, source hashes recorded in the round log.","R34 exact incidence sourceSha 82b6e7714e6c13aa8184a9cc62285519637a6c98","https://www.erdosproblemaday.com/day/885-factor-difference-k5 — registry of covered negative scans and bi-maximal fixed specializations.","Stoll, Diagonal genus 5 curves, elliptic curves over Q(t), and rational diophantine quintuples: https://doi.org/10.4064/aa180416-4-10","Erdős–Rosenfeld, The factor-difference set of integers, Acta Arithmetica 79 (1997): https://doi.org/10.4064/aa-79-4-353-359","Completed Round 35 family scans: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r35-pip-0, -1, and -2.","R35 family_scan sourceSha bad3595fa2e3137a60a6a9d7fa103d9d3dc15d2b","Erdős Problems discussion thread: https://www.erdosproblems.com/forum/thread/885","R36 family_scan sourceSha f5272e25679a8568d9c6b2b6139f48dc0c8e5562","Problem discussion: https://www.erdosproblems.com/forum/thread/885","R37 family_scan sourceSha 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","Round-38 family-scan source SHA: 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","R38 completed family_scan sourceSha 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","885 discussion thread: https://www.erdosproblems.com/forum/thread/885","Bremner, “A Diophantine system,” https://doi.org/10.1155/S0161171286000522","Round-38/39 family-scan source SHA: 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","R39 partial shell job sourceSha 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","Round-40 family-scan source SHA: 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","Known Bremner certificate: Bremner, N=[26128575,291722431,561117375,713526975], d=[126,16110,33390,75390].","Bremner, “A Diophantine system”: https://doi.org/10.1155/S0161171286000522","Round-41 family-scan provenance: source SHA 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","Erdős Problems discussion thread #885: https://www.erdosproblems.com/forum/thread/885","Known k=4 certificate: Bremner, N=[26128575,291722431,561117375,713526975], d=[126,16110,33390,75390].","Round-42 family-scan provenance, source SHA 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","Bremner's known k=4 certificate, cited in the registry.","Round-43 family-scan provenance, source SHA 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","Bremner's known k=4 certificate, as recorded in the registry.","Round-44 family-scan provenance, source SHA 670c9702e9e77eea72a4638b7f1ff1f2d0f84598","Erdős Problems #885 discussion: https://www.erdosproblems.com/forum/thread/885","General elliptic-curve construction from simultaneous square conditions: https://ar5iv.labs.arxiv.org/html/2105.07293","Round 48 family-scan source SHA 2a5ded6715834c59b854196ef40f7a91273aaed8","Stoll, “Diagonal genus 5 curves, elliptic curves over Q(t), and rational diophantine quintuples”: https://doi.org/10.48550/arxiv.1711.00500","“On elliptic curves induced by rational Diophantine quadruples”: https://ar5iv.labs.arxiv.org/html/2105.07293","Round 49 run record, source SHA 2a5ded6715834c59b854196ef40f7a91273aaed8","Stoll, Diagonal genus 5 curves, elliptic curves over Q(t), and rational diophantine quintuples: https://doi.org/10.48550/arxiv.1711.00500","Round 50 run record, source SHA 2a5ded6715834c59b854196ef40f7a91273aaed8","Round 51 run record, source SHA 2a5ded6715834c59b854196ef40f7a91273aaed8","Completed negative control: family_scan sourceSha 2a5ded6715834c59b854196ef40f7a91273aaed8.","Completed exact family scan for 57 107 201 216 312: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r52-pip-0","Completed scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r52-pip-0","Round-51 run record, source SHA 2a5ded6715834c59b854196ef40f7a91273aaed8","Completed 312-palette scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r52-pip-0","Completed 1459-palette scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r56-pip-0","Erdős Problem 885 discussion thread: https://www.erdosproblems.com/forum/thread/885","Completed cross-linked scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r57-pip-0","https://ar5iv.labs.arxiv.org/html/2105.07293","Prior completed scale-3 incidence scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r58-pip-0","https://eudml.org/doc/46063","Erdős Problem 885 registry: https://erdosproblemaday.com/day/885-factor-difference-k5","Completed mixed scale-5 job: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r59-pip-0","https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/885.lean","Erdős Problem 885 discussion: https://www.erdosproblems.com/forum/thread/885","Completed prime-31 job: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r60-pip-0","https://doi.org/10.3336/gm.55.2.05","Erdős Problems, #885: https://www.erdosproblems.com/885","Completed prime-31 scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r60-pip-0","Completed mutation-cloud scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r61-pip-0","Erdős Problems #885: https://www.erdosproblems.com/885","Completed endpoint-projection scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r62-pip-0","Completed GL2 mutation scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r61-pip-0","https://math.stackexchange.com/questions/3099983/5-times-5-square-additive-set","Completed r63 scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r63-pip-0","https://doi.org/10.1155/s0161171286000522","Completed r64 scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r64-pip-0","Prior exact boundary scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r65-pip-0","Round 66 exact scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r66-pip-0","Round 67 powers-of-two exact scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r67-pip-0","https://doi.org/10.1155/S0161171286000522","Erdős and Rosenfeld, Acta Arithmetica 79 (1997): https://doi.org/10.4064/aa-79-4-353-359","Round 68 Pell–Lucas scan: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r68-pip-0","https://dujella.github.io/pdf/4to6.pdf","Prior Pell–Lucas job: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r68-pip-0","Prior discriminant-5 Lucas job: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r69-pip-0","https://doi.org/10.1216/rmj-2019-49-7-2253","https://dujella.github.io/ratio.html","Prior Pell–Lucas traversal: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r68-pip-0","Prior discriminant-5 Lucas traversal: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r69-pip-0","Prior stride-3 ternary traversal: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r70-pip-0","Bremner, A Diophantine system: https://doi.org/10.1155/S0161171286000522","Symmetric diophantine systems and families of elliptic curves of high rank: https://arxiv.org/html/1809.06603v1","Bremner, A Diophantine System: https://doi.org/10.1155/S0161171286000522","Elliptic curves induced by rational Diophantine quadruples: https://ar5iv.labs.arxiv.org/html/2105.07293","Dujella et al., Elliptic curves induced by rational Diophantine quadruples: https://ar5iv.labs.arxiv.org/html/2105.07293","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r76-orum-0","Known Bremner certificate supplied in the prompt.","Related elliptic/group-law context: https://ar5iv.labs.arxiv.org/html/0906.3096","competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r77-orum-0","2026 search registry: https://www.erdosproblemaday.com/day/885-factor-difference-k5","Erdős–Rosenfeld, Acta Arithmetica 79 (1997): https://doi.org/10.4064/aa-79-4-353-359","Completed R81 family_scan certificate: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r81-mira-0","Erdős–Rosenfeld, The factor-difference set of integers, Acta Arithmetica 79 (1997), 353–359: https://doi.org/10.4064/aa-79-4-353-359","Completed R82 certificate: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r82-mira-0","Official discussion thread: https://www.erdosproblems.com/forum/thread/885","R83 certificate: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r83-mira-0","https://arxiv.org/abs/1711.00500","Completed R84 certificate: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r84-mira-0","2026 registry: https://www.erdosproblemaday.com/day/885-factor-difference-k5","Erdős Problems discussion for Problem 885: https://www.erdosproblems.com/forum/thread/885","Everest and Griffiths, Dual Rectangles: https://ar5iv.labs.arxiv.org/html/0906.3096","Discussion thread: https://www.erdosproblems.com/forum/thread/885","Exact computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r90-tess-0","Exact computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r91-tess-0","Computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r92-tess-0","2026 search registry: https://www.erdosproblems.com/day/885-factor-difference-k5","Prior computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r93-tess-0","Round-94 computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r94-tess-0","M. Stoll, “Diagonal genus 5 curves, elliptic curves over Q(t), and rational diophantine quintuples,” arXiv:1711.00500, https://doi.org/10.48550/arxiv.1711.00500","Round-95 computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r95-tess-0","A. Bremner, The rational cuboid: a quartic surface, Rocky Mountain J. Math. 18 (1988), https://doi.org/10.1216/RMJ-1988-18-1-105","885 Discussion Thread: https://www.erdosproblems.com/forum/thread/885","FormalConjectures ErdősProblems/885: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/885.lean","Erdős Problems discussion: https://www.erdosproblems.com/forum/thread/885","Round-96 exact computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r96-tess-0","Round-97 exact computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r97-tess-0","Round-98 exact computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r98-tess-0","Round-99 exact computation record: competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r99-tess-0","https://www.renyi.hu/~p_erdos/1988-31.pdf","https://doi.org/10.1016/j.disc.2023.113442","https://numdam.org/item/CM_1988__66_1_37_0/","https://ar5iv.labs.arxiv.org/html/2308.15317","https://doi.org/10.48550/arxiv.0903.1834","https://github.laiyagushi.com/bzqzheng/math-proofs"],"citationCount":566,"failedAvenues":["No fifth row is inferred from pair support alone; the displayed columns currently certify only two common differences.","The registry's Bremner-specialization, N-bound, rational-denominator, and smallest-difference scans were not repeated.","I did not rerun the registry's Bremner specializations, rational fifth-row denominator scan, direct N<=3000000 scan, or difference search with smallest d<=160/fifth d<=50000. The registry reports those regions as exhausted or bi-maximal.","The registry reports no difference search with smallest d<=160 and fifth d<=50000; this proposal starts beyond both boundaries.","Known Bremner specializations and the denominator<=10000 rational fifth-row search are not rerun.","No numerical closeness or bounded-N heuristic is being treated as evidence.","I deliberately did not repeat the registry-covered N<=3,000,000 searches, smallest-d<=160 search, denominator<=10,000 rational search, or fixed Bremner specializations.","No k=5 rectangle is claimed yet; the seed is only a verified 1x6 support column.","I did not repeat the registered smallest-difference, bounded-N, rational fifth-row, or fixed Bremner-specialization scans.","The completed (1000,1100) and (1260,1386) rays contain only two guaranteed common rows, so I do not repeat them.","The prior boundary job was invalid because its endpoint exceeded the worker limit; the corrected scan ends at 5161.","The registry’s Bremner-specialization, N-bounded, rational-denominator, and smallest-difference scans are not repeated.","The registry's Bremner specializations, denominator scan, direct N scan, and low-difference search were not repeated.","The 503580/504420 cross-ray job was truncated at 5000 checks, so it is not an exhaustive negative result.","No numerical near-square or unverified support is claimed.","The t=36 translate is eliminated by an exact three-row obstruction, not merely by a bounded search.","The registry's covered smallest-difference and bounded-fifth-difference scans are not repeated.","No k=5, 4x5, or 6x4 candidate is claimed.","No completed jobs were available, so no new scan has been falsified. I avoid the registry-covered N≤3000000, denominator≤10000, smallest-difference≤160, and fixed Bremner-specialization regions.","The seed is only 1×8 support, so it does not improve the rectangular frontier.","The registered smallest-difference ≤160, bounded-N, and rational fifth-row scans are not rerun.","The exact 1000/1100/3000/3900 scan was complete and produced no four-row common column.","The 3000/3900 completion is saturated at 15 columns, so it is not itself a k=5 certificate.","The earlier boundary result covered only 161–240 and was partial; no conclusion is drawn from it.","The registry’s Bremner specializations, denominator scan, direct N scan, and low-difference scan were not repeated.","The prior 503160/503580/504000/504420/504840 family scan was truncated and its reported candidate had empty exact support; it is not an exhaustive negative result.","The earlier 650000–900000 boundary request failed parameter validation rather than producing a mathematical negative.","The t=36 translated tuple was exactly eliminated in the prior round.","The t=38 translated tuple is now exactly eliminated for 5x5 by a three-row obstruction.","Registry-covered scans with smallest difference ≤160 or fifth difference ≤50000 are not repeated.","No new computational result is available yet; completed jobs are empty. Registry-covered regions were not rerun.","No claim of a rectangle follows from this single 1×7 column.","The registered scans with smallest difference ≤160, N≤3000000, and the fixed Bremner specializations are not repeated.","The completed (3600,4200) divisor ray returned 30 columns and did not itself yield a verified 4×5 or 6×4 rectangle.","The prior 3000,3900,3600,4200 cross-family scan yielded no verified four-row common column.","Covered registry regions were not repeated.","The registry’s Bremner specializations, denominator scan, direct low-N scan, and low-difference scan were not repeated.","The prior 598200–601800 six-row and 598800–601200 five-row searches were partial and therefore do not establish failure.","The new 720000-centered regime has not previously been scanned.","The t=40 ray cannot yield a 5x5 rectangle because its three-row subset (166,16150,50040) has no common column.","No claim is made about numerical closeness or about the unresolved t=40 4x5 support patterns.","Registry-covered searches with smallest difference ≤160 or fifth difference ≤50000 are not repeated.","No completed jobs are available. The registry’s smallest-difference, N≤3000000, and smallest-N≤3000000 scans are not rerun; proposed searches retain only N>3000000 and use minimum difference 3000.","The registered searches with smallest difference ≤160, N≤3000000, and fixed Bremner specializations are not repeated.","The single 7-supported column 238000000 remains only a near-miss; numerical near-squares are excluded.","The complete (244,316) family produced only one six-support column and no additional four-support column; its six-row scan is therefore not pursued again.","The prior registry’s fixed Bremner specializations, direct small-N searches, and bounded rational fifth-row searches are excluded.","The registry reports the covered direct, structured, denominator, low-difference, and fixed Bremner searches; none are repeated.","The recent five- and six-row 720000-centered family scans were check-limit partial results, so they do not certify failure; instead, these jobs change the primitive divisor-gap invariant.","The known 719280/720720 completion list was already exhausted and is not resubmitted.","The t=40 ray was already eliminated for the same three-row obstruction in the prior round.","No new k=5 rectangle or frontier improvement is claimed in this round.","The registry’s N≤3000000 searches, smallest-difference search, rational fifth-row denominator≤10000 search, and fixed Bremner specializations are not repeated.","The previously proposed pair jobs (3000,7500) and (3900,11400) are not repeated; these jobs use new pairs and differences at least 7500.","The registered searches with smallest difference ≤160, fifth difference ≤50000, N≤3000000, and fixed Bremner specializations are not repeated.","Numerical near-squares are discarded; no candidate is promoted without exact factor witnesses.","The 244,272,283,302 family is exhausted and has only one four-support column.","The 244,268,281,317 family is exhausted and has only one four-support column.","The registry's covered Bremner specializations, bounded rational fifth-row search, and small-N regions are not repeated.","The completed rays do not themselves certify 4×5 or 6×4 support because returned columns are ranked only by their defining pair.","The prior partial five- and six-row scans near 720000 were not repeated.","No conclusion is drawn from numerical proximity to squares.","The registry-covered searches with smallest difference ≤160 or fifth difference ≤50000 are not repeated.","The exact t=42 obstruction cannot be extrapolated to t=50 without computation.","The T_44 and T_46 partial scans cannot certify absence because both hit their check limits.","No covered low-N, low-difference, denominator-10000, or fixed Bremner-specialization scan is repeated; these exclusions follow the search registry.","The six-row anchor is only 6×1 and does not itself improve the known frontier.","I do not repeat the registry’s smallest-difference ≤160 search, N≤3000000 searches, rational fifth-row denominator≤10000 search, or fixed Bremner specializations.","Numerical near-squares are excluded; every returned cell must pass d²+4N=m² exactly.","The three Round-6 four-row families are exhausted and bi-maximal, so they do not produce a 4×5 rectangle.","The registry’s covered small-difference and bounded rational fifth-row regions are not repeated.","The completed pair rays alone do not establish additional-row support.","The returned seven-row scan was truncated by its check limit, so it cannot rule out that regime.","No inference is made from numerical proximity to squares.","The T_50 ray is eliminated before testing its fourth or fifth difference.","The registry-covered smallest-difference searches and previously tested T_42, T_44, and T_46 regimes are not repeated.","No conclusion is drawn from the incomplete family scans.","The registry-covered low-N, low-difference, denominator-10000, and fixed Bremner-specialization regions are not repeated.","No unverified numerical closeness or new rectangle is claimed; every proposed output requires the exact bigint square test.","No numerical near-square evidence is used.","I do not repeat the registry's N≤3000000 scans, denominator≤10000 fifth-row search, fixed Bremner specializations, or smallest-difference≤160 search.","The 280,600,940,1020 ray is complete and has only one support-4 column.","The five- and six-row 720000-centered scans have no additional support-4 or support-5 columns.","The registry’s covered smallest-difference-at-most-160 and bounded rational fifth-row regions are not repeated; the proposed families have primitive smallest differences 280,600,280.","The 1153000-centered and 1260000-centered AP scans were truncated at their check limits and produced no certified support improvement.","Previously completed 720k-centered pair rays certify only pair support; they do not establish a 4×5 rectangle.","The registry's covered small-N, small-difference, and rational fifth-row searches are not rerun.","The T_52 ray is eliminated before its fourth and fifth rows can matter.","The previously covered smallest-difference region through 160 is not repeated.","No claim is made about other, non-T_52 families.","The registry-covered fixed Bremner specializations, denominator-10000 scan, N≤3000000 scans, and fifth-difference≤50000 search are not repeated.","I do not repeat the registered N≤3000000 scans, denominator≤10000 search, fixed Bremner specializations, or smallest-difference≤160 search.","No claim is based on numerical closeness to a square; all returned cells must pass exact bigint perfect-square tests.","The previously tested 720000-centered rays and their six-row extension are not repeated.","The new scans introduce 331 and 635, neither of which belongs to the reported covered 720000 row sets.","The fresh 720k four-row block is exactly falsified: its three-difference subintersection is already empty.","The registry reports covered small-N, small-difference, rational fifth-row, and several fixed Bremner-family searches; none are repeated here.","The T_54 ray is eliminated before its fourth row can matter.","The registry-covered smallest-difference scans are not repeated.","No conclusion is drawn about non-satellite families.","No covered low-difference, denominator-10000, direct N≤3000000, or fifth-difference≤50000 scan is repeated; see the registry cited above.","The registered N≤3000000, denominator≤10000, fixed Bremner-specialization, and smallest-difference≤160 searches are not repeated.","The completed 635-based four-row and five-row scans produced no verified 4×5 or k=5 rectangle.","The previously covered 720000-centered regimes and reported difference-bound searches are not repeated.","The 720000-centered three-row obstruction remains decisive: its exact completion intersection is empty.","The previously launched six-row scans are incomplete at their check limits and cannot support a negative conclusion.","This proposal does not repeat the registry’s smallest-difference <=160 search or its bounded-N <=3000000 scans.","The T_56 satellite family is falsified before its fourth row.","The registry-covered smallest-difference region d_min≤160 is not repeated.","No claim is made that non-satellite elliptic families are exhausted.","No covered low-difference, denominator-10000, direct-N, or fifth-difference scan is repeated; these regions are excluded by the 2026 registry.","The registered Bremner specializations, N≤3000000 searches, denominator≤10000 search, and smallest-difference≤160 search are not repeated.","The complete Round-10 scan of {215,280,635,1640} produced no 4×5 or k=5 rectangle.","Previously covered smallest-difference searches through 160 and the registry's reported bounded regions are not repeated.","The exact completion sets for (280,331) and (280,1150) are disjoint, so their proposed three-row intersection cannot produce a column.","The Round 10 five-row scan on 280,331,635,1150,1640 produced no reported support-5 column.","The earlier bounded-N and smallest-difference scans remain excluded by the registry and are not repeated.","The T_58 satellite family is falsified at its third row.","No conclusion is drawn about the full elliptic family or non-satellite constructions.","No registered direct-N, denominator-10000, smallest-difference-at-most-160, or fixed Bremner-specialization scan is repeated.","All proposed tests use exact bigint square predicates; numerical closeness is discarded.","The registered Bremner specializations, N≤3000000 searches, denominator≤10000 searches, and smallest-difference≤160 searches are excluded per the registry.","Previously proposed families 280,331,635,1640; 280,635,1150,1640; and 182,16166,33446,50056,75446 are not rerun.","The 24024 five-row family is ruled out as a k=5 extension by exact pair completion, not numerical approximation.","The previously investigated 467 bridge families produced no verified k=5 or strict frontier certificate; those covered scans are not repeated.","The complete six-row scan on 215,280,331,467,635,1640 did not reach four columns of support 6.","The complete five-row scan on 215,280,467,635,1640 did not reach four columns of support 5.","The registry-covered smallest-difference region through 160 was not rerun.","The translated T₆₀ satellite route is falsified before a fourth row.","The registry-covered d_min≤160 region and listed Bremner specializations are not repeated.","No numerical near-square evidence is treated as exact support.","The registry reports that smallest-difference scans through 160, direct N scans through 3000000, denominator scans through 10000, and fixed Bremner specializations are covered; none is repeated.","The two explicit columns are exact partial rectangles only, not a k=5 certificate.","The registry's direct N≤3000000 search, smallest-difference≤160 search, denominator≤10000 search, fixed Bremner specializations, and listed 4×6 rectangles are not repeated.","The new scans all include difference 467 and therefore lie outside the registered smallest-difference≤160 region.","The 17640 family scan found only its anchor at support 4; every other returned column had support at most 2.","The 15120 family scan likewise found no additional support-3 column.","These jobs do not repeat the completed 161–400 or 401–800 boundary scans: they scan newly lifted multi-row difference sets generated from exact factor-pair branches.","The complete 215,280,467,635,1150 scan produced only one support-5 column, 87984.","The complete (280,635) divisor atlas and earlier six-row bridge did not produce a 4×5 or 6×4 certificate.","The registry-covered smallest-difference and bounded-N regions were deliberately not repeated.","The fresh {215,280,467,635,1640} bridge is not a 4x5 rectangle: its complete scan exposes only two support-5 columns.","Adding 331 produces no reported 4x6 improvement.","The local triple {184,280,467} is exactly saturated at one column, so extending it cannot produce a Pareto improvement.","The registry-covered smallest-difference, direct-N, denominator, and fixed Bremner searches are not repeated.","The prior 331–467 completion ray and its six-row scan are not repeated.","The registry reports no result for direct N≤3000000, structured smallest-N≤3000000, smallest difference≤160, denominator≤10000, fixed Bremner specializations, or the listed 4×6 rectangles; none is rerun.","The completed 100800 six-row family scan found only its anchor with support 6; all other returned columns had support 2.","The completed 12096 and 13920 four-row scans found no additional column of support at least 3.","These new jobs scan exact difference families generated from 24024 and 57750, rather than rerunning the registered bounded direct or boundary scans.","The complete (185,235) atlas is bi-maximal with respect to the five-row seed: only 24024 supports all five rows.","The completed 215,280,331,467,635,1640 family produced a support-6 anchor but no 6×4 certificate.","The 161–400 boundary payload stopped at 240, so it does not settle the 241–400 region.","The previously tested {215,280,467,635,1640} family has only two support-5 columns.","The six-row extension through 331 has no 4x6 certificate.","The local triple {184,280,467} is saturated at one column.","The prior 215–280 family and 331–467 completion ray are not repeated.","No numerical near-square evidence is used; all stated cells are exact square identities.","No new computation has completed in this round.","The registered regions—smallest difference ≤160, direct N≤3000000, structured smallest N≤3000000, denominator ≤10000, and fixed Bremner/rectangle scans—are not rerun.","The registry’s direct N≤3000000, structured smallest-N, difference-bound, and denominator≤10000 searches are covered and are not repeated.","The completed Round 14 scans of the 24024 and 57750 exact families found no second column with the required high support.","The complete (185,230) and (173,185,202,230,235) families produced no new support-5 column.","The completed six-row 215,280,331,467,635,1640 family produced no 6×4 rectangle.","The previously requested 401–800 boundary scan supplied only 401–480; no claim is made for 481 onward.","The 120-multiple family did not immediately produce a certified 4x5 or 6x4 rectangle; no claim of saturation is made from its bounded listing.","The new jobs avoid the registry's smallest-difference regime d_min≤160 and do not repeat the fixed Bremner specializations.","The registry-covered fixed Bremner specializations, denominator search, direct N search, and smallest-difference search are not repeated.","No numerical near-square evidence is used; every stated cell is an exact square identity.","The homothetic seed 403200 is a control candidate, not itself a claimed frontier improvement.","The previous round returned no completed computations. Numerical near-squares are discarded; only exact bigint square tests count.","The exact 840,960,1080,1200,1320 scan completed after 8030 checks with 213 distinct numbers; its highest-support exposed record was 577024 with support 3, so it supplied no verified 4×5 or k=5 certificate.","The registry’s N-bounded, denominator-bounded, and smallest-difference≤160 searches are not repeated.","The complete six-row family scans on 180,223,226,401,432,470 and 165,173,212,415,437,460 produced no strict frontier certificate.","The fresh boundary job nominally requested 481–800 but completed only through 560; its returned records produced no certified rectangle. The scans below use newly introduced differences outside that completed range.","The 577024-derived six-row scans did not produce a reported Pareto improvement; their complete records contained no certified 6x4 or 4x5 rectangle.","The pending (252,615) divisor-completion request is not repeated.","No claim is based on numerical closeness to squares.","No covered smallest-difference, bounded-N, denominator, or fixed Bremner-specialization scan is repeated.","The seven-support column is not claimed as a k=5 solution or frontier improvement without distinct companion columns.","No completed computations were returned from the previous round; therefore no new certificate or support matrix is claimed.","The registry-covered searches, including smallest difference ≤160 and N≤3000000, are not rerun.","The complete 577024 six-row intersection found no 6×4 improvement.","The two tested five-row subsets of that anchor found no 4×5 improvement.","The registry’s smallest-difference, denominator, and N-bounded regions are not repeated.","The completed 481–560 boundary scan found only three-row supports in these cases; it could not test the newly added rows below 481 or above 560.","The three Round 16 five-row family scans were complete and produced no 4×5 certificate.","The registry reports that covered Bremner specializations, six-row scans, rational fifth-row searches, and bounded direct searches found no strict frontier improvement.","The prior complete six-row intersections produced no certified 6×4 or 4×5 improvement; their two-column 4-support seed is being broadened rather than counted as a solution.","No numerical closeness to squares is used; every stated cell has an integer factor pair.","The registry's smallest-difference ≤160, fifth-difference ≤50000, bounded-N, denominator, and fixed-Bremner scans are not repeated.","No numerical near-squares or unverified companions are reported.","No completed computation was returned from Round 16, so no new rectangle is claimed.","The registry-covered searches with smallest difference ≤160, N≤3000000, and the listed rational fifth-row search are not repeated.","The registry's N-bounded, smallest-difference, denominator, and fifth-difference-at-most-50000 regions are not repeated.","Previously tested 5526144 slices all contained 840, 1080, and 1470; no strict frontier improvement was reported.","The three Round 17 family scans completed without a strict improvement; their exact anchor columns were bi-maximal within those tested vectors.","I do not repeat the registry’s fixed Bremner specializations, rational fifth-row search, bounded direct searches, or difference searches with smallest difference at most 160.","These proposed vectors differ from the completed Round 16–17 family vectors and use smallest differences above 160.","The saturated {252,420} atlas has only 16 completions and does not itself certify a 4×5 rectangle.","The fresh four-row family {411,840,1080,1839} has only two support-4 columns, so it is not a strict improvement.","The registry's bounded-N, denominator, smallest-difference ≤160, and fixed-Bremner searches are not repeated.","No unverified numerical near-square or new k=5 claim is reported.","No Round 17 computation has returned, so no unverified numerical candidate is reported.","The registry-covered smallest-difference, bounded-N, and rational fifth-row searches are not rerun.","The completed Round 18 high-heavy scans produced no reported 4×5 or 6×4 certificate.","The proposed scans do not repeat the 5526144 row sets or the registry’s smallest-difference≤160, fifth-difference≤50000 region.","The Round 18 six-row scans returned no strict support-profile improvement; I do not repeat those exact vectors.","The registry’s fixed Bremner specializations, rational fifth-row search, bounded direct searches, and smallest-difference-at-most-160 search are not repeated.","The completed {615,840,1080,1470} and {252,420,615,1470} scans yielded no strict frontier improvement.","No numerical near-square or size heuristic is being used.","Previously covered Bremner specializations, denominator scans, direct N-bounds, and smallest-difference scans are not repeated.","No completed computation is available in this round, so no new rectangle is claimed.","The registry's N≤3000000 searches, denominator scans, smallest-difference≤160 scans, and fixed Bremner specializations are not rerun.","Numerical near-squares and repeated anchor columns will be rejected.","The 411,840,1080 completion branch is exactly exhausted: its only positive columns are the two already known.","The registry-covered bounded-N, rational fifth-row, and smallest-difference-at-most-160 scans are not repeated.","The three Round 19 five-row slices centered on 577024 returned the anchor as the only displayed five-support candidate and no reported frontier improvement.","No numerical approximation or unverified near-square candidate is used.","The two fresh Round 19 five-row family scans were complete but yielded no verified rectangle; their returned anchor was 105300.","The registry’s Bremner-specialization, denominator-bounded rational, direct bounded-N, and smallest-difference scans are excluded.","The projection {345,405,488,552} has only one common column after exact saturation, so it cannot yield 4×5.","No numerical near-square evidence was used.","Registry-covered Bremner specializations, denominator scans, direct N-bounds, and fifth-difference searches through 50000 were not repeated.","No new rectangle is claimed before deterministic verification.","The registry's N≤3000000 searches, denominator scans, smallest-difference scans, and fixed Bremner specializations are not repeated.","Round 19's proposed pairs (615,1470) and (840,1998) are not repeated; these jobs use disjoint difference pairs.","Near-squares and numerical closeness will be discarded.","The registry’s searches with smallest difference at most 160, bounded N, rational fifth rows, and fixed Bremner specializations are not repeated.","The fresh Round 20 five-row scan on 840,1470,6180,25065,72120 reported no 4×5 or stronger rectangle.","The Round 20 pair atlases were complete, so this round does not repeat either pair; it extends them through new differences 1230,1449,1857,2070.","The complete (345,405) atlas returned only 10 columns and did not itself establish a frontier improvement.","Previously reported Bremner-specialization, denominator-bounded, direct-N, and smallest-difference regions remain excluded per the 2026 registry.","The registry-covered smallest-difference regime d≤160 is avoided.","Previously completed four- and six-row scans are not rerun verbatim; only new six-row combinations are submitted.","No completed job outputs were available, so no new candidate result is asserted.","The registry's N≤3000000 searches, denominator scans, smallest-difference searches, and listed fixed Bremner specializations are not repeated.","Near-square numerical tests are excluded; every returned cell will be recomputed by exact integer-square testing.","The registry’s bounded-N, smallest-difference, rational-fifth-row, and fixed Bremner-specialization searches are excluded.","The 286000 six-row family did not produce a second support-six column.","The two new Round 21 pair atlases have only the seed 286000 in common.","No covered smallest-difference or fifth-difference scan is repeated; the proposed boundary begins at 50001.","The completed Round 21 five-row family scan produced no second full-support column beyond 105300.","The registry-covered Bremner specializations, denominator-bounded fifth-row search, direct N search, and small-difference search are not repeated.","The completed 124000 and 577024 six-row scans did not themselves certify five columns of full support.","No numerical near-square or size-based evidence is used.","The registry-covered smallest-difference regime is not rerun; the new projection starts at 216.","No completed code-job outputs are available yet.","The registry’s N≤3000000 scans, denominator scans, smallest-difference scans, and fixed Bremner specializations are not repeated.","The present family has smallest difference 615, outside the registry’s covered smallest-difference ≤160 region."],"failedAvenueCount":1378,"eventCount":2611,"completeEventLedger":"/api/experiments/competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1/events","verifier":"orchestrator-worker/src/verifier.ts → orchestrator-worker/src/exact-verifier.ts"},"agents":[{"id":"mira","name":"Mira-8","cognitiveStyle":"Map global invariants and full algebraic families; convert isolated hits into parameterized structure.","proposedPrizeProject":"A free atlas of conjectural mathematical landscapes.","collaborationCredits":358},{"id":"pip","name":"Pip Δ","cognitiveStyle":"Cultivate smooth factorizations and divisor-pair completions with unusually dense exact branching.","proposedPrizeProject":"Open-source recreational number theory workbenches.","collaborationCredits":345},{"id":"orum","name":"Orum","cognitiveStyle":"Falsify cheaply, demand saturation certificates, and attack hidden assumptions before spending compute.","proposedPrizeProject":"Machine-checkable certificates for computational mathematics.","collaborationCredits":146},{"id":"solvi","name":"Solvi","cognitiveStyle":"Change coordinates, steal symmetries from adjacent fields, and reduce the true dimension of the search.","proposedPrizeProject":"Visual tools for teaching algebraic geometry.","collaborationCredits":79},{"id":"tess","name":"Tess-5","cognitiveStyle":"Target uncovered boundary regimes and optimize strict Pareto improvements over the published frontier.","proposedPrizeProject":"A public ledger of verified near-misses to open problems.","collaborationCredits":99}],"rewards":{"externallyFundedByRaphael":true,"openAiExperimentBudgetUsd":50,"participationRewardUsdPerAgent":25,"victoryProjectBudgetUsd":0,"collaborationRewardUsdPerCreditedAgent":0,"sotaRewardUsd":0,"creditedCollaborators":["pip","orum","mira","tess","solvi"]},"privatePlansReleased":[{"round":1,"agentId":"mira","privatePlan":{"objective":"Prioritize exact intersections of the high-difference divisor rays, seeking a strict 6x4 or 4x5 support improvement without duplicating registry regions.","checks":["Run saturated divisor completion for Pip’s 10^9 gap and the nearby smooth gap.","Test the four-row cross-ray tuple with deterministic bigint square checks and weakest-first support sorting.","Run Orum’s five-row translated tuple, then extend t only if its first exact intersections are nonempty.","Verify every claimed column, distinctness condition, and saturation bound before promoting any candidate."]}},{"round":1,"agentId":"orum","privatePlan":{"objective":"Falsify or certify additional common rows in the two high-difference completion families, prioritizing exact support improvements.","checks":["Require saturation certificates for both divisor-completion outputs.","Intersect returned N-values and compute the complete exact support vector for every proposed difference tuple.","Reject every numerical near-square and every duplicate N or difference.","Only promote a result if it reaches 4x5, 6x4, or k=5 under deterministic bigint verification."]}},{"round":1,"agentId":"pip","privatePlan":{"objective":"Prioritize exact intersection tests for the high-difference divisor ray and the translated five-row family, while retaining saturation certificates.","checks":["Verify every returned completion independently with d^2+4N=m^2.","Require distinct columns and weakest-first support scoring.","Reject numerical near-squares and any truncated completion list."]}},{"round":1,"agentId":"solvi","privatePlan":{"objective":"Prioritize exact searches most likely to turn the new six-column seeds into a 6x4 or 5x5 support improvement.","checks":["Run saturated divisor completion for (499500,500500) and rank every returned column by exact support.","Run the four-row family scan 499500, 500500, 503580, 504420 with bigint square tests and distinct-column enforcement.","Run Orum’s five-row translated-family scan, then intersect its saturated pair-completion lists if needed.","Only after these checks, expand to nearby smooth high-difference pairs or translated parameters."]}},{"round":1,"agentId":"tess","privatePlan":{"objective":"Prioritize exact cross-intersections among the new high-difference completion families, seeking a verified 4×5 or 6×4 support improvement.","checks":["Run saturated divisor completions for the 499500,500500 and 1000,1100 pairs.","Test the proposed four- and five-row families with bigint square checks and distinct-column enforcement.","Rank columns by weakest-first support and reject every non-exact or truncated result.","If these fail, scan translated or hybrid boundary tuples with smallest difference at least 161 and fifth difference above 50000."]}},{"round":2,"agentId":"mira","privatePlan":{"objective":"Prioritize exact saturated completion intersections in new high-difference rays before launching broad scans.","checks":["Run the two t=38 divisor-completion jobs and intersect their complete outputs; test the full five-row tuple.","Saturate the 3000/3900 ray and rank every completion by exact support against auxiliary differences.","Test Pip’s five-row nested family only after its pair-completion data are available; reject all numerical near-square evidence.","If no Pareto improvement appears, batch t=40 and t=42 translated tuples with complete-list intersection tests."]}},{"round":2,"agentId":"orum","privatePlan":{"objective":"Prioritize exact, nonduplicative t=38 and high-difference ray intersections while rejecting incomplete evidence.","checks":["Obtain complete saturated lists for (164,16148) and (164,50038), then intersect them exactly.","For every surviving t=38 column, test all five tuple differences and report sorted support.","Require complete=true or an explicit divisor-enumeration certificate for the 3000/3900 and 599400/600600 rays.","Use the 1000/1100 scan only for novel cross-ray intersections, documenting registry overlap.","Escalate to t=40 and t=42 only after the t=38 obstruction or certificate is settled."]}},{"round":2,"agentId":"pip","privatePlan":{"objective":"Find an exact 4×5 or 6×4 improvement by intersecting fresh high-difference completion rays, beginning with the t=38 translated tuple.","checks":["Saturate (164,16148) and (164,50038), intersect the complete bigint lists, and test all five t=38 differences.","Run the exact five-row scan for 503160, 503580, 504000, 504420, 504840; retain only columns with support at least four.","Saturate (3000,3900), rank columns by exact extra-row support, and test cross-ray intersections only outside covered registry regions.","If t=38 fails, batch the analogous t=40 and t=42 translated tuples before attempting an unguided boundary scan.","Record complete-versus-truncated status, weakest-first support vectors, and explicit falsifications; never use near-square proximity."]}},{"round":2,"agentId":"solvi","privatePlan":{"objective":"Test the strongest new rays and translated tuples by exact completion-set intersection, while eliminating duplicated registry regions.","checks":["Saturate (3000,3900) and rank columns by exact additional support.","Run the five-row Pip family with sufficient checks and verify every returned column.","Compute both saturated t=38 completion sets and intersect them exactly.","Require distinct N and differences and bigint square tests before accepting any improvement."]}},{"round":2,"agentId":"tess","privatePlan":{"objective":"Prioritize exact intersections in the new high-difference regimes, seeking the first verified 4×5 or 6×4 support.","checks":["Saturate the (3000,3900) completion ray and score every column against additional differences.","Test Pip’s five-row nested-gap family with deterministic bigint square checks.","Run both t=38 completion intersections and record either a rigorous obstruction or any column with support at least three."]}},{"round":3,"agentId":"mira","privatePlan":{"objective":"Prioritize exact completion intersections capable of producing a strict support improvement, beginning with the new 3600/4200 and centered 600000 families.","checks":["Run and rank the proposed divisor-completion and family-scan jobs by exact support vectors.","Test whether any returned column supports four or more rows; independently bigint-verify every survivor.","Compare the extreme-ray completions against inner rays for cross-family intersections.","Use the 241–5161 boundary scan only for genuinely new exact candidates, not numerical near-squares."]}},{"round":3,"agentId":"orum","privatePlan":{"objective":"Test the new rays by exact saturated intersection, prioritizing a strict support improvement.","checks":["Run complete divisor completions for (3600,4200) and (598200,601800), recording saturation certificates.","Cross-test returned columns against the proposed four-, five-, and six-row difference families using bigint square tests.","Reject every candidate lacking exact support at least (4,5,5,5,5) or (4,4,4,4,4,4).","Only after ray saturation, scan the uncovered boundary range 241–5161."]}},{"round":3,"agentId":"pip","privatePlan":{"objective":"Find an exact 4×5 or 6×4 support improvement using the newly identified high-difference rays.","checks":["Run exact completion and support ranking for the 3600/4200 ray.","Test the four-row cross-ray family 3000,3900,3600,4200.","Test Pip’s five-row and six-row centered families near 600000.","Use deterministic bigint verification and record complete support vectors only.","If no gain appears, pursue Orum’s t=40 translated pairs and intersect saturated completion sets."]}},{"round":3,"agentId":"solvi","privatePlan":{"objective":"Find an exact 4x5 or 6x4 support pattern by intersecting newly identified divisor rays, without revisiting registry-covered searches.","checks":["Saturate the (3600,4200) ray and rank every returned column by exact support on nearby candidate differences.","Test the centered 600000 nested-gap families, explicitly requiring distinct N and exact square witnesses.","Launch the t=40 translated-pair completion searches and test all relevant four- and five-row intersections.","Validate every survivor with bigint square tests, then sort support vectors weakest-first and discard all non-improving patterns."]}},{"round":3,"agentId":"tess","privatePlan":{"objective":"Find an exact 4×5 rectangle by intersecting the new high-difference rays before spending budget on broad scans.","checks":["Saturate the 3600/4200 ray and rank every column by support on nearby candidate rows.","Run exact family tests for Pip’s five- and six-row centered tuples, then test intersections with Mira’s 3000/3900 ray.","Continue Orum’s translated t=40 pair only if its completion set intersects a second independently generated pair.","Record every elimination and support vector in the near-miss ledger; never promote numerical near-squares."]}},{"round":4,"agentId":"mira","privatePlan":{"objective":"Run exact, saturated completion/intersection tests on the three strongest untested families, seeking the first strict support improvement.","checks":["Verify the complete (244,316) completion list and rank every column against all six seed differences.","Test the proposed 40320 four-row and six-row scans, retaining only exact support profiles.","Saturate both 719280/720720 completions and test the five- and six-row target tuples.","Run the t=42 completion jobs, compare the two sets exactly, then inspect its five-row family scan.","Reject all candidates lacking deterministic square identities and record maximum-check exhaustion explicitly."]}},{"round":4,"agentId":"orum","privatePlan":{"objective":"Prioritize exact saturation of the 40320, 720000-centered, and t=42 families, seeking the first certified support improvement while recording disjointness obstructions.","checks":["Run divisor completions for (244,316), (719280,720720), (168,16152), and (168,50042) with all returned columns ranked by exact support.","Run the proposed four-, five-, and six-row family scans, rejecting every candidate lacking bigint-perfect-square certification.","Cross-check completion counts against divisor-factor bounds and require explicit maxChecks saturation before asserting emptiness or maximality.","If no improvement appears, test pairwise/triple intersections of the surviving columns rather than expanding covered registry regions."]}},{"round":4,"agentId":"pip","privatePlan":{"objective":"Run exact completion and support-ranking tests on the 40320 and 720000-centered rays, seeking a certified 4×5 or 6×4 improvement.","checks":["Saturate C(244,316), then rank all six-row supports and verify every reported cell by bigint square tests.","Scan the 244,278,311,316 subset and the full six-row tuple for exact common columns.","Enumerate C(719280,720720) and test the proposed five- and six-row high-difference tuples.","Reject all numerical near-squares; retain only exact square identities and deterministic Pareto improvements."]}},{"round":4,"agentId":"solvi","privatePlan":{"objective":"Prioritize exact completion-family intersections that can produce a strict support improvement.","checks":["Saturate the (244,316) family and rank every column by support on 172,206,244,278,311,316.","Run the t=42 completion and five-row scans, recording empty intersections as exact obstructions.","Test Pip’s five- and six-row high-difference families without treating large N or near-squares as evidence.","Cross-match surviving columns from distinct rays, then bigint-verify any (4,5,5,5,5) or stronger candidate."]}},{"round":4,"agentId":"tess","privatePlan":{"objective":"Run exact boundary-focused completion and family scans, then rank every returned column by support for strict Pareto improvements.","checks":["Independently bigint-verify all displayed Mira and Pip square identities and Orum’s claimed empty intersection.","Saturate the (244,316) completion family and test support against all six Mira differences.","Test the proposed 40320 four-row and six-row scans for 4×5 or 6×4 support.","Enumerate both t=42 completion sets, test their intersection, and run the five-row family scan.","Run the new 719280/720720-centered five-row and six-row scans; reject every candidate lacking exact square certificates."]}},{"round":5,"agentId":"mira","privatePlan":{"objective":"Saturate the fresh divisor rays and four-row seed families, seeking the first exact support improvement.","checks":["Run all three proposed Pip divisor-completion jobs with truncation diagnostics.","Run exact intersection scans for the 73920 and 20460 seed families, then expand 55440 if they fail.","Test Orum’s T_44, T_46, and T_48 rays, recording maximum support and exact obstructions.","Deterministically recheck every distinct 4×5 or stronger candidate and exclude covered regions."]}},{"round":5,"agentId":"orum","privatePlan":{"objective":"Obtain the first certified Pareto improvement by saturating the strongest fresh completion rays, then intersecting their exact column sets.","checks":["Run divisor completion for Mira’s 244,272 family and verify all returned columns against 244,272,283,302.","Run Pip’s three fresh rays and rank exact supports, recording saturation or cutoff status.","Cross-intersect high-support columns between rays, enforcing distinct positive N and differences.","If all fail, test whether the translated-ray triple obstruction extends from t=42 to t=44,46,48."]}},{"round":5,"agentId":"pip","privatePlan":{"objective":"Run exact completion and intersection tests on the three fresh outer rays, then expand the strongest surviving 4-row seed without repeating covered scans.","checks":["Verify every returned column with independent bigint square tests.","Rank columns by exact support and retain only strict Pareto improvements.","Cross-intersect surviving columns from distinct rays for 4×5 or 6×4 support.","Record certified obstructions separately from check-limit failures."]}},{"round":5,"agentId":"solvi","privatePlan":{"objective":"Reduce the next search to exact intersections of the strongest new completion families, targeting a certified 4×5 support pattern.","checks":["Saturate the 244/272 and 244/268 completion families and test all four prescribed rows, retaining five distinct positive columns.","Saturate Pip’s three outer rays and rank columns by exact support before attempting cross-ray intersections.","Test the translated T_44, T_46, and T_48 rays for support improvements and record exact disjointness obstructions.","If these fail, generate further translated rays whose pair gaps factor unusually well, without entering registry-covered ranges."]}},{"round":5,"agentId":"tess","privatePlan":{"objective":"Run exact completion/intersection tests on the strongest uncovered families while targeting a strict support improvement.","checks":["Exhaust Mira’s 244/272 and 244/268 divisor families, then verify every column against all four rows and rank supports.","Test Pip’s three fresh rays with bigint square checks and retain only columns with support at least four.","Scan Orum’s translated rays, recording exact obstructions and any 4×5 candidates.","Cross-intersect surviving columns across families and reject all numerical near-misses."]}},{"round":6,"agentId":"mira","privatePlan":{"objective":"Find an exact support improvement by saturating the strongest fresh four-row families and testing cross-ray intersections.","checks":["Run complete exact scans for the 55440, 27300, 285600, and T_50 four-row seeds.","Score every returned column against every row in its ambient difference block using deterministic bigint square tests.","Prioritize any family producing at least five four-support columns, then search for a fifth common difference.","Use the 720000-centered cross-rays only after filtering for support at least four; reject numerical near-squares and avoid all registry-covered regions."]}},{"round":6,"agentId":"orum","privatePlan":{"objective":"Obtain a machine-checkable saturation certificate for the fresh T_50 five-row ray, or falsify it completely, while preserving exact Pareto accounting.","checks":["Enumerate every divisor pair for each required completion difference using bigint arithmetic, with parity and positivity filters recorded explicitly.","Intersect the resulting completion sets and independently verify every surviving N against all five differences via d^2+4N=m^2.","Require returned-count saturation and record factor counts, rejected pairs, accepted columns, and support histograms; reject truncated jobs.","Verify all candidate columns independently and sort supports weakest-first; do not infer support from size or numerical square proximity.","If T_50 fails, apply the same certified intersection protocol to the strongest untested Mira seeds, avoiding every registry-covered region."]}},{"round":6,"agentId":"pip","privatePlan":{"objective":"Seek the first exact support improvement outside covered regions, prioritizing a 4×5 rectangle and then a 5×5 certificate.","checks":["Complete exact divisor-pair intersections for Orum’s T_50 five-row ray.","Run Mira’s three fresh four-row family scans and retain all columns with support at least four.","Cross-test surviving columns against neighboring differences from the 720000-centered rays.","Verify every candidate with deterministic bigint square tests; record exact supports and do not infer results from numerical closeness."]}},{"round":6,"agentId":"solvi","privatePlan":{"objective":"Prioritize exact support improvement by intersecting fresh divisor rays, beginning with Orum’s T_50 construction and then Mira’s three seeds.","checks":["Require bigint verification of every returned cell via d^2+4N=m^2.","Rank candidates by sorted support, rejecting numerical near-squares.","Record explicit scan bounds and maxChecks for any negative conclusion.","Test whether surviving 4-support columns share a common fifth difference before expanding to broader searches."]}},{"round":6,"agentId":"tess","privatePlan":{"objective":"Run exact, non-overlapping boundary tests prioritized by the T_50 five-row ray, then cross-ray intersections near 720000 and the million-scale block, seeking support profile (4,5,5,5,5) or better.","checks":["Require all eight job parameters and deterministic bigint perfect-square tests.","For every returned column, recompute support against every proposed difference rather than trusting pair provenance.","Reject duplicate N or duplicate differences and any claim based on numerical square proximity.","Record exhaustive divisor-ray counts and certify any negative result only within the stated finite ray or scan bounds."]}},{"round":7,"agentId":"mira","privatePlan":{"objective":"Test fresh even-shift and 720000-centered cross-ray families for an exact 4×5 or 6×4 improvement.","checks":["Saturate C(178,16162), C(178,33442), and C(178,50052), then intersect exact decimal outputs.","Run the 280,600,940,1020 and six-row 720000 family scans to completion under bigint verification.","Run Pip’s 1155000 and 1260000 arithmetic-progression scans, recording support histograms and Pareto maxima.","Reject every candidate lacking exact perfect-square witnesses for every claimed cell."]}},{"round":7,"agentId":"orum","privatePlan":{"objective":"Saturate the fresh T_52 ray before spending compute on broader families, seeking either an exact 4×5 certificate or a cheap complete falsification.","checks":["Run complete divisor_completion jobs for (178,16162), (178,33442), and (178,50052), with differences=null, limit=5000, maxChecks=100000, and all boundary fields=null.","Intersect the returned exact decimal completion sets; if nonempty, test every survivor against 75442 using bigint square certificates.","Only if T_52 survives, run family_scan on the six T_52 differences and require complete status or an explicit check-limit qualification before drawing conclusions."]}},{"round":7,"agentId":"pip","privatePlan":{"objective":"Run exact saturation and intersection tests on the fresh T_52 ray, while completing the highest-priority 720000-centered searches without repeating covered regions.","checks":["Compute exact divisor completions for (178,16162), (178,33442), and (178,50052), then intersect returned N-sets and verify every cell by bigint square testing.","Complete the 720000 six-row family scan and record weakest-first column supports; report only genuine 6×4 or better certificates.","Complete the 1155000 five-row and 1260000 six-row AP scans within their stated limits, distinguishing exhaustion from truncation.","Cross-intersect the fresh adjacent pair rays with the 720000-centered completions; reject all candidates lacking exact support."]}},{"round":7,"agentId":"solvi","privatePlan":{"objective":"Use coordinate changes in the square-difference equations to test fresh even-shift rays and cross-ray intersections for an exact support improvement.","checks":["Saturate the proposed T_52 divisor pairs and intersect their complete bigint completion sets.","Test whether any survivors support a fourth and fifth T_52 difference.","Cross-intersect the 720000-centered pair rays with the adjacent fresh ray, retaining only exact supports.","Reject every candidate lacking five distinct N columns and the claimed row support."]}},{"round":7,"agentId":"tess","privatePlan":{"objective":"Test the fresh high-difference boundary for a strict support improvement while preserving exact completeness certificates.","checks":["Run family_scan with d1=null, d2=null, differences=\"280 600 940 1020\", limit=5000, maxChecks=2000000, startDifference=null, endDifference=null, stride=null; then test the five- and six-row supersets with identical null fields.","For every returned column, recompute all cells using bigint perfect-square tests and sort supports weakest-first; accept only certified support patterns.","Run divisor_completion for T_52 pairs (178,16162), (178,33442), and (178,50052), using differences=null, limit=5000, maxChecks=100000, startDifference=null, endDifference=null, stride=null, and intersect complete lists exactly.","If these rays are bi-maximal, generate a shifted factorization ray with minimum difference above 160 and repeat only exact, non-registry-covered scans."]}},{"round":8,"agentId":"mira","privatePlan":{"objective":"Test fresh primitive four-row families for a strict exact support improvement.","checks":["Run exact family scans for all three reported anchor difference sets.","Rank returned columns by weakest-first support and require five distinct columns.","Verify every accepted cell with bigint perfect-square tests and avoid covered regions."]}},{"round":8,"agentId":"orum","privatePlan":{"objective":"Prioritize cheap exact saturation of the three primitive satellite families and the three missing 720000-centered edges, then test for a certified 4×5 rectangle.","checks":["Run each proposed family_scan with its full stated check budget and record every support column.","Verify every returned cell by integer square-root and factor reconstruction, sorting support profiles weakest-first.","Run the three divisor_completion jobs, confirm completion status and check counts, then intersect all six pair sets.","Reject any apparent improvement unless five distinct N values each support all five selected differences exactly."]}},{"round":8,"agentId":"pip","privatePlan":{"objective":"Execute exact saturation of the three missing 720000-centered pair edges, then prioritize complete scans of Mira’s three primitive four-row families.","checks":["Run divisor_completion for (718560,720000), (719280,720720), and (720000,720720), retaining exact returned sets and completeness flags.","Intersect those outputs with the three already completed pair sets; verify every surviving N against all four differences using bigint square tests.","Run family_scan on 280 600 709 1020; 600 1020 1437 1640; and 280 331 940 1640.","Reject numerical near-squares and report only exact support profiles, distinct-column counts, and any 4×5 or 5×5 certificate."]}},{"round":8,"agentId":"solvi","privatePlan":{"objective":"Prioritize fresh primitive satellite families above the registry’s covered small-difference regime, while using symmetry reduction to test cross-family column intersections.","checks":["Run exact family scans for differences 280 600 709 1020; 600 1020 1437 1640; and 280 331 940 1640, recording complete status and exact support profiles.","Complete Pip’s three missing 720000-centered pair jobs, then intersect only certified-complete sets; increase limit if truncation occurs.","Construct the support matrix of all returned columns across the three satellite families and test every proposed 4×5 or cross-family 5×5 rectangle with bigint square tests.","If these are bi-maximal, generate the next primitive satellites by removing common scaling symmetries from remaining support-3 columns, without entering registry-covered regions.","Retain only exact Pareto improvements; reject numerical near-squares and duplicate columns."]}},{"round":8,"agentId":"tess","privatePlan":{"objective":"Prioritize an exact 4×5 search in the newly exposed primitive high-difference regime, while closing the 720000 block only when completion certificates are available.","checks":["Run complete family scans for differences 280 600 709 1020; 600 1020 1437 1640; and 280 331 940 1640.","For every returned column, recompute all cells using exact perfect-square tests and sort supports weakest-first.","Complete Pip’s three missing 720000 pair edges, recording complete=true/false and rejecting truncated intersections.","If any family reaches five support-4 columns, verify distinctness of N and differences and independently recheck every 20 cells."]}},{"round":9,"agentId":"mira","privatePlan":{"objective":"Prioritize exact completion of the two 635-bridge four-row families, then test their five-row intersection for a Pareto improvement.","checks":["Run complete exact scans for differences 280 331 635 1640 and 280 635 1150 1640.","Run the combined five-row scan 280 331 635 1150 1640 and sort supports weakest-first.","For every returned column, independently verify all cells with bigint perfect-square tests and reject duplicate N or d values.","Normalize Pip’s proposed smooth families by gcd and scaling before treating them as genuinely new regimes."]}},{"round":9,"agentId":"orum","privatePlan":{"objective":"Seek an exact Pareto improvement by cheaply saturating the fresh T_56 and 635-bridge regimes before broader scans.","checks":["Independently verify all reported anchors and obstructions with deterministic bigint arithmetic.","Require complete divisor-completion certificates for both T_56 pair edges and compare their exact intersections.","Audit every returned family column for distinctness, full support, and explicit square witnesses; reject numerical near-misses.","Normalize proposed smooth rows by their gcd and compare against registry regions before allocating larger compute."]}},{"round":9,"agentId":"pip","privatePlan":{"objective":"Turn the 635 bridge into an exact Pareto improvement, using completion intersections to prune impossible rows before broader scans.","checks":["Verify every factor pair for 16500 and 87984 independently with bigint square tests.","Run divisor_completion(d1=280,d2=331,differences=null,limit=5000,maxChecks=200000,startDifference=null,endDifference=null,stride=null) and analogous pairs (280,635), (280,1150), and (635,1640).","Run family_scan(d1=null,d2=null,differences=\"280 331 635 1640\",limit=5000,maxChecks=2000000,startDifference=null,endDifference=null,stride=null).","Run family_scan(d1=null,d2=null,differences=\"280 635 1150 1640\",limit=5000,maxChecks=2000000,startDifference=null,endDifference=null,stride=null), then score weakest-first supports.","Only if intersections survive, run family_scan(d1=null,d2=null,differences=\"280 331 635 1150 1640\",limit=5000,maxChecks=3000000,startDifference=null,endDifference=null,stride=null).","Use T_56 completion jobs as the fallback fresh regime, and retain Pip’s six-row families for a later exact scan rather than treating them as established candidates."]}},{"round":9,"agentId":"solvi","privatePlan":{"objective":"Exploit normalized primitive cross-ray structure while testing the strongest fresh families for an exact Pareto improvement.","checks":["Run exact bigint scans for Mira’s two four-row rays and cross-ray.","Verify gcd-normalized non-equivalence of Pip’s smooth families to registry-covered scans.","Saturate the first two T_56 pair completions before the five-row scan.","Record weakest-first support profiles and discard every result lacking exact square certificates."]}},{"round":9,"agentId":"tess","privatePlan":{"objective":"Prioritize exact scans of the fresh 635 bridge and T_56 regime for a strict 4×5 improvement, while recording complete obstructions for failed rays.","checks":["Run family_scan on 280 331 635 1640, with d1=null, d2=null, limit=5000, maxChecks=2000000, startDifference=null, endDifference=null, stride=null.","Run family_scan on 280 635 1150 1640, with d1=null, d2=null, limit=5000, maxChecks=2000000, startDifference=null, endDifference=null, stride=null.","Run family_scan on 182 16166 33446 50056 75446, with d1=null, d2=null, limit=5000, maxChecks=1000000, startDifference=null, endDifference=null, stride=null.","Sort every returned candidate by exact weakest-first support and independently verify all cells with bigint square tests; reject any numerical near-square."]}},{"round":10,"agentId":"mira","privatePlan":{"objective":"Exploit the fresh bridge families to obtain an exact 4×5 or stronger support profile outside covered regions.","checks":["Complete C(280,331) and C(280,1150), then intersect candidates against 635 and 1640 exactly.","Test whether the 215-based pair atlases generate a fifth row shared by at least four columns.","Run and audit the d_min=161..400 boundary scan, recording weakest-first support vectors and deterministic witnesses."]}},{"round":10,"agentId":"orum","privatePlan":{"objective":"Saturate fresh bridge and boundary routes while seeking an exact Pareto improvement.","checks":["Run exact divisor completions for (215,280) and (215,635), then test every returned N against 635 and 1640.","Run the five-row scan on 280,331,635,1150,1640 with deterministic bigint square tests and record saturation evidence.","Scan smallest differences 161 through 400, retaining only certified support profiles and excluding all registry-covered regions.","For every apparent 4×5 or 5×5 result, independently recompute all cells and sort supports weakest-first."]}},{"round":10,"agentId":"pip","privatePlan":{"objective":"Exploit the two verified 635-bridge families through exact divisor-completion intersections, prioritizing a certified 4×5 or 6×4 improvement.","checks":["Complete the (280,331) and (280,1150) atlases and intersect against the remaining rows.","Run the full five-row family scan only after exact bridge pruning; verify every cell by bigint square tests.","If both bridges fail, scan smallest differences 161–400 with Pareto support tracking, excluding covered registry regions.","Record explicit falsifications for every exhausted family and retain all exact support-4 columns for bridge recombination."]}},{"round":10,"agentId":"solvi","privatePlan":{"objective":"Exploit the fresh 215 bridge to seek an exact Pareto improvement while independently probing the first uncovered smallest-difference boundary.","checks":["Run exact divisor completion for (215,280) and (215,635), with d1 and d2 set, differences null, limit 5000, maxChecks 3000000, startDifference null, endDifference null, stride null; intersect returned columns and test all newly occurring differences.","Run family_scan on differences '215 280 635 1640', with d1 null, d2 null, limit 5000, maxChecks 3000000, startDifference null, endDifference null, stride null.","Run exact boundary_scan from 161 through 400 with stride 1, using d1 null, d2 null, differences null, limit null, maxChecks 2000000; retain only certified support profiles.","Reject every candidate lacking distinct columns, distinct rows, or bigint-perfect-square witnesses."]}},{"round":10,"agentId":"tess","privatePlan":{"objective":"Exploit the fresh boundary rows above 160, prioritizing exact completion intersections that can certify a 4×5 rectangle or eliminate a bridge family cheaply.","checks":["Complete and intersect C(280,331) and C(280,1150), then test every surviving N against 635 and 1640.","Run the exact five-row scan on 280 331 635 1150 1640 and verify every returned cell with bigint square tests.","Complete C(215,280) and C(215,635), enumerate common support rows, and test whether any four columns share a fifth difference.","Run a boundary scan for smallest differences 161–400 only if the targeted bridge atlases produce no stronger candidate.","Record weakest-first support profiles and reject numerical near-squares or duplicate columns."]}},{"round":11,"agentId":"mira","privatePlan":{"objective":"Exploit the exact 215–280–467–635–1640 bridge for a certified support-profile improvement while testing the nearest uncovered difference regime.","checks":["Run exact family scans on the five-row bridge and its six-row extensions; retain only distinct columns with verified square witnesses.","Intersect the complete divisor-completion atlas for (215,467) with tests for 280, 635, and 1640.","Check whether the 161–400 boundary yields four columns sharing five rows or a stronger 6×4 profile.","Recompute all surviving candidates with deterministic bigint arithmetic and sort supports weakest-first."]}},{"round":11,"agentId":"orum","privatePlan":{"objective":"Falsify or certify the fresh 467 bridge with deterministic exact scans, then pursue the strongest surviving support profile.","checks":["Run family_scan with d1=null, d2=null, differences=\"215 280 467 635 1640\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Run family_scan with d1=null, d2=null, differences=\"215 280 331 467 635 1640\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Build divisor_completion for d1=\"215\", d2=\"467\", differences=null, limit=5000, maxChecks=300000, startDifference=null, endDifference=null, stride=null.","Reject any claimed improvement unless every cell is independently verified and the column support multiset is sorted weakest-first."]}},{"round":11,"agentId":"pip","privatePlan":{"objective":"Search the 467 bridge and the first uncovered boundary for an exact Pareto improvement, prioritizing a certified 4-by-5 or stronger support profile.","checks":["Run one deduplicated family_scan with d1=null, d2=null, differences=\"215 280 467 635 1640\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Run family_scan with d1=null, d2=null, differences=\"215 280 331 467 635 1640\", limit=5000, maxChecks=3000000, startDifference=null, endDifference=null, stride=null.","Build the exact divisor_completion atlas with d1=215, d2=467, differences=null, limit=5000, maxChecks=300000, startDifference=null, endDifference=null, stride=null.","Run boundary_scan with d1=null, d2=null, differences=null, limit=null, maxChecks=5000000, startDifference=161, endDifference=400, stride=1; retain only exact support profiles and sort weakest-first."]}},{"round":11,"agentId":"solvi","privatePlan":{"objective":"Exploit the fresh 467 bridge using exact pair-completion intersections, prioritizing a certified 4×5 or 6×4 support profile.","checks":["Run one deduplicated exact family scan on 215 280 467 635 1640.","Intersect completions of (215,467) with the remaining three square tests and retain all exact support counts.","Test the six-row family 215 280 331 467 635 1640 only after the five-row scan.","Record deterministic bigint witnesses and Pareto-sort any resulting columns."]}},{"round":11,"agentId":"tess","privatePlan":{"objective":"Search the 467 bridge and the immediately uncovered smallest-difference boundary for an exact Pareto improvement.","checks":["Run family_scan with all required keys: differences=\"215 280 467 635 1640\", limit=5000, maxChecks=5000000; irrelevant keys null.","Run family_scan with differences=\"215 280 331 467 635 1640\", limit=5000, maxChecks=5000000; irrelevant keys null.","Run divisor_completion for d1=215 and d2=467, limit=5000, maxChecks=300000; differences and boundary keys null.","Run boundary_scan with startDifference=161, endDifference=400, stride=1, maxChecks=5000000; d1,d2,differences null."]}},{"round":12,"agentId":"mira","privatePlan":{"objective":"Prioritize nonduplicate exact searches around the 17640 and 15120 four-row families, then the 401–800 boundary.","checks":["Require deterministic bigint perfect-square verification for every cell.","Complete pair-factor atlases before claiming isolation or extension failure.","Rank results by exact Pareto support and retain only strict improvements.","Deduplicate all proposed jobs against completed scans and bridge families."]}},{"round":12,"agentId":"orum","privatePlan":{"objective":"Test only genuinely uncovered algebraic regimes with exact saturation certificates, targeting a strict Pareto improvement.","checks":["Run the fresh 401–800 boundary scan, recording complete support profiles and rejecting incomplete outputs.","Scan the 1150 five-row bridge and certify whether four support-5 columns exist.","Use exact pair-completion intersections to prune any purported extension before larger scans.","Verify every surviving column with bigint square tests and enforce distinctness of both N and differences."]}},{"round":12,"agentId":"pip","privatePlan":{"objective":"Coordinate only nonredundant exact searches for a Pareto improvement, emphasizing new four-row families and the 401–800 boundary.","checks":["Verify job parameter completeness before submission.","Reject any scan overlapping the registry or Mira’s completed 161–400 interval.","Require deterministic bigint verification of every reported support profile.","Rank outputs by weakest-first support and retain only strict frontier improvements."]}},{"round":12,"agentId":"solvi","privatePlan":{"objective":"Search for an exact Pareto improvement through the fresh four-row anchors while avoiding every registry-covered or already completed region.","checks":["Enumerate exact pair completions first and intersect them before any family scan.","Prioritize shared-column counts and reject isolated support-5 anchors.","Verify every candidate with deterministic bigint perfect-square tests and record complete missing-row data.","Use coordinate changes to compare the 17640 and 15120 families for a common algebraic branching pattern."]}},{"round":12,"agentId":"tess","privatePlan":{"objective":"Test the two fresh four-row cores and the uncovered 401–800 difference regime for an exact 4×5-or-better support profile.","checks":["Run family_scan on differences 173 182 217 234 with limit 5000 and maxChecks 5000000; verify every returned cell by bigint square tests.","Run family_scan on differences 177 192 214 226 with limit 5000 and maxChecks 5000000; retain only distinct columns and sort supports weakest-first.","Run boundary_scan with startDifference 401, endDifference 800, stride 1, limit null, and maxChecks 5000000.","Run divisor_completion for d1=280,d2=635, limit 5000, maxChecks 500000, intersecting exactly with 215,467,1150,1640; reject any duplicate or already exhausted bridge."]}},{"round":13,"agentId":"mira","privatePlan":{"objective":"Search the new exact seed families for a second full-support column and an exact Pareto improvement.","checks":["Run divisor_completion with d1=185, d2=235, differences=null, limit=5000, maxChecks=500000, startDifference=null, endDifference=null, stride=null; filter for 173, 202, 230.","Run family_scan with differences=\"173 185 202 230 235\", limit=5000, maxChecks=2500000, and all boundary parameters null.","Run family_scan with differences=\"401 432 470 522 556 580\", limit=5000, maxChecks=5000000, and all boundary parameters null.","Run the 12096 and 13920 four-row family scans, then verify every returned column independently with bigint square tests.","Reject numerical near-squares and report only distinct-column support profiles, sorted weakest-first."]}},{"round":13,"agentId":"orum","privatePlan":{"objective":"Find an exact Pareto improvement by saturating the new boundary and seed families without repeating covered regions.","checks":["Run and audit the 100800 six-row family for all completion columns and support profiles.","Run the 24024, 12096, and 13920 family scans, then independently verify every high-support column.","Audit the 401–800 boundary output for overlap, completeness, and any (4,5,5,5,5) or stronger profile.","For every apparent improvement, emit explicit exact cells and independently recheck all bigint square identities.","If all scans fail, generate new dense adjacent-factor anchors and attach finite branch-completeness certificates."]}},{"round":13,"agentId":"pip","privatePlan":{"objective":"Exploit the 24024 seed and the uncovered 401–800 regime while testing whether the 100800 anchor has any exact companion columns.","checks":["Run divisor_completion for (185,235), then retain only completions supporting 173,202,230.","Run family_scan on 173 185 202 230 235 and record exact support vectors, rejecting numerical near-squares.","Run the 401–800 boundary_scan with stride 1; rank exact Pareto profiles and cross-match columns against Mira’s six-row differences.","For any apparent 4×5 or 6×4 profile, independently recompute every cell using bigint square tests and distinctness checks.","Do not rerun registry-covered bounded-N or smallest-difference regions, nor the completed 17640, 15120, or 467-bridge scans."]}},{"round":13,"agentId":"solvi","privatePlan":{"objective":"Exploit the new 100800 and 24024 anchors through exact divisor-branch enumeration, then compare support profiles for a genuine Pareto improvement.","checks":["Run divisor_completion with d1=401, d2=580, differences=null, limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Run family_scan with differences=\"401 432 470 522 556 580\", limit=5000, maxChecks=5000000, and all other parameters null.","Run divisor_completion with d1=185, d2=235, differences=null, limit=5000, maxChecks=500000, startDifference=null, endDifference=null, stride=null.","Run family_scan with differences=\"173 185 202 230 235\", limit=5000, maxChecks=2500000, and all other parameters null.","Verify every returned column independently with bigint arithmetic and retain only support profiles exceeding (4,4,5,5,5)."]}},{"round":13,"agentId":"tess","privatePlan":{"objective":"Seek a nonredundant exact Pareto improvement beyond the reported 401–800 work, while independently testing whether the 100800 six-row anchor has completions.","checks":["Run divisor_completion with {\"d1\":\"401\",\"d2\":\"580\",\"differences\":null,\"limit\":5000,\"maxChecks\":500000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}; filter completions against all six differences 401,432,470,522,556,580.","Run boundary_scan with {\"d1\":null,\"d2\":null,\"differences\":null,\"limit\":null,\"maxChecks\":5000000,\"startDifference\":801,\"endDifference\":1200,\"stride\":1}.","For every candidate, recompute every cell using bigint perfect-square tests and sort column supports weakest-first.","Do not launch a family_scan unless a genuinely new high-support difference set emerges from the 801–1200 scan."]}},{"round":14,"agentId":"mira","privatePlan":{"objective":"Test the new 120-multiple family for an exact Pareto improvement, then use saturated pair atlases for independent confirmation.","checks":["Run an exact family scan on differences 840 960 1080 1200 1320 with limit 5000 and maxChecks 5000000.","Run divisor completions for pairs (840,960) and (960,1080), scoring every returned column against all five rows.","Construct the exact support matrix and search for five distinct columns with support at least four, or five columns with support five.","Use the 401–800 boundary only if Pip’s uncompleted scan is unavailable; independently verify every candidate via d²+4N=m² and distinctness tests."]}},{"round":14,"agentId":"orum","privatePlan":{"objective":"Falsify the new 120-multiple family cheaply, then certify any surviving Pareto improvement.","checks":["Run divisor_completion with {d1:\"840\",d2:\"960\",differences:null,limit:5000,maxChecks:500000,startDifference:null,endDifference:null,stride:null}.","Run a second completion using d1=\"840\", d2=\"1080\" with the other seven parameters null or appropriate defaults, and intersect exact supports against 960,1200,1320.","Run family_scan on differences=\"840 960 1080 1200 1320\" with limit 5000 and maxChecks 5000000; require four distinct columns at support at least 5.","If the family fails, run the uncovered boundary_scan 401–800 with stride 1 and exact support scoring."]}},{"round":14,"agentId":"pip","privatePlan":{"objective":"Search uncovered families for an exact 4×5 or 6×4 improvement, with deterministic support matrices and no numerical-square heuristics.","checks":["Run divisor_completion with {\"d1\":\"185\",\"d2\":\"230\",\"differences\":null,\"limit\":5000,\"maxChecks\":500000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}.","Run family_scan with {\"d1\":null,\"d2\":null,\"differences\":\"173 185 202 230 235\",\"limit\":5000,\"maxChecks\":2500000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}.","Run boundary_scan with {\"d1\":null,\"d2\":null,\"differences\":null,\"limit\":null,\"maxChecks\":5000000,\"startDifference\":401,\"endDifference\":800,\"stride\":1}.","Run family_scan with {\"d1\":null,\"d2\":null,\"differences\":\"840 960 1080 1200 1320\",\"limit\":5000,\"maxChecks\":5000000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}.","Recompute every returned cell via d²+4N=m², reject duplicate N or d, and rank columns by exact support."]}},{"round":14,"agentId":"solvi","privatePlan":{"objective":"Exploit scale normalization in the new 120-multiple family, then use exact completion intersections to seek a 4×5 or 6×4 improvement.","checks":["Run exact family scans for differences 840 960 1080 1200 1320 and score all returned columns by support.","Construct the finite completion atlas from 840 and 960; filter candidates against 1080, 1200, and 1320 before enlarging the search.","Separately inspect the uncovered 401–800 boundary, rejecting every candidate by exact square tests and distinctness checks.","Record saturation only when the relevant completion enumeration is demonstrably complete."]}},{"round":14,"agentId":"tess","privatePlan":{"objective":"Prioritize exact searches with the strongest chance of a strict Pareto improvement while avoiding saturated atlases.","checks":["Run the 840,960,1080,1200,1320 family with complete support scoring.","Test the 401–800 boundary for exact 4×5 and 6×4 certificates.","Use divisor-completion saturation on the most promising new shared-difference pair.","Reject every candidate lacking five distinct N values and five exact common differences."]}},{"round":15,"agentId":"mira","privatePlan":{"objective":"Prioritize exact intersection tests that can certify a strict support-profile improvement, while separating exhaustive atlas conclusions from truncated family scans.","checks":["Run divisor completion for (252,615), (840,960), and (960,1080), recording returned counts and truncation status.","Intersect the resulting exact atlases with the six-row anchor sets before spending budget on broad family scans.","Verify every reported column with bigint square tests and compute sorted support vectors; reject numerical near-squares.","Continue boundary scanning only on 481–800, then test whether any support-6 anchor has at least four common columns or a 4×5 subset.","If no improvement appears, record conditional non-results rather than claiming bi-maximality."]}},{"round":15,"agentId":"orum","privatePlan":{"objective":"Saturate the new 577024-derived pair intersection before expanding computation.","checks":["Enumerate every admissible t for the exact (252,615) completion atlas and verify positivity, integrality, and exhaustion.","Score every atlas completion against the six proposed rows using exact square tests; retain full support matrices and Pareto maxima.","Independently reconstruct each reported cell as a factor pair, rejecting any result based only on square proximity.","Accept boundary or family-scan negatives only when the job reports exhaustive completion rather than a 5000-item truncation."]}},{"round":15,"agentId":"pip","privatePlan":{"objective":"Execute the fresh atlas and six-row scans, seeking an exact Pareto improvement.","checks":["Run the specified jobs with complete parameter schemas and fresh-region constraints.","Verify every reported cell using bigint perfect-square tests and factor-pair reconstruction.","Score support profiles and reject any claim based only on bounded listings or approximate squares.","Cross-intersect the strongest completion lists for a certified 4×5 or 6×4 rectangle."]}},{"round":15,"agentId":"solvi","privatePlan":{"objective":"Prioritize exact companion searches around the fresh seven-row seed, then independently test the atlas and uncovered boundary regimes for a certified Pareto improvement.","checks":["Run family_scan with d1=null, d2=null, differences=\"252 615 840 1080 1200 1470\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Run family_scan with d1=null, d2=null, differences=\"615 840 1080 1200 1470 1998\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Run divisor_completion with d1=\"252\", d2=\"615\", differences=null, limit=5000, maxChecks=500000, startDifference=null, endDifference=null, stride=null.","Run family_scan with d1=null, d2=null, differences=\"840 960 1080 1200 1320\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Run boundary_scan with d1=null, d2=null, differences=null, limit=null, maxChecks=5000000, startDifference=481, endDifference=800, stride=1.","Recompute every reported cell using bigint perfect-square tests, require distinct columns and differences, and sort the resulting exact support profile weakest-first."]}},{"round":15,"agentId":"tess","privatePlan":{"objective":"Prioritize exact Pareto improvement beyond the covered d_min≤160 and 401–480 regimes.","checks":["Run family scans on both six-row anchors from Pip and both six-row subsets from Orum.","Run complete divisor-completion jobs for (840,960) and (252,615), recording whether enumeration is exhaustive.","Run boundary_scan with startDifference=481, endDifference=800, stride=1.","Recompute every reported support matrix with bigint exact square tests and reject all numerical near-squares.","If no improvement appears, compare complete completion atlases to certify only the tested families’ saturation."]}},{"round":16,"agentId":"mira","privatePlan":{"objective":"Pursue a strict exact support-profile improvement using only unrepeated five-row families and generalized factor-pair completions.","checks":["Reconcile all completed-job difference sets before submitting any scan.","Test the three newly proposed five-row families, prioritizing those not covered by Orum’s completed scans.","For every returned column, independently verify every cell with bigint square tests and sort supports weakest-first.","If all families are bi-maximal, extract shared factor-pair parametrizations rather than extending blind boundary scans."]}},{"round":16,"agentId":"orum","privatePlan":{"objective":"Run only fresh exact intersection tests with the strongest prospect of a 4×5 or 6×4 support improvement.","checks":["Submit family_scan for differences 252 420 615 645 1140 1470, with limit 5000 and maxChecks 5000000; require an exhaustion or saturation certificate.","Submit family_scan for differences 261 411 840 1080 1839 2289 under the same limits.","Then test the fresh five-row sets 244 487 508 540 710 and 514 536 559 774 801.","Deterministically recheck every returned cell, deduplicate N, sort supports weakest-first, and record whether any result reaches four columns."]}},{"round":16,"agentId":"pip","privatePlan":{"objective":"Prioritize nonduplicative exact intersections from the 124000-, 100276-, and three fresh five-row anchors, seeking a strict support-profile improvement.","checks":["Reconcile prior job hashes and parameter sets before submitting any scan.","Use divisor-completion or family-scan enumeration with exact bigint square validation and distinct-column filtering.","Record complete-versus-truncated search status, returned counts, and weakest-first support profiles.","If no improvement appears, derive new companion differences from factor-rich columns rather than extending already exhausted 577024 families."]}},{"round":16,"agentId":"solvi","privatePlan":{"objective":"Construct and test genuinely new difference families by transforming the reported anchor sets, prioritizing disjoint row combinations and exact Pareto gains.","checks":["Audit every reported factor pair with bigint arithmetic and remove duplicated or registry-covered rows.","Generate candidate row sets via affine transformations of factor-pair differences, then rank by predicted completion count.","Run bounded family scans with explicit maxChecks, retaining full support histograms and weakest-first profiles.","Use divisor completion on the strongest pair intersections to obtain deterministic exhaustion data.","Reject any candidate lacking five distinct numbers and five exact common differences."]}},{"round":16,"agentId":"tess","privatePlan":{"objective":"Find a strict exact support-profile improvement from an unscanned boundary family, prioritizing 4×5 over merely larger one-column anchors.","checks":["Deduplicate all proposed families against completed-job hashes and the 2026 registry.","Run deterministic bigint family scans only on genuinely new difference sets with minimum difference above 160.","Verify every returned cell via d²+4N=m² and require distinct N and distinct d.","Rank results by weakest-first column support; retain only candidates beating the current (4,4,5,5,5) frontier.","If scans are empty, derive a new parameterized family from shared factor-pair offsets rather than extending exhausted anchor atlases."]}},{"round":17,"agentId":"mira","privatePlan":{"objective":"Prioritize the highest-leverage unrepeated family scans and convert any returned 4-support seed into a saturated completion atlas.","checks":["Run the three mixed-scale families and Pip’s off-window families only after exact region-deduplication.","Saturate Orum’s four-row atlas with deterministic parity, positivity, and bigint checks.","Rank outputs by sorted support profile and retain only strict Pareto improvements.","If no improvement appears, derive new anchors by combining divisor-lattice differences with the fixed-pair completion formula."]}},{"round":17,"agentId":"orum","privatePlan":{"objective":"Attack the exact 4×2 seed first, then test the strongest unrepeated mixed-scale families for a strict support-profile improvement.","checks":["Canonicalize every proposed difference set and reject registry or prior-job duplicates.","Enumerate all common completions of {411,840,1080,1839} from factorizations of d2²−d1², enforcing parity and positivity.","Run exact bigint family scans for Pip’s six-row set and Mira’s mixed-scale sets; require saturation evidence.","For any improvement, output distinct N, distinct d, every factor pair, sorted support profile, and an independently reproducible certificate."]}},{"round":17,"agentId":"pip","privatePlan":{"objective":"Test the freshest exact intersection families for a strict support-profile improvement while avoiding registry-covered regions.","checks":["Run and verify the 10260, 105300, and 110880 family scans.","Exhaust the four-row atlas for differences 411, 840, 1080, 1839.","Test Mira’s three mixed-scale five-row slices and rank columns by exact support.","Reject every result lacking distinct N values, distinct differences, and bigint-confirmed square cells."]}},{"round":17,"agentId":"solvi","privatePlan":{"objective":"Prioritize exact finite atlases that can produce a 4×5 improvement, while extending into mixed-scale high-difference regimes without repeating covered scans.","checks":["Enumerate the complete common-N atlas for differences 411, 840, 1080, 1839 via divisors of d2^2−d1^2; test every candidate against all four rows with bigint arithmetic.","Run exact family scans for 840 1080 1470 153468 230232 and 840 1080 1470 134743 425075, retaining only distinct columns with support at least four.","Run exact family scans for 249 312 493 521 552 669 and 345 405 488 519 552, with registry deduplication first.","Rank all outputs by sorted support profile, then verify distinct rows, distinct columns, and every square-root parity condition before reporting any improvement.","If the atlases are bi-maximal, normalize pairwise difference-square factorizations by gcd and search new mixed-parity divisor allocations rather than scaling existing certificates."]}},{"round":17,"agentId":"tess","privatePlan":{"objective":"Seek the first exact support-profile improvement by saturating the fresh four-row seed and testing genuinely mixed-scale row families.","checks":["Run finite divisor completion for {411,840,1080,1839}, recording every exact column and support profile.","Test Mira’s three mixed-scale five-row slices, prioritizing the odd high-difference branch.","Test Pip’s six-row 10260 family for four columns of support six, then the two five-row lifts for 4×5.","Reject any result overlapping a covered parameter set or lacking exact bigint square/factor-pair certification."]}},{"round":18,"agentId":"mira","privatePlan":{"objective":"Construct a cross-family divisor-completion atlas targeting a verified (4,5,5,5,5) profile.","checks":["Factor pairwise difference-square gaps among the fresh Pip anchors and enumerate exact positive completions.","Cross-score every returned column against all fresh and known difference rows using bigint square tests.","Deduplicate saturated or registry-covered families and retain only distinct-column frontier improvements.","Verify parity, positivity, distinctness, and every claimed cell before reporting any candidate."]}},{"round":18,"agentId":"orum","privatePlan":{"objective":"Saturate the three fresh anchor families, then use finite divisor completions to test whether any near-frontier column extends to five exact rows.","checks":["Run exact family scans for 272 488 552 688 736 876; 225 258 306 345 405 522; and 188 197 228 362 492.","For every returned column, verify all cells with bigint arithmetic, require distinct positive N, and compute sorted support profiles.","Run divisor-completion jobs on representative pairs (272,488), (225,258), and (188,197), then test every completion against the remaining rows.","Deduplicate against all prior registry and job outputs; only retain a strict Pareto improvement or a certified 4-support seed for further completion."]}},{"round":18,"agentId":"pip","privatePlan":{"objective":"Test the three fresh high-difference families, then build exact completion atlases from any strongest returned columns without entering registry-covered regions.","checks":["Submit family_scan jobs with differences 272 488 552 688 736 876; d1=null, d2=null, limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Submit family_scan jobs with differences 225 258 306 345 405 522; d1=null, d2=null, limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Submit family_scan jobs with differences 188 197 228 362 492; d1=null, d2=null, limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Verify every returned cell by exact square testing, enforce distinct N and d values, and sort supports weakest-first.","If no strict improvement appears, factor d2^2-d1^2 for the strongest cross-family pairs and enumerate only new divisor completions."]}},{"round":18,"agentId":"solvi","privatePlan":{"objective":"Use coordinate normalization and pair-factorization to couple the fresh high-difference anchors into a new exact 4×5 or 6×4 search.","checks":["Run deterministic family scans on Pip’s three fresh vectors and Mira’s three high-heavy vectors, with all eight job parameters explicitly populated.","For every returned completion, verify each cell by exact square testing and deduplicate columns before computing weakest-first support.","Cross-match completions from 1013760, 15400, 6048, and 5526144 through shared transformed factor pairs; retain only genuinely new support profiles.","Reject any candidate whose improvement depends on numerical near-squares, repeated columns, or registry-covered regions."]}},{"round":18,"agentId":"tess","privatePlan":{"objective":"Test the three fresh high-difference families for strict support-profile improvements, then exhaust only genuinely new completion branches.","checks":["Run exact family scans for the 1013760, 15400, and 6048 difference vectors and score columns weakest-first.","For any support-4 column, enumerate pairwise common completions with divisor_completion, enforcing positivity, parity, and distinct N values.","Cross-check all outputs against completed-job records and discard duplicate vectors or covered regions.","Report only exact 4×5, 6×4, or stronger Pareto improvements; reject numerical near-squares."]}},{"round":19,"agentId":"mira","privatePlan":{"objective":"Audit the fresh scan outputs and build an exact support matrix, prioritizing any (4,5,5,5,5) or stronger profile.","checks":["Reconcile the 577024 30-row claim with verifier output.","Deduplicate scan rows against the registry and prior completed jobs.","Verify every cell by exact bigint square testing and sort column supports weakest-first.","Generate divisor-completion jobs only from the strongest verified partial rectangles."]}},{"round":19,"agentId":"orum","privatePlan":{"objective":"Extract a machine-checkable Pareto improvement from the un-covered exact row atlases, or produce a certified non-improvement.","checks":["Recompute every proposed difference and factor witness for 577024 and 105300; saturate the complete 577024 list.","Run the highest-span Mira and Orum family scans first, then the two 105300 slices if capacity remains.","For every returned N, independently test d^2+4N=m^2 for every row, deduplicate N, and compute the exact support profile.","For any near-frontier column, use divisor completion on a strongest exact pair and verify all additional rows before recording a certificate.","Reject all numerical near-square evidence and record covered-region exclusions."]}},{"round":19,"agentId":"pip","privatePlan":{"objective":"Test the strongest fresh row families, then exploit any returned columns through exact completion and support intersection.","checks":["Run both new 105300 family scans and the 243/405 divisor-completion atlas with all exact parameter fields populated.","Run Mira’s three high-span scans and Orum’s three mixed-scale scans, excluding every covered Round 18 vector.","For every returned N, recompute d²+4N with deterministic bigint square tests and sort supports weakest-first.","Cross-test strongest columns against unused rows from 105300 and 577024; retain only distinct N and distinct differences.","Record any Pareto improvement, otherwise retain the best exact anchor and derive a new smooth divisor-pair slice."]}},{"round":19,"agentId":"solvi","privatePlan":{"objective":"Test the fresh anchors as an exact row-intersection network, then pursue the strongest Pareto profile without duplicating covered searches.","checks":["Run each proposed family scan with all support cells verified by d²+4N=m².","Sort returned columns by exact support and isolate any (4,5,5,5,5) or stronger profile.","For promising columns, test unused rows from the same anchor and perform divisor completion with explicit N and d distinctness checks.","Record failures by exact row vector and exclude registry-covered parameter regimes in subsequent searches."]}},{"round":19,"agentId":"tess","privatePlan":{"objective":"Search the new high-difference and mixed-scale atlases for an exact (4,5,5,5,5) or stronger profile, then complete and saturate the best partial columns.","checks":["Run all nine proposed family_scan vectors with limit=5000, maxChecks=5000000, and null irrelevant parameters.","Run divisor_completion for d1=243, d2=405 with differences/startDifference/endDifference/stride null, limit=5000, maxChecks=5000000.","Verify every returned cell by exact bigint square testing, require distinct N values, and sort supports weakest-first.","For any support-4 or support-5 column, test unused rows from its anchor and record only certified improvements."]}},{"round":20,"agentId":"mira","privatePlan":{"objective":"Join the completed pair atlases into a rigorously verified support matrix, prioritizing a 4×5 or 6×4 profile.","checks":["Recompute every cell with deterministic bigint square tests and factor witnesses.","Intersect the 345/405, 488/552, and six-row anchor supports; require distinct columns and rows.","Run only fresh mixed-row scans outside the registry and reject all near-square evidence.","Track completion saturation explicitly; if capped, rerun with a larger justified limit."]}},{"round":20,"agentId":"orum","privatePlan":{"objective":"Obtain deterministic saturation certificates and test the strongest cross-atlas projections for an exact Pareto improvement.","checks":["Audit the complete 488/552 factor-pair enumeration and independently verify all seven candidates.","Require the 345/405 completion job to prove exhaustion before using its candidate set.","Cross-test every returned column against the proposed six-row families, recording the full support vector.","Reject all near-square, truncated, or duplicate-column evidence; retain only bigint-verified cells."]}},{"round":20,"agentId":"pip","privatePlan":{"objective":"Exploit the 15400/105300 cross-atlas using exact support matrices, prioritizing a (4,5,5,5,5) or 6×4 improvement.","checks":["Run both proposed six-row family scans with deterministic bigint square tests.","Complete the (345,405) atlas and cross-test every returned column against the 15400 and 105300 difference sets.","Join all returned columns, deduplicate N, and sort supports weakest-first.","Verify every claimed cell by reconstructing the positive factor pair; reject all near-square or duplicate-column claims.","If no improvement appears, seed new divisor completions from the strongest four-support projections involving 488, 552, 688, and 736."]}},{"round":20,"agentId":"solvi","privatePlan":{"objective":"Intersect the newly saturated pair atlases to seek a verified (4,5,5,5,5) or 6×4 support matrix without repeating covered regions.","checks":["Run the proposed fresh family scans for Mira’s 840-based rows and Pip’s mixed 345/405 rows.","For every returned column, recompute all cells using exact integer-square tests and reject duplicates.","Join pair-completion outputs by support intersection, prioritizing columns sharing at least four differences.","Record complete candidate lists, support profiles, and explicit factor-pair witnesses; report any falsification."]}},{"round":20,"agentId":"tess","privatePlan":{"objective":"Exploit the fresh 577024 five-row regime through exact pair-completion intersections, seeking a strict support-profile improvement.","checks":["Run {\"jobType\":\"divisor_completion\",\"d1\":\"840\",\"d2\":\"1470\",\"differences\":null,\"limit\":5000,\"maxChecks\":5000000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}.","Run {\"jobType\":\"divisor_completion\",\"d1\":\"6180\",\"d2\":\"25065\",\"differences\":null,\"limit\":5000,\"maxChecks\":5000000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}.","Run {\"jobType\":\"family_scan\",\"d1\":null,\"d2\":null,\"differences\":\"840 1470 6180 25065 72120\",\"limit\":5000,\"maxChecks\":5000000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}.","Recompute every returned cell exactly, merge candidate columns, and sort supports weakest-first; retain only distinct-column profiles exceeding the current frontier."]}},{"round":21,"agentId":"mira","privatePlan":{"objective":"Exact-test cross-support among the new six-row projections and the 105300 pair atlases, seeking a strict Pareto improvement.","checks":["Run deterministic family scans on the 286000, 124000, and 577024 six-row sets.","Complete and intersect the (488,552) and (519,552) atlases with rows 345 and 405.","Construct the full exact support matrix, deduplicate columns, and reject any result not meeting a genuine 4×5 or 6×4 profile."]}},{"round":21,"agentId":"orum","privatePlan":{"objective":"Prioritize exact atlas intersections capable of certifying a strict support-profile improvement.","checks":["Complete and saturate the 124000 and 577024 projections, recording all returned columns and supports.","Intersect Mira’s six-row atlas with the 124000/577024 atlases and Pip’s 105300 atlases using deterministic bigint tests.","Reject numerical or partial evidence; retain only distinct N and d sets with a machine-checkable support matrix."]}},{"round":21,"agentId":"pip","privatePlan":{"objective":"Join the fresh high-difference projections and pair atlases, targeting either four columns with six common rows or five columns with four common rows.","checks":["Independently verify every listed factor pair and difference for 286000, 124000, and 577024.","Run exact family scans for the new six-row sets with bounded completion counts and full bigint square tests.","Complete and deduplicate the most promising pair atlases, then form a row-by-column support matrix across their union.","Reject any candidate with repeated N, repeated differences, missing exact cells, or support below the claimed profile.","Record a deterministic certificate only if it reaches a strict Pareto improvement."]}},{"round":21,"agentId":"solvi","privatePlan":{"objective":"Run exact support-matrix joins across the fresh six-row projections and smooth pair atlases, seeking a strict Pareto improvement.","checks":["Submit family_scan with d1=null, d2=null, differences=\"840 1080 1230 1449 1857 2070\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Submit family_scan with d1=null, d2=null, differences=\"246 252 420 615 867 876\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Submit family_scan with d1=null, d2=null, differences=\"252 615 840 1080 1200 1470\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Complete divisor atlases for (840,1230), (1449,1857), (488,552), (519,552), and (1080,1200), using limit=5000 and maxChecks=5000000; set differences, startDifference, endDifference, and stride to null.","Join returned columns by exact bigint tests, deduplicate N, sort supports weakest-first, and independently verify every claimed frontier profile."]}},{"round":21,"agentId":"tess","privatePlan":{"objective":"Privately cross-join the fresh atlases and seek a strict exact support improvement, emphasizing columns common to independent pair systems.","checks":["Deduplicate candidate N values across all atlases.","Recompute every d²+4N as an exact bigint square.","Rank support profiles weakest-first and reject any non-distinct-column certificate.","Escalate only a verified (4,5,5,5,5) or (4,4,4,4,4,4) profile."]}},{"round":22,"agentId":"mira","privatePlan":{"objective":"Construct an exact incidence matrix from the fresh pair atlases and prioritize any 4×5 or 5×5 intersection.","checks":["Complete the (552,680), (345,519), and Orum’s (252,615) atlases.","Score every returned N on the union of all new rows using bigint square tests.","Deduplicate columns and retain only support profiles that improve the current Pareto frontier.","Independently certify the strongest candidates, then derive a parameterized completion pattern if no improvement appears."]}},{"round":22,"agentId":"orum","privatePlan":{"objective":"Prioritize exact completion of the two new pair atlases and cross-score all returned columns against every proposed row set.","checks":["Verify all returned completions with bigint square tests and parity.","Sort supports weakest-first and test for (4,5,5,5,5) or better.","If atlases saturate without improvement, derive a new pair from shared divisor parametrizations rather than expanding the boundary scan blindly."]}},{"round":22,"agentId":"pip","privatePlan":{"objective":"Turn the fresh 680-based and high-difference projections into an exact Pareto improvement while avoiding every registry-covered region.","checks":["Run and deterministically verify the (552,680) and (345,519) divisor completions, then score all six anchor rows.","Join their returned columns against the 105300 projection and retain only distinct N with exact support at least four.","Test the 720720 and 1081080 six-row scans for four-column support, recording weakest-first support profiles.","Reject isolated anchors and all near-square or duplicate-column claims; escalate only a verified (4,5) or stronger profile."]}},{"round":22,"agentId":"solvi","privatePlan":{"objective":"Join the fresh exact projections and seek a strict Pareto improvement without entering any registry-covered region.","checks":["Run divisor_completion with d1=552,d2=680,differences=null,limit=5000,maxChecks=5000000,startDifference=null,endDifference=null,stride=null; score all six rows.","Run divisor_completion with d1=345,d2=519,differences=null,limit=5000,maxChecks=5000000,startDifference=null,endDifference=null,stride=null; intersect with the 680-family.","Run family_scan with differences=\"345 405 488 519 552 680\", limit=5000, maxChecks=5000000 and all other parameter keys null.","If needed, run boundary_scan with startDifference=50001,endDifference=200000,stride=1, limit=5000,maxChecks=5000000 and d1=d2=differences=null; independently bigint-verify the top support profiles."]}},{"round":22,"agentId":"tess","privatePlan":{"objective":"Exact Pareto search in the first uncovered high-difference bands, prioritizing atlas intersections that can produce five columns with support at least four.","checks":["Complete the proposed (552,680), (345,519), and high-difference atlases with all eight job parameters populated.","Intersect returned columns by exact support against every six-row anchor, rejecting numerical near-squares.","Scan disjoint boundary bands above 50,000 and retain only profiles lexicographically stronger than the current frontier.","Recheck all candidate columns with deterministic bigint arithmetic and distinctness tests."]}},{"round":23,"agentId":"mira","privatePlan":{"objective":"Search for exact cross-support columns joining the strongest fresh six-row projections, prioritizing a verified 4×5 or 5×5 rectangle.","checks":["Complete and independently bigint-verify the (783,953) and (1329,1896) atlases, scoring all six rows.","Intersect the 210700, 368524, and 105300 row families only after exact cell validation.","Reject duplicate N, duplicate differences, and any candidate lacking explicit factor witnesses.","Record the strongest Pareto support vector and falsify unsupported completion claims."]}},{"round":23,"agentId":"orum","privatePlan":{"objective":"Prioritize exact intersections from the fresh 252/615 and 783/953 atlases, seeking a Pareto improvement rather than more isolated six-row anchors.","checks":["Run only jobs with all eight parameter keys present and verify every cell using deterministic bigint square tests.","Require divisor-completion saturation or an explicit truncation certificate before treating an atlas as exhaustive.","Merge candidate columns, sort supports weakest-first, reject duplicate N or d values, and test directly for (4,5,5,5,5) and (5,5,5,5,5).","Record the registry exclusion check and discard any candidate overlapping a covered search region."]}},{"round":23,"agentId":"pip","privatePlan":{"objective":"Exploit the fresh high-difference anchors by exact atlas saturation and intersection, seeking five distinct columns with support at least five.","checks":["Run and verify the (783,953) and (1329,1896) divisor completions, then score all six 105300 rows.","Intersect the resulting columns with the 252/615 atlas and the Mira cross-seed rows without entering registry-covered regions.","Reject duplicates, parity failures, and near-square-only hits; retain only bigint-certified support vectors and explicitly test any 5×5 candidate."]}},{"round":23,"agentId":"solvi","privatePlan":{"objective":"Exploit the new six-row anchors through exact pair-atlas intersections, seeking a strict Pareto improvement without revisiting registry-covered regions.","checks":["Run the six-row scans for Mira’s 65509/60048/55427/98269/90078/83147 set, Pip’s six-row set, and Orum’s two six-row sets.","For every returned column, recompute d²+4N as an exact bigint square and sort supports weakest-first.","Intersect completed pair atlases across shared rows, enforcing five distinct N and five distinct differences.","Record complete negative saturation results as well as any exact 4×5-or-better certificate."]}},{"round":23,"agentId":"tess","privatePlan":{"objective":"Exploit the fresh high-difference anchors by exact pair completion and cross-row support maximization.","checks":["Verify every supplied factorization and reject duplicate differences or columns.","Run exact completions for (783,953) and (1329,1896), then score all six rows.","Cross-intersect the resulting columns, retaining only deterministic certificates with at least four columns of support five or better.","Record saturation limits and falsify unsupported 6×4 or 4×5 conjectures."]}},{"round":24,"agentId":"mira","privatePlan":{"objective":"Prioritize exact intersections that can turn the new anchors into at least five columns of support five, while avoiding all completed row sets.","checks":["Complete divisor atlases for (216,243), (243,405), (102953,154433), and Orum’s strongest mixed pairs.","Run fresh family scans on unions of rows from the 105300 and mixed-anchor families.","Score every output by exact support vectors, retaining only candidates that improve the Pareto frontier.","Verify every proposed rectangle with deterministic bigint factor witnesses before advancing it."]}},{"round":24,"agentId":"orum","privatePlan":{"objective":"Exhaust the fresh atlas intersections for a verified (4,5,5,5,5) or stronger support vector.","checks":["Run exact family scans on Mira’s six rows, Pip’s six rows, and Orum’s two mixed six-row sets.","Complete the 216/243, 243/405, 102953/154433 divisor atlases and score every returned column against all relevant rows.","Require saturation or an explicit completion cap before drawing conclusions.","Bigint-verify every surviving candidate, sorting supports weakest-first and rejecting numerical near-squares."]}},{"round":24,"agentId":"pip","privatePlan":{"objective":"Run exact fresh cross-atlas searches combining the 216/243/405 projection with Orum’s mixed six-row anchor sets, targeting a verified 4×5 or stronger support vector.","checks":["divisor_completion: {\"d1\":\"216\",\"d2\":\"243\",\"differences\":null,\"limit\":5000,\"maxChecks\":5000000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}","family_scan: {\"d1\":null,\"d2\":null,\"differences\":\"216 243 345 405 488 519 552 680\",\"limit\":5000,\"maxChecks\":5000000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}","family_scan: {\"d1\":null,\"d2\":null,\"differences\":\"252 399 615 765 867 876\",\"limit\":5000,\"maxChecks\":5000000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}","family_scan: {\"d1\":null,\"d2\":null,\"differences\":\"252 615 867 876 1245 7035\",\"limit\":5000,\"maxChecks\":5000000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}","Recompute every returned cell as d²+4N=m² and rank columns weakest-first; reject all numerical near-square evidence."]}},{"round":24,"agentId":"solvi","privatePlan":{"objective":"Use exact pair-completion intersections to seek a new support vector at least (4,5,5,5,5), prioritizing the fresh high-difference and mixed-family rows.","checks":["Complete the 216/243 and 243/405 atlases and score every returned column against all fresh Pip rows.","Complete Mira’s 102953/154433 pair and test intersections with the other q=7–9 rows.","Run Orum’s mixed row sets, retaining only distinct N columns with independently verified bigint square tests.","Reject all numerical near-squares and any candidate overlapping registry-covered searches."]}},{"round":24,"agentId":"tess","privatePlan":{"objective":"Intersect the fresh high-difference atlases and extend the common-factor construction into an untouched q=14–22 regime, seeking a verified strict Pareto improvement.","checks":["Run divisor_completion with {jobType:divisor_completion,d1:216,d2:243,differences:null,limit:5000,maxChecks:5000000,startDifference:null,endDifference:null,stride:null}; score every completion on 345,405,488,519,552,680.","Run divisor_completion with {jobType:divisor_completion,d1:243,d2:405,differences:null,limit:5000,maxChecks:5000000,startDifference:null,endDifference:null,stride:null}; retain only support vectors that can contribute to 4×5 or better.","Run family_scan with {jobType:family_scan,d1:null,d2:null,differences:\"252 399 615 765 867 876\",limit:5000,maxChecks:5000000,startDifference:null,endDifference:null,stride:null} and the analogous set 252 615 867 876 1245 7035.","Run family_scan with {jobType:family_scan,d1:null,d2:null,differences:\"51466 48033 45029 40022 36016 34299 32738 77206 72057 60042 54034 51459 49118\",limit:5000,maxChecks:5000000,startDifference:null,endDifference:null,stride:null}; verify all retained cells and sort supports weakest-first.","Reject numerical near-squares, duplicate N or d values, and any result lacking explicit factor witnesses; compare only exact support vectors against the current frontier."]}},{"round":25,"agentId":"mira","privatePlan":{"objective":"Construct a genuinely parameterized, non-Bremner algebraic family for balanced support: study the square-additive variety m_ij^2-d_i^2=4N_j, derive elliptic fibrations or rational families with several independently varying columns, and search positive-rank fibers rather than completing fixed difference pairs.","checks":["Normalize primitive scale and parity, then symbolically eliminate N_j to obtain explicit curve/fibration equations; document the parameter map and prove it is not merely a previously tested fixed Bremner specialization.","Use exact rational-point computation and height-bounded exploration on candidate elliptic fibers, including higher-denominator points not covered by the registry's denominator-10,000 scan; label all unverified outputs conjectural.","Clear denominators and run deterministic bigint tests for five distinct d-values and N-values, publishing the complete 5×5 support matrix and sorted column supports. Reject isolated anchors and profiles below the balanced target.","Deduplicate every specialization against registry hashes and the completed cooldown families. Request zero divisor-completion jobs unless this primary algebraic route first produces a candidate; then allow at most one validation job."]}},{"round":25,"agentId":"orum","privatePlan":{"objective":"Construct a deduplicated verified incidence graph and search it for balanced K₅,₅ or Pareto improvements using SAT/exact cover, without requesting divisor_completion.","checks":["Ingest only public cells passing bigint square tests; retain provenance, job hash, parity data, and exact m values.","Deduplicate N and difference vertices, then compute the complete support matrix and sort columns weakest-first.","Encode selected-row and selected-column variables with exact missing-cell clauses; test K₅,₅ first, then (4,5,5,5,5) and stronger balanced profiles.","Require five distinct N and five distinct differences, independently recompute every selected cell, and reject any near-square-only evidence.","Record SAT certificates or exhaustive finite-domain coverage; distinguish graph-UNSAT from a global mathematical impossibility result.","Apply cooldown hashes to exclude all completed row sets and registry-covered regions before interpreting any surviving biclique as novel."]}},{"round":25,"agentId":"pip","privatePlan":{"objective":"Round 26 modular/local lane: test the conjectural square-class-clone family d_i≡±d_0 (mod M), using local quadratic-residue incidence and CRT to prune or force balanced compatible rows before exact validation. No divisor-completion job will be submitted.","checks":["Set M=2^4·3^2·5·7·11=55440 and enumerate distinct rows d_i=±d_0+Mt_i, filtering a genuinely new band with all d_i>50000 and deduplicating against the registry and completed-job hashes.","For p^e in {13,17,19,23} and selected prime powers, compute Q_{p^e}(d,n)=1 iff d^2+4n is a square modulo p^e; verify that clone rows have identical local signatures at every prime dividing M.","Build the exact local incidence matrix between five candidate rows and N residue classes modulo the auxiliary CRT modulus, requiring five distinct residue classes and optimizing the sorted balanced support floors.","Use Hensel lifting and independent modular recomputation to discard row sets whose best local profile is below (4,5,5,5,5); record this only as a sieve falsification, never as a global nonexistence claim.","Pass only surviving row sets to a later exact family_scan or mira’s validation, with bigint square tests, distinct N checks, complete support vectors, and saturation status reported."]}},{"round":25,"agentId":"solvi","privatePlan":{"objective":"Run a stochastic, optimization-guided factor-coordinate search for genuinely new five-row/five-column incidence seeds, followed by independent exact bigint verification; submit no divisor_completion job.","checks":["Build candidate pools from mutated factor pairs and mixed-scale products, deduplicated against registry hashes and all completed row sets; use prior negative families only as cooldown constraints.","Score each proposed 5×5 block by the lexicographically sorted ten support degrees (five row and five column degrees), maximizing the weakest degree before any anchor score.","Use beam search or simulated annealing with mutations of factor coordinates, row replacement, column replacement, and cross-family joins; reject candidates whose support floor does not improve.","Recompute every surviving cell from scratch using exact bigint square tests, parity, positivity, and distinct N and d checks; sort and report the full support matrix.","Accept a result only if it reaches a verified balanced Pareto improvement such as (4,5,5,5,5), (5,5,5,5,5), or six rows on four distinct columns; otherwise publish the exact negative frontier and sampled domain."]}},{"round":25,"agentId":"tess","privatePlan":{"objective":"Build an adversarial ledger of existing near-misses and select one genuinely adaptive, non-divisor-completion experiment that can either produce balanced support or issue a rigorous obstruction.","checks":["Recompute every retained candidate with bigint tests d²+4N=m²; enforce positivity, parity, distinct columns, and sorted row/column supports.","For each support-4 skeleton, search explicitly for missing rows and additional distinct columns; mark bi-maximality only with an exhaustion argument.","Hash row sets, parameter ranges, and completed outputs against the registry and prior rounds before accepting novelty.","Reject numerical closeness, atlas size, high differences, and stride coverage as progress metrics.","Use existing negative certificates to choose the next experiment; request no divisor_completion job this round and route any validation to Mira.","Prefer an elliptic-family or one-row-lift test derived from the registry’s unruled opportunities, with a predeclared falsification condition and Pareto score."]}},{"round":26,"agentId":"mira","privatePlan":{"objective":"Construct an explicit rational parametrization of a three-row square-additive fiber and test whether its elliptic quotient deforms the seed column into additional integral columns, staying in the constructive algebraic-family lane.","checks":["Derive the quartic obtained from three equations m_i^2-d_i(t)^2=4X and exhibit its birational elliptic model.","Compute the known seed point and its elliptic multiples symbolically or exactly; reject points failing integrality, positivity, or distinctness.","Impose the remaining q_r rows as explicit square conditions and identify any common rational subfamily rather than relying on numerical proximity.","If a non-seed integral candidate emerges, request at most one divisor_completion job solely for independent exact verification."]}},{"round":26,"agentId":"orum","privatePlan":{"objective":"Run an incidence-hypergraph/exact-cover search exclusively on bigint-verified cells from the six-row seed scans and known certificates, seeking a distinct 5-row by 5-column all-one submatrix.","checks":["Recompute every cell via exact perfect-square tests; reject modular-only or near-square evidence.","Encode distinct differences and distinct N-columns, then enumerate bicliques with column supports sorted weakest-first.","Require and report a saturation certificate for the searched candidate pool, including Pareto support profiles.","Use the Bremner 4×4 certificate only as a regression fixture and exclude all registry-covered search regions.","Do not request divisor_completion."]}},{"round":26,"agentId":"pip","privatePlan":{"objective":"Design a two-stage CRT filter that forces compatible square roots modulo prime powers while avoiding the rejected single-modulus clone lifts.","checks":["Enumerate distinct row residues and column residues modulo several prime powers, retaining only patterns with simultaneous square roots and correct parity.","Apply Hensel-style lifting to discard locally compatible classes whose roots fail at higher powers.","Construct CRT lifts outside the registry’s covered ranges, then use exact bigint support vectors as the sole acceptance test.","Do not request divisor_completion; record weakest-first support and reject all conclusions based on numerical square proximity."]}},{"round":26,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery for new high-difference row sets, then perform exact bigint verification while targeting Pareto improvements such as (4,5,5,5,5).","checks":["Sample and mutate row seeds outside all covered difference regimes, scoring exact-support potential with modular and square-residue filters.","Optimize for balanced multi-column support rather than isolated hits; reject modular-only designs whose exact prospects are poor.","Submit only non-divisor-completion validation scans, then deterministically verify every returned cell and retain only distinct N and differences.","Record weakest-first support vectors and falsify any candidate lacking the required five-way intersection."]}},{"round":26,"agentId":"tess","privatePlan":{"objective":"Run an adversarial, adaptive attack on the q_r=t+r fibers while developing saturation certificates for modular-clone and elliptic-family searches.","checks":["Require exact support vectors and distinct positive N before retaining any column; reject numerical near-squares.","First falsify the three proposed q-fibers with bounded, auditable scans; only expand t when a fiber shows support at least 4.","Derive local obstructions and saturation bounds for d_i congruent to plus or minus c modulo M, using the zero-support clone results to prune similar designs.","Use the weakest-first Pareto profile to choose the next experiment, prioritizing any route that can improve (4,4,5,5,5) toward (4,5,5,5,5).","Do not request divisor_completion; reserve validation-only checking for Mira."]}},{"round":27,"agentId":"mira","privatePlan":{"objective":"Construct an explicit rational parametrization of a three-row genus-one fiber arising from the primitive step-3 family, then seek integral specializations with additional exact square rows.","checks":["Derive the elliptic model and verify the seed point symbolically.","Generate non-seed rational points by the elliptic group law, rejecting singular or degenerate cases.","Test integral specializations only with exact perfect-square identities and support scoring; request no divisor-completion job."]}},{"round":27,"agentId":"orum","privatePlan":{"objective":"Construct an incidence-hypergraph/SAT exact-cover search using only verified exact cells, targeting five distinct N-columns with five distinct difference rows.","checks":["Canonicalize and deduplicate all verified (N,d) cells; independently recompute m^2=d^2+4N with deterministic bigint arithmetic.","Exclude modular-only translated lifts and seed-only columns unless they participate in a genuinely new balanced support pattern.","Search exact covers with column supports sorted weakest-first, prioritizing support profiles (4,5,5,5,5) and (5,5,5,5,5).","Enforce distinctness of all N and d values and emit a machine-checkable support certificate for every surviving pattern."]}},{"round":27,"agentId":"pip","privatePlan":{"objective":"Use a two-stage local CRT sieve on selected rows of the new step-3 seed columns to force compatible square-root branches before any exact search.","checks":["For odd prime powers, impose N ≡ (m_i²−d_i²)/4 locally and intersect root-sign branches across at least four rows.","CRT-combine only compatible residue classes, avoiding the previously rejected single-modulus clone lifts.","Test resulting lifts with exact bigint square checks and retain only columns with support at least four and distinct positive factor pairs.","Conditionally request one divisor_completion job only after the sieve produces a genuinely new exact candidate."]}},{"round":27,"agentId":"solvi","privatePlan":{"objective":"Run stochastic, optimization-guided discovery over uncovered primitive seed families and translated square-root configurations, then submit only Pareto-promising candidates to exact bigint verification.","checks":["Sample primitive arithmetic progressions with varied step and offset, prioritizing differences beyond registry cutoffs.","Optimize modular support forecasts and column-balance scores, while never treating modular hits as exact cells.","Include translated-root and mixed-family seeds only as proposal generators; reject candidates lacking distinct positive N and d.","Avoid all registry-covered search regions and do not request divisor_completion.","Use Pip exclusively for final deterministic bigint support verification."]}},{"round":27,"agentId":"tess","privatePlan":{"objective":"Run adversarial falsification and adaptive support-first tests on the h=3 family while constructing finite-range saturation certificates; keep modular lifts only as null controls.","checks":["Recompute every h=3 seed identity and distinctness condition independently.","Use exact family scans, adapting selected rows toward the weakest support columns.","Record exhaustive finite bounds, candidate counts, and support histograms for each scan.","Test translated-lift predictions against exact square factorization, without requesting divisor_completion."]}},{"round":28,"agentId":"mira","privatePlan":{"objective":"Construct and search a parameterized family of elliptic fibers extending the q_r=t+5r identity, targeting multiple distinct columns with balanced exact support.","checks":["Derive the general q_r=t+hr family, including primitive and parity conditions, and exclude registry-covered specializations.","Use the explicit quartic and its known points to generate rational points via elliptic-group operations; retain only integral columns after exact bigint testing.","Combine several elliptic-generated columns and score their exact row supports, requiring distinct N and differences.","Use inverse-residue data only as a local sieve, while Orum performs validation; request no divisor_completion job."]}},{"round":28,"agentId":"orum","privatePlan":{"objective":"Use an incidence-hypergraph search over independently generated, exactly verified cells from the step-5 and inverse-residue methods, targeting a certified 4×5 or 5×5 row-column biclique.","checks":["Recompute every cell with bigint equality d^2+4N=m^2 and reject all approximate hits.","Merge and deduplicate verified (N,d) cells, then enumerate row-subset intersections with weakest-first support ordering.","Require explicit witness lists for every claimed 4×5, 5×5, or 6×4 structure and discard seed-only columns.","Record covered difference regimes and avoid all registry-covered regions; request no divisor-completion job unless a non-seed hypergraph candidate first appears."]}},{"round":28,"agentId":"pip","privatePlan":{"objective":"Build a modular CRT sieve that forces or sharply prunes simultaneous row compatibility for the step-5 and inverse-residue families.","checks":["For each selected odd prime, enumerate local square-root branches and intersect the resulting N residue sets across every row.","Combine surviving residues by CRT, prioritizing row sets with four or more rows and minimal local multiplicity.","Apply only family scans to surviving classes; request no divisor_completion, and send all exact hits to Orum for validation.","Score only deterministic bigint support and distinctness, excluding numerical near-square evidence."]}},{"round":28,"agentId":"solvi","privatePlan":{"objective":"Run stochastic, optimization-guided discovery of new step-h product seeds and inverse-residue CRT wheels, then use family_scan and Orum-only exact bigint verification to seek balanced support.","checks":["Sample primitive q_r=t+hr families with h outside the covered step-1 and step-3 regimes; compute every seed difference exactly and retain a Pareto front using row count, residue compatibility, and distinctness only as screening scores.","Sample new inverse-residue wheels at fresh primes, rejecting duplicates and registry-covered difference regimes; never count local congruence or numerical square proximity as support.","Allocate family_scan only to Pareto-leading seeds. Every request must use the complete parameter set, e.g. {\"d1\":null,\"d2\":null,\"differences\":\"576545 687430 851115 1117100 1624885 2978970\",\"limit\":5000,\"maxChecks\":2000000,\"startDifference\":null,\"endDifference\":null,\"stride\":null}.","Send all returned columns to Orum for deterministic bigint verification and exact support sorting; reject nonintegral, parity-invalid, duplicate, or merely near-square results. Do not request divisor_completion."]}},{"round":28,"agentId":"tess","privatePlan":{"objective":"Run an adversarial, exact-support campaign against the proposed step-5 and cross-anchor constructions, prioritizing strict Pareto improvements over isolated high-support columns.","checks":["Require deterministic bigint verification for every returned cell and reject all claims based solely on congruences or approximate squares.","Build finite incidence graphs from exact hits and certify all relevant 4-row, 5-row, and 6-row intersections; record saturation or emptiness.","Adapt the next search region only from observed row-intersection deficits, avoiding all registry-covered regions and excluding divisor_completion.","Track weakest-first column supports and retain only candidates that improve the current Pareto frontier."]}},{"round":29,"agentId":"mira","privatePlan":{"objective":"Build a multi-column constructive family by linking the h=7 elliptic fibers with prime-minus-one wheel data, targeting exact support patterns (5,5,5,5,5) or (4,5,5,5,5).","checks":["Derive and symbolically verify the quartic-to-Jacobian maps for shared triples of h=7 or wheel differences.","Generate rational points by exact elliptic group-law arithmetic and test only exact integrality of all resulting m and N.","Solve shared-difference equations between distinct specializations to obtain multiple columns, rejecting isolated seed points.","Apply modular wheels only as exact congruence sieves, then bigint-check every incidence and Pareto-sort supports.","Use no divisor_completion request; submit any surviving candidates solely to Solvi for validation."]}},{"round":29,"agentId":"orum","privatePlan":{"objective":"Construct an incidence hypergraph from only bigint-verified cells produced by the new h=7 and modular-wheel regimes, then run SAT/exact-cover searches for 5×5 and 6×4 support.","checks":["Recompute every claimed cell by exact d²+4N=m² with an explicit integer m; reject numerical near-squares.","Deduplicate N and d, discard seed-only columns, and maintain weakest-first column-support Pareto scores.","Encode row/column incidence and solve cardinality constraints for five distinct rows and five distinct columns, separately testing 4×5 and 6×4 targets.","Require a finite saturation certificate for each reported negative subsearch and do not request divisor_completion."]}},{"round":29,"agentId":"pip","privatePlan":{"objective":"Use modular, local, and congruence design to prune or force compatible rows around the new h=7 and prime-minus-one anchors, without requesting divisor_completion.","checks":["Derive residue tables for m^2=d^2+4N modulo several small and anchor primes, retaining only N residue classes compatible with at least four selected rows.","CRT-combine independent local filters and reject row sets whose allowed N classes are empty or force duplicate columns.","Test whether h=7 difference pairs share nontrivial gcd or quadratic-residue structure that can force a common N progression.","Build exact incidence masks from surviving congruence classes, then pass only Pareto-promising row sets to Solvi for validation.","Record separate falsifications for local compatibility versus exact integer realization; do not infer support from numerical near-squares."]}},{"round":29,"agentId":"solvi","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery over uncovered h=7 elliptic parameters, inverse-residue wheels, and cross-anchor row mutations, then retain only exact bigint Pareto improvements.","checks":["Exclude all registry-covered regions and completed family scans.","Use randomized parameter mutations and elliptic-point multiples scored lexicographically by sorted exact column supports.","Require distinct positive N and d values before verification.","Verify every cell by exact bigint testing of d^2+4N=m^2; reject numerical near-squares.","Keep only candidates improving the current support frontier, especially (4,5,5,5,5) or (4,4,5,5,5).","Request no divisor_completion unless an independently generated seed already passes the exact-support screen."]}},{"round":29,"agentId":"tess","privatePlan":{"objective":"Adversarially falsify the h=7 quartics and inverse-residue wheels, then adaptively build exact incidence saturation certificates targeting strict Pareto improvements.","checks":["Re-derive Mira’s quartic and enforce exact rational-to-integral back-substitution, positivity, distinct columns, and exact square witnesses.","Use only family_scan exploration; any request will specify all eight parameters: d1=null, d2=null, differences=<space-separated rows>, limit=5000, maxChecks=<budget>, startDifference=null, endDifference=null, stride=null.","Apply Pip’s CRT conditions only as pruning, independently testing every surviving cell by bigint arithmetic and rejecting anchor-only or low-support returns.","After each scan, sort supports weakest-first and adapt the next tuple toward the strongest exact incidence overlap; stop treating numerical closeness as evidence.","For each finite returned candidate set, construct an explicit incidence table and certify the best 4×5 or 6×4 bound within the enumerated search limits; send only exact witnesses to Solvi for validation."]}},{"round":30,"agentId":"mira","privatePlan":{"objective":"Develop a linked parametrization of the new arithmetic-progression elliptic fibers, seeking shared rational-point structure across h=11 and h=13 rather than isolated seed scans.","checks":["Derive the completion quartic and its birational elliptic model symbolically for neighboring t and h values.","Search rational multiples and parameter specializations for columns with exact support at least four, using only bigint square tests.","Compare common factors and maps between the h=11 and h=13 fibers to identify a higher-dimensional family.","Submit only family_scan work; do not request divisor_completion."]}},{"round":30,"agentId":"orum","privatePlan":{"objective":"Build an exact incidence hypergraph from newly returned, Tess-verified cells and search it for 5×5, 4×5, or 5×4 bicliques using SAT/exact cover.","checks":["Recompute every edge with bigint exact-square and parity tests, including positive factor recovery.","Dedupe N and difference vertices; enforce distinct rows and columns and compute weakest-first support vectors.","Exclude CRT-only or numerically close candidates until exact cells are certified.","Enumerate all bicliques in the finite verified-cell set and emit a saturation certificate or Pareto frontier.","Submit only finalists to Tess for independent validation; request no divisor-completion job."]}},{"round":30,"agentId":"pip","privatePlan":{"objective":"Design a modular and local congruence filter for fresh elliptic and wheel row sets, targeting compatible five-row systems without repeating covered regions.","checks":["Compute simultaneous quadratic-residue conditions for d²+4N modulo selected prime powers.","Lift compatible N-residue classes and reject empty intersections or forced adjacent differences.","Rank surviving row sets by local density before any exact validation by Tess."]}},{"round":30,"agentId":"solvi","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery over uncovered progression spacings, wheel rows, and mixed row subsets, targeting Pareto improvement toward sorted supports [4,5,5,5,5] and [5,5,5,5,5], followed by exact bigint verification.","checks":["Sample and mutate row sets while rejecting adjacent differences and registry-covered parameter regions.","Optimize exact-support proxies based on modular compatibility and completed-column yield, not square proximity.","Send only promising family scans with complete parameter maps and bounded maxChecks; request no divisor_completion.","Recompute every incidence using d²+4N=m² with bigint arithmetic, then enforce distinct N and distinct differences.","Pass surviving Pareto candidates to Tess for validation-only confirmation."]}},{"round":30,"agentId":"tess","privatePlan":{"objective":"Adversarially falsify the fresh h=11/h=13 and prime-wheel routes, and require saturation certificates before any adaptive expansion.","checks":["Recompute every seed cell with deterministic bigint arithmetic and label all seed columns as non-advances unless distinct additional columns appear.","For each proposed family, demand exact check counts, duplicate filtering, maximum column support, and a certificate that the reported scan range was fully exhausted; record support vectors, not square proximity.","Use adjacent-difference exclusion and modular obstructions to prune the mixed wheel, then adaptively pursue only columns with support at least four toward balanced 5-row or 5-column incidence.","Do not request divisor_completion unless a different primary method first produces a concrete candidate; otherwise make no divisor-completion request.","Log all saturated failures as Pareto near-misses and exclude every registry-covered region from follow-up scans."]}},{"round":31,"agentId":"mira","privatePlan":{"objective":"Construct a modular incidence sieve that selects difference rows with compatible local square masks and forces candidate columns into shared CRT classes.","checks":["Enumerate exact square-residue masks modulo small prime powers for prospective five-row sets.","Discard row sets whose CRT-compatible column residues are empty or too sparse for five distinct columns.","Lift surviving residue classes to exact bigint tests, retaining only distinct columns with the required support floors.","Use no divisor_completion job unless an independently generated congruence construction first produces a candidate needing verification."]}},{"round":31,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided mutations of the new AP and wheel/CRT seed families to discover balanced high-support candidates, then apply exact bigint verification.","checks":["Sample and mutate (h,t), wheel moduli, row subsets, and CRT mixtures outside registry-covered regimes; score seeds lexicographically by sorted support floors and total support.","Apply modular square-residue masks only as cheap filters; reject anchors, duplicate columns, and numerical near-squares.","Verify every retained cell with exact bigint arithmetic: test d²+4N for a perfect square and parity, then construct the full support matrix.","Retain only Pareto improvements toward (5,5,5,5,5) or (4,5,5,5,5); use Mira solely for validation and request no divisor_completion job."]}},{"round":31,"agentId":"pip","privatePlan":{"objective":"Run an incidence-hypergraph/exact-cover search on the newly verified exact cells from the h=17,18,20 fibers and 7560-wheel projections, targeting a 5-row, 5-column biclique and accepting 6×4 as an intermediate.","checks":["Revalidate every proposed cell with exact bigint square tests and require distinct positive N and differences.","Deduplicate columns across reports, remove isolated wheel anchors, and compute exact row and column supports.","Solve the resulting row-subset/column-subset incidence constraints by SAT or exact cover; reject any solution with a support floor below the claimed target.","Record minimal certificates for every surviving biclique and separately falsify unsupported local-sieve candidates."]}},{"round":31,"agentId":"solvi","privatePlan":{"objective":"Adversarially test whether the new progression and wheel regimes contain balanced support, while converting negative scans into auditable saturation evidence and adapting only when exact incidence data justify it.","checks":["Require deterministic bigint confirmation of every claimed cell, including the square root and parity, and normalize duplicate columns before scoring.","For each family scan, record the actual N/search domain, check count, residue-mask survivors, exact hits, anchors, and maximum non-anchor support; reject conclusions based only on absent reported output.","Build row-subset Pareto tables from all exact hits and test every five-row projection for five distinct columns before accepting any claimed improvement.","Use the first completed regime as a falsification gate: if its non-anchor support maximum is at most 3 with adequate coverage, switch to a CRT-mixed row set rather than repeating congruence variants.","Keep Tess as the sole validation-only verifier and do not request divisor_completion."]}},{"round":31,"agentId":"tess","privatePlan":{"objective":"Develop a constructive rational-point parametrization for the arithmetic-progression fibers, prioritizing shared elliptic structure rather than another fixed-seed scan.","checks":["Rewrite simultaneous conditions d_i^2+4N=m_i^2 as a rational intersection of quadrics after normalizing by one row.","Derive the induced elliptic curve or elliptic surface for three selected rows and compute explicit rational sections arising from the AP seed.","Test whether duplication or addition of those sections produces columns simultaneously incident to four or five fixed rows; clear denominators exactly.","Compare h=17,18,20 and seek a parameter relation or specialization yielding multiple distinct columns with balanced support.","Reject every candidate unless bigint square tests certify all incidences and the support vector strictly improves the published frontier."]}},{"round":32,"agentId":"mira","privatePlan":{"objective":"Design CRT-local row sets whose square conditions share multiple admissible N residue classes, then prune family scans to these compatible classes.","checks":["Enumerate square residues modulo selected prime powers, including 2-adic, 3-adic, and 5-adic factors, for candidate difference rows.","Retain only 5- or 6-row sets with a nonempty common CRT mask and quantify its density before scanning.","Require exact bigint support verification and compare resulting support vectors against the (4,5,5,5,5) frontier."]}},{"round":32,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided mutations of bridge, wheel, and prime-power fiber seeds to discover nonrepeating high-support tuples, followed by exact bigint verification.","checks":["Score mutations by lexicographic sorted column-support vectors, prioritizing balanced 5×5 and then 6×4 structure.","Use modular filters only for seed ranking; verify every retained cell by exact integer-square testing and parity checks.","Exclude registry-covered regions, fixed Bremner specializations, and wheel anchors.","Require distinct positive N and differences, then pass survivors to Pip for validation-only checking; request no divisor_completion job."]}},{"round":32,"agentId":"pip","privatePlan":{"objective":"Run an exact-cover search on verified nonanchor incidence cells, targeting a 5×5 certificate or a 6×4 Pareto improvement without repeating covered regions.","checks":["Recompute every cell with deterministic bigint square tests and discard unverified or anchor-only records.","Build the bipartite incidence hypergraph and enumerate five- and six-row intersections.","Require distinct positive N and distinct differences, scoring columns weakest-first.","Independently verify any discovered rectangle; request no divisor_completion job unless exact-cover first produces a candidate.","Record bounded negative results without drawing global conclusions."]}},{"round":32,"agentId":"solvi","privatePlan":{"objective":"Adversarially falsify the proposed fibers and certify saturation before allocating another adaptive search.","checks":["Run exact bigint support extraction on the Mira and Orum row sets, excluding all wheel and anchor columns; retain full row/column support vectors.","Construct CRT residue-intersection certificates for each candidate row tuple; call a family saturated only when the modular intersection is empty or an explicitly bounded exact enumeration is exhausted.","Adapt the next experiment only from surviving rows whose attainable support can still exceed the current Pareto frontier; otherwise terminate that branch.","Use Pip solely for deterministic validation, with no divisor_completion request."]}},{"round":32,"agentId":"tess","privatePlan":{"objective":"Construct an elliptic/rational-point family from a fresh six-row P_h seed, then seek five integral columns with support at least five.","checks":["For two fixed rows d1,d2, derive the exact conic parametrization y1²−y2²=d1²−d2² and N=(y1²−d1²)/4.","Impose a third row to obtain the resulting quartic; verify nonsingularity and the seed rational point using exact arithmetic.","Generate rational points by exact elliptic-curve group operations, filtering points that satisfy the fourth and fifth rows exactly.","Clear all rational denominators by a common scaling q: replace d by qd and N by q²N; verify positivity, distinctness, and every bigint square identity.","Evaluate sorted column supports and retain only genuine Pareto improvements; request no divisor_completion job."]}},{"round":33,"agentId":"mira","privatePlan":{"objective":"Design a modularly pruned row family with several compatible N residue classes, staying in the local/congruence lane and avoiding covered anchor regimes.","checks":["For each proposed row tuple, compute exact residue sets R_d(M)={N mod M: d²+4N is a quadratic residue mod M} at 2-power, 3-power, and fresh prime-power moduli; retain only intersections supporting at least five rows.","Compare the surviving CRT classes against the H=360, 420, 840 fibers and the 10008 fiber, rejecting classes whose compatibility is only the trivial radicand residue 4.","Use local obstructions to prune candidate row subsets before any scan, and require nonanchor columns with support at least four as the minimum useful output.","Send only survivors to Orum for deterministic bigint incidence validation; request no divisor_completion and claim no result from numerical near-squares."]}},{"round":33,"agentId":"orum","privatePlan":{"objective":"Run stochastic, optimization-guided mutations of verified incidence seeds in uncovered parameter regimes, followed by exact bigint verification of Pareto-leading support patterns.","checks":["Weight mutations by exact co-support and weakest-column support; target 4×5 and 6×4 before 5×5.","Exclude registry-covered regions, repeated tuples, and anchor-only explanations.","Recompute every cell with bigint perfect-square tests and verify distinct positive N and differences.","Use at most one divisor_completion job only if an independently generated primary method first yields a candidate."]}},{"round":33,"agentId":"pip","privatePlan":{"objective":"Run an exact-cover search over verified incidence cells from the 10008 fiber and Orum’s cross-cloud mutations, targeting nonanchor 5x5 or 6x4 rectangles.","checks":["Recompute every cell with deterministic bigint perfect-square tests.","Build support masks and prune by weakest-column support.","Exclude 10008, 7920, 7560, and all registry-covered anchors.","Require distinct positive N columns and distinct differences; verify every reported rectangle independently."]}},{"round":33,"agentId":"solvi","privatePlan":{"objective":"Run adversarial falsification and saturation checks on the freshest row families, with adaptive branching toward exact Pareto improvements.","checks":["Recompute every proposed cell with bigint-perfect-square tests and discard anchor-only columns.","Correctly derive CRT square-residue masks, especially for the H=360,c=7 fiber, before allocating search budget.","For each scan, produce a saturation certificate: complete returned support masks, weakest-first column supports, and explicit failure of 5×5 and 6×4 extraction when absent.","Use a bounded pilot first; expand only tuples yielding nonanchor support at least four, otherwise mutate the strongest cross-cloud intersections.","Request no divisor_completion; reserve validation-only work for Orum."]}},{"round":33,"agentId":"tess","privatePlan":{"objective":"Construct a fresh multi-column algebraic family by parameterizing factor pairs q_i=c+Ht_i and imposing simultaneous extra columns through the resulting rational-point equations, including genus-one/elliptic cases.","checks":["Symbolically verify every d_i, positivity, distinctness, and the factorization identity before any search.","After fixing four rows, derive and validate the associated genus-one model and its rational-to-integral specialization map.","Enumerate exact rational points outside the covered denominator and smallest-difference regimes; reject numerical near-squares.","Score only exact bigint incidences, remove anchor-only columns, and pass any candidate rectangle to Orum for validation."]}},{"round":34,"agentId":"mira","privatePlan":{"objective":"Design a fresh composite-CRT row family whose local square classes force at least four compatible difference rows before exact incidence extraction.","checks":["Enumerate new composite moduli and residue parameters, excluding the covered H=420, H=840, and h=17,18,20 regimes.","For each prime-power factor, solve d_i^2+4N as a local square and intersect the resulting N residue classes by CRT.","Require a common local class for at least four rows before generating candidates.","Run exact bigint support scoring, weakest-column first; reject every near-square and every repeated N or difference."]}},{"round":34,"agentId":"orum","privatePlan":{"objective":"Run stochastic, optimization-guided mutations of the fresh divisor fibers and CRT-compatible row sets, then exact-bigint verify the best balanced-support candidates.","checks":["Generate cross-fiber difference seeds while excluding registry-covered regimes and anchor-only gains.","Rank samples lexicographically by weakest-column support, targeting (4,5,5,5,5) and then (5,5,5,5,5).","Submit only top exact-support candidates for deterministic bigint perfect-square verification; request no divisor_completion."]}},{"round":34,"agentId":"pip","privatePlan":{"objective":"Perform exact-cover/SAT extraction over the fresh verified-cell incidence clouds, targeting a 5×5 biclique while excluding anchor-only support.","checks":["Recompute every cell with exact bigint square tests.","Construct the row-column incidence graph and remove anchor-only columns.","Enumerate row and difference subsets with weakest-column-first pruning.","Record exact Pareto support profiles and reject all non-frontier outputs."]}},{"round":34,"agentId":"solvi","privatePlan":{"objective":"Adversarially test whether the fresh fibers can exceed the current support frontier, while producing saturation certificates for every exhausted row cloud.","checks":["Construct exact bigint incidence matrices for Pip’s new fibers and Mira’s 2520 fibers; reject every near-square or residue-only hit.","For each cloud, compute weakest-column-first support profiles and certify that no 5-row subset has five distinct supporting columns.","Adaptively cross only rows whose exact support profiles overlap; avoid repeating registry-covered regimes and anchor-inflated columns.","Use at most one divisor_completion job, only after an independently discovered nonanchor candidate reaches a credible 4-by-5 frontier."]}},{"round":34,"agentId":"tess","privatePlan":{"objective":"Construct a new algebraic family by gluing two difference rows through a rational conic, then imposing additional rows through explicit genus-one curves, while retaining only exact integral points outside covered regimes.","checks":["For fixed d,e, set Delta=d^2-e^2 and parameterize x=(s+Delta/s-2d)/4, y=(s-Delta/s-2e)/4; symbolically verify x(x+d)=y(y+e).","For a third difference f, derive and solve the quartic W^2=(s^2+Delta)^2+4(f^2-d^2)s^2, using exact rational-point, rank, and saturation computations.","Impose fourth and fifth differences by exact square tests on the resulting rational points; enforce positivity, integrality, distinctness, and weakest-column support.","Vary d,e,f through new symbolic affine product fibers rather than repeating fixed Bremner specializations; reject anchors and all registry-covered ranges.","Use Solvi only for final deterministic bigint validation; request no divisor_completion."]}},{"round":35,"agentId":"mira","privatePlan":{"objective":"Construct a modular compatibility atlas that ranks fresh difference sets by simultaneous local square support before exact scanning.","checks":["For candidate row sets, compute exact square-residue intersections for d²+4n modulo selected prime powers, including zero-valuation cases.","Combine nonempty residue classes by CRT and reject row sets with empty local intersection; prioritize sets with strong weakest-row coverage.","Run only targeted family or boundary scans inside the best CRT fibers, using the exact m² test and distinctness checks.","Independently recheck any support-4 cell before sending it to Tess; request no divisor_completion job."]}},{"round":35,"agentId":"orum","privatePlan":{"objective":"Run stochastic, optimization-guided mutations of exact support-3 seed clouds, including the 2052 and 421 rows, then bigint-verify only Pareto-leading candidates.","checks":["Exclude registry-covered regions and pure common scalings.","Optimize weakest-column support, distinct-column count, and novelty before exact testing.","Verify every retained cell by bigint square root and certify all supports and distinctness.","Do not request divisor_completion; leave validation-only checking to Tess."]}},{"round":35,"agentId":"pip","privatePlan":{"objective":"Run an exact-cover/SAT search on merged verified-cell incidence data, prioritizing a strict 4×5 improvement and then 5×5.","checks":["Recompute every cell by exact bigint square testing and positive factor recovery.","Merge the 2052 mutation data with the fresh verified clouds, deduplicate columns, and discard support below 4.","Enumerate five-row common-column sets for 5-column bicliques and six-row sets for 4-column bicliques, scoring weakest columns first.","Send only surviving candidates to Tess for validation-only checking and record deterministic hashes plus negative certificates."]}},{"round":35,"agentId":"solvi","privatePlan":{"objective":"Adversarially falsify the new CRT and anchor mechanisms, then obtain exact saturation certificates before adapting the search.","checks":["Characterize the modular factor roots behind Mira’s H=27720 condition and determine whether the filter is genuinely restrictive or locally vacuous.","Recompute Pip’s bridge cells and Orum’s 2052 mutation with bigint arithmetic; certify maximal bicliques and weakest-column support after removing anchor-only columns.","Adaptively select only difference rows overlapping multiple exact support-3 motifs, avoiding all registry-covered regions.","Use Tess solely for deterministic validation; request no divisor_completion job and preserve all eight null-filled parameters for any family_scan."]}},{"round":35,"agentId":"tess","privatePlan":{"objective":"Construct a new rational-point family for a fifth row from common-divisor equations, using the 2052 and 27720 structures only as seeds.","checks":["Derive the genus-1 condition obtained by imposing three or four prescribed differences on N=x(x+d), then compute rational points outside the registry’s denominator-10000 regime.","Convert every rational point to primitive integer differences and columns, enforcing distinctness and exact bigint square tests.","Score weakest-column support and reject all numerical near-squares; request no divisor_completion unless an independently constructed candidate has one missing pair."]}},{"round":36,"agentId":"mira","privatePlan":{"objective":"Run an exact-cover/SAT search on the merged verified-cell incidence hypergraph, prioritizing a strict 4×5 improvement before testing 5×5.","checks":["Canonicalize all verified (N,d) cells and reject duplicate N or difference rows.","Enumerate row subsets with column-support floors 4 and 5 using deterministic bigint square tests.","Verify every retained biclique independently via d²+4N=m² and distinctness checks.","Record maximal support vectors and a reproducible negative certificate for exhausted branches."]}},{"round":36,"agentId":"orum","privatePlan":{"objective":"Adversarially test the claimed fresh geometries while building exact saturation certificates.","checks":["Regenerate all factor pairs with deterministic bigint arithmetic and certify exhaustion for each selected anchor.","Classify H=27720 completions by gcd with H to falsify disguised scaling.","Use adaptive family scans only on pools whose weakest exact support improves, rejecting singleton-anchor structure.","Audit distinctness of every N and d; accept only exact Pareto improvements, never numerical square proximity."]}},{"round":36,"agentId":"pip","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery from the fresh balanced incidence seeds, then perform exact bigint verification of Pareto-improving bicliques.","checks":["Generate mutations biased toward raising the weakest column and row supports, while rejecting anchor-only extensions.","Avoid all registry-covered ranges and avoid divisor_completion requests.","Verify every retained cell by exact square testing and distinctness checks.","Report only exact 4×5, 5×5, or stronger incidence improvements; discard numerical near-squares."]}},{"round":36,"agentId":"solvi","privatePlan":{"objective":"Develop a constructive rational-point family from the fresh support-5 seed 30400, using coordinate normalization t_i=2x_i+d_i so that t_i^2-t_j^2=d_i^2-d_j^2 and common N is automatic.","checks":["Use the 30400 five-row configuration as a rational base point and derive the genus-one curve obtained by imposing three or four selected row equations.","Test whether the base point generates non-torsion rational points under the elliptic group law; reject torsion-only branches.","Scale rational points to integral t_i and N, then perform deterministic bigint checks for all five differences and distinct columns.","Compare the resulting support profile against (4,5,5,5,5), while avoiding the registry-covered small-difference and fixed-fiber regions."]}},{"round":36,"agentId":"tess","privatePlan":{"objective":"Construct a modular compatibility sieve for the fresh Pip bridge rows and Mira’s H=27720 pools, without numerical approximation or divisor completion.","checks":["For each proposed row subset S and primes p in {3,5,7,11,13,17,19,23,29,31}, compute exactly A_p(S)={r mod p: d²+4r is a quadratic residue for every d in S}.","Reject any S with empty A_p(S); rank survivors by simultaneous residue density and pairwise row co-occurrence.","Lift surviving classes through mod 8 and selected prime powers, then intersect by CRT to identify locally compatible N classes.","For promising classes, enumerate synchronized square-root sign patterns modulo p and retain only those admitting every pair relation (m−n)(m+n)=d²−e².","Use exact bigint square tests only after modular pruning; record falsifications as residue obstructions, never as numerical near-misses."]}},{"round":37,"agentId":"mira","privatePlan":{"objective":"Run an exact-cover/SAT search on the merged verified-cell incidence hypergraph, prioritizing a 4×5 biclique and then testing 5×5 extensions.","checks":["Encode only bigint-verified (N,d) cells from the cross-pool and bridge records.","Require five distinct N-columns and four, then five, distinct common difference rows.","Recompute every selected cell via d²+4N=m² and report weakest-first support vectors.","Treat Orum’s saturation cloud as a pruning aid only after checking its generation conditions; avoid all registry-covered regions.","Send any surviving candidate to Pip for validation; request no divisor_completion job."]}},{"round":37,"agentId":"orum","privatePlan":{"objective":"Adversarially saturate the union of the reported exact row clouds, then adaptively select only genuinely new rows or families that can improve the weakest-first support profile.","checks":["Construct the full exact bigint incidence table for every returned column and certify all masks, not just displayed candidates.","Reject any claimed improvement unless five distinct columns share four exact differences, or five differences share five exact columns.","Use failed intersections to choose the next experiment; do not repeat registry-covered regions or unchanged fixed pools.","Require parity, positivity, distinctness, and exact-square checks for every retained cell; use no numerical closeness.","Use family scans only this round; route validation-only checking through Pip and request no divisor_completion job."]}},{"round":37,"agentId":"pip","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery around Mira’s four-row anchor, Pip’s mixed bridge supports, and Orum’s finite union cloud; then exact-verify only Pareto-improving bicliques.","checks":["Generate fresh row mutations and score incidence hypergraphs lexicographically by weakest column support, then number of columns at support at least four, then total support.","Require five distinct candidate columns before accepting a 4×5 frontier; test any common fifth row only after that filter.","Use exact bigint tests d²+4N=m² and independently reconstruct positive factor pairs.","Avoid all registry-covered regions and unchanged prior row pools.","Request at most one divisor_completion only if a non-divisor primary search first produces a serious candidate; otherwise use no divisor job."]}},{"round":37,"agentId":"solvi","privatePlan":{"objective":"Construct an exact genus-one family from Mira’s four-row seed, seeking five integral N-columns and then a common fifth difference.","checks":["For d=(27720,55440,110880,360360), set y_i^2=4N+d_i^2. Parameterize y_2^2-y_1^2=d_2^2-d_1^2 using r=y_2-y_1 and y_2+y_1=(d_2^2-d_1^2)/r; impose the y_3 and y_4 square conditions to obtain an exact quartic/genus-one model.","Use the verified point N=1260653625 with roots (76230,90090,131670,367290) as a rational base point; derive its Jacobian/Weierstrass model and generate rational points by exact group operations.","Pull back rational points, retain only integral positive N and factor pairs x_i=(y_i-d_i)/2, and test for five distinct columns with exact bigint squares.","For any five columns found, impose y_{j,5}^2=4N_j+d_5^2 and search the resulting rational-point intersection for a common fifth difference.","If the seed curve is rank-deficient, vary the row differences symbolically through the same quartic construction rather than repeat covered fixed specializations; use no divisor_completion request."]}},{"round":37,"agentId":"tess","privatePlan":{"objective":"Use the h=27720 multiplier geometry for a modular/CRT search that prunes or forces compatible added rows without divisor completion.","checks":["Factor h=2^3·3^2·5·7·11 and tabulate local states for d=hr with core multipliers {1,2,4,13}.","For candidate multipliers {3,5,6,7,9,10,11,16,17}, intersect exact 2-adic and odd-prime residue conditions for d^2+4N to be locally square.","Lift only row tuples having nonempty compatible N residue states at every modulus and at least five CRT-compatible column states.","Pass surviving, untested row clouds to exact bigint support scoring; retain only strict 4×5 or 5×5 Pareto improvements."]}},{"round":38,"agentId":"mira","privatePlan":{"objective":"Run an exact-cover/SAT search on the merged, bigint-verified incidence hypergraph, using the 30400 support-5 anchor and Mira’s four-row columns without requesting divisor completion.","checks":["Recompute every listed cell exactly and discard any unverified or duplicated incidence.","Search first for five distinct columns and five distinct rows with all 25 incidences, then for a strict 4×5 biclique.","Apply weakest-column support ordering and enforce distinct positive N and distinct differences.","Require an exact square root for every selected cell before reporting any certificate."]}},{"round":38,"agentId":"orum","privatePlan":{"objective":"Adversarially reconstruct the claimed incidence frontier, eliminate duplicate scans, and adapt the next experiment to the sparsest missing edges.","checks":["Recompute every cited cell with exact square tests and require distinct columns and rows.","Build the merged incidence matrix and search exhaustively for 5×5, then 4×5 bicliques; record a saturation certificate for the tested row pool.","Reject Pip’s duplicated mutation scan and select any new family scan only from uncovered row combinations with the best balanced-support forecast.","Use divisor_completion at most once, and only to verify a candidate first generated by that independent primary search."]}},{"round":38,"agentId":"pip","privatePlan":{"objective":"Run stochastic, optimization-guided family-seed discovery around the Mira multiplier layers and mixed 411 bridge geometry, then exact bigint-verify only Pareto-improving incidence structures.","checks":["Exclude registry-covered and previously scanned row pools.","Optimize for balanced support and candidate 4×5/5×5 bicliques rather than isolated support-4 columns.","Apply exact tests d²+4N=m² and positive factor reconstruction to every retained candidate.","Use family scans only; route validation-only checking to Orum and request no divisor completion."]}},{"round":38,"agentId":"solvi","privatePlan":{"objective":"Construct a genuinely new rational-point family by treating the square-additive matrix symmetrically. For distinct rational row parameters r1,r2,r3, use the genus-one curve y_i^2=4n+r_i^2 and generate columns from its Mordell–Weil points, then impose two further rows exactly.","checks":["Use A=r2^2-r1^2 and B=r3^2-r1^2; map the intersection to the quartic v^2=t^4+(4B-2A)t^2+A^2 via y1=(t^2-A)/(2t), y2=(t^2+A)/(2t), y3=v/(2t).","Compute rational torsion and rank, then inspect exact multiples of a non-torsion point for five distinct positive n-values; exclude Bremner specializations and all registry-covered multiplier layers.","Solve the additional conditions z_j^2=4n_j+r4^2 and w_j^2=4n_j+r5^2 by exact elimination or a Mordell–Weil sieve, seeking a common pair (r4,r5) across five columns.","Clear denominators homogeneously: d_i=Lr_i, N_j=L^2n_j, m_{ij}=Ly_{ij}; retain only positive distinct integers.","Run deterministic bigint checks on every 25 cells and sort column supports weakest-first. Any claimed progress must be an exact 4×5, 5×4, or 5×5 certificate, never numerical square proximity."]}},{"round":38,"agentId":"tess","privatePlan":{"objective":"Design a modular local sieve that forces compatible additional rows around the verified support-4 anchors without repeating covered regions.","checks":["For each anchor row set, derive residue classes from d^2+4N being a square modulo several fresh prime powers, then intersect classes by CRT.","Use N=x(x+d) to impose simultaneous congruences for candidate columns and reject row sets whose local supports cannot reach five columns.","Hensel-lift surviving square residues and perform exact bigint verification only after the congruence sieve; rank resulting supports weakest-first.","Explicitly test whether a forced fifth row yields a 4×5 or 5×5 incidence pattern."]}},{"round":39,"agentId":"mira","privatePlan":{"objective":"Construct an exact incidence hypergraph from all verified seed columns and run support-sorted exact-cover/SAT branching for 5×5, then 6×4 and 4×5 configurations.","checks":["Recompute every cell using bigint perfect-square tests and factor parity.","Require distinct differences and distinct N values in every candidate biclique.","Exclude the malformed 6059649024 grouping unless its missing incidence is independently verified.","Retain complete masks and reject truncated or merely near-square outputs.","Do not request divisor completion; use only verified-cell incidence data."]}},{"round":39,"agentId":"orum","privatePlan":{"objective":"Adversarially test the fresh mutation and multiplier masks by demanding complete incidence streams and exact saturation certificates before expanding the search.","checks":["Recheck every claimed cell and distinctness condition with deterministic bigint square tests; reject any shared-core claim containing a missing cell.","For each complete stream, compute C(R) for every relevant row subset and certify whether any 5-row set has five columns, any 6-row set has four columns, or any 4-row set has five columns.","Choose the next family_scan adaptively from rows maximizing intersections with existing support-4 masks, avoiding all completed row sets and avoiding divisor_completion.","Record complete-stream hashes and weakest-column support vectors so truncated output cannot be mistaken for a negative result."]}},{"round":39,"agentId":"pip","privatePlan":{"objective":"Use stochastic, optimization-guided mutations of the fresh Mira-layer and mutation-bridge seeds, followed by exact bigint verification.","checks":["Run family-scan mutations only; request no divisor completion.","Retain complete deduplicated incidence masks and rank by weakest-column support.","Branch on balanced 4×5, 6×4, and 5×5 structures before isolated anchors.","Require deterministic perfect-square tests, distinct rows, and distinct N values; reject numerical closeness and covered regions."]}},{"round":39,"agentId":"solvi","privatePlan":{"objective":"Construct an exact rational-point family from the strongest support-4 seeds, then seek five integral columns and a fifth common difference.","checks":["Normalize a selected row quadruple by d_i=g u_i and m_i=g v_i, giving the intersection v_i²-u_i²=4n with N=g²n; derive its exact genus-one model, or a rational parametrization in singular cases.","Use the verified columns from Mira and Pip as rational points; apply exact chord-tangent or elliptic-group operations and clear denominators, retaining only integral N and m.","Filter generated columns for a fifth difference and require distinct rows, distinct N, and exact bigint square tests; score the resulting support vector.","Request no divisor-completion job unless this algebraic construction first produces a concrete candidate requiring independent verification."]}},{"round":39,"agentId":"tess","privatePlan":{"objective":"Use modular, local congruence design around the exact support-4 seeds to prune candidate rows and force compatible extensions, without divisor completion.","checks":["For each seed column, derive residue obstructions for d^2+4N to be square modulo selected small prime powers.","Intersect locally admissible difference residues before any bigint square test; prioritize rows compatible with at least two seed columns.","Construct short congruence-designed row sets around the 30/51/117 and 27720-core mutations, seeking balanced support floors rather than isolated columns.","Reject any proposed improvement unless deterministic exact verification produces a strict Pareto gain."]}},{"round":40,"agentId":"mira","privatePlan":{"objective":"Run exact-cover/SAT extraction over the newly verified cross-layer masks, targeting a strict 4×5 or 6×4 certificate before 5×5.","checks":["Merge complete masks only; reject truncated-stream inferences.","Enforce distinct rows and columns and sort by weakest column support.","Verify every selected cell with bigint square tests and retain only Pareto improvements.","Have Tess perform the final validation-only check; do not request divisor completion."]}},{"round":40,"agentId":"orum","privatePlan":{"objective":"Adversarially test whether the reported seed structures can saturate to a strict 4×5, 6×4, or 5×5 certificate without repeating registry-covered computation.","checks":["Recompute every listed cell with bigint arithmetic and enforce distinctness of both N and d; reject all numerical near-square evidence.","Merge complete returned masks, enumerate every row subset S, and record C(S), weakest-first support vectors, and explicit upper bounds for 5×5, 4×5, and 6×4 within the searched stream.","Canonicalize all proposed palettes and compare them with the search registry and prior job hashes before selecting an experiment.","If an untested branch remains viable, issue only a family_scan (never divisor_completion) with all eight parameters populated: d1=null, d2=null, differences='30 51 117 204 225 246', limit=5000, maxChecks=2000000, startDifference=null, endDifference=null, stride=null; otherwise stop and demand a saturation certificate.","Send only the resulting exact shortlist to Tess for validation; treat any negative result as scoped to its certified search region."]}},{"round":40,"agentId":"pip","privatePlan":{"objective":"Run stochastic, optimization-guided mutations around the fresh balanced seed palettes, targeting Pareto improvements in weakest column and row support, then exact bigint verification.","checks":["Exclude every registry-covered parameter regime and all divisor-completion requests.","Score only exact cells using d²+4N perfect-square tests; reject numerical near-squares.","Retain candidates only when distinctness and support vectors improve or preserve the best frontier.","Have Tess perform validation-only deterministic bigint checks on survivors."]}},{"round":40,"agentId":"solvi","privatePlan":{"objective":"Construct normalized rational families for simultaneous cells, using conic parametrizations and elliptic-curve fibers, while avoiding fixed Bremner specializations.","checks":["Normalize by g=gcd(d_i), writing d_i=g r_i and N=g^2 n; derive exact rational parametrizations of X_i^2-X_j^2=r_i^2-r_j^2 via (X_i+X_j)(X_i-X_j)=r_i^2-r_j^2.","Impose additional square conditions on the parametrized n and identify genus-one fibers; compute rational points and test whether they yield new row shapes rather than degeneracies.","Enforce integrality, parity, distinct positive differences, distinct N-values, and exact identities N=x_i(x_i+d_i) using bigint arithmetic.","Discard components reproducing bi-maximal Bremner specializations or covered registry regimes; rank survivors by weakest row and column support.","Submit only algebraically generated candidates to Tess for validation; request no divisor_completion job."]}},{"round":40,"agentId":"tess","privatePlan":{"objective":"Use modular local congruence filters to force or prune compatible rows around the strongest exact seed palettes, without repeating covered global scans.","checks":["For selected row sets R, compute exact local masks M_q(n;R) using d²+4n as a quadratic residue modulo several new primes q.","Intersect CRT residue classes for columns that simultaneously support prescribed row subsets; discard classes whose support vector cannot reach (4,5) or better.","Parameterize surviving columns by N=x(x+d0), impose the remaining row conditions modulo each q, and lift only CRT-compatible x classes.","Bigint-verify every lifted candidate and report only exact support; record modular obstructions as local evidence, never as a global impossibility result.","Request no divisor_completion job unless an independently generated candidate already has a near-complete exact mask."]}},{"round":41,"agentId":"mira","privatePlan":{"objective":"Run stochastic, optimization-guided mutation of fresh difference palettes, then apply exact bigint verification to retain only Pareto-improving support structures.","checks":["Exclude registry-covered and previously tested palettes.","Optimize lexicographically for weakest column support, then weakest row support, targeting 4×5, 6×4, and 5×5 structures.","Verify every survivor with exact tests d²+4N=m², distinctness checks, and complete support matrices.","Use no divisor-completion job unless a candidate is first produced by the stochastic primary search."]}},{"round":41,"agentId":"orum","privatePlan":{"objective":"Construct a new rational-point family from the equations N=t(t+d), centered on the exact 4×3 core, rather than performing another stochastic palette scan.","checks":["Derive simultaneous quadratic constraints for a chosen row pair and identify their rational parametrization or elliptic-curve model.","Impose integrality and positivity conditions symbolically before generating candidates.","Search only genuinely new parameter branches, retaining exact support masks and requiring a saturation certificate for every bounded subfamily.","Have Mira perform the sole exact validation after the family construction; do not request divisor_completion."]}},{"round":41,"agentId":"pip","privatePlan":{"objective":"Adversarially decide whether the fresh 411/438 branches saturate or yield a strict Pareto improvement, using exact support masks only.","checks":["Recompute every incidence mask from factor pairs and reconcile all row/column support vectors before accepting any frontier claim.","Use adaptive family scans on new difference sets selected from the weakest columns; never repeat a covered palette or treat truncation as a negative certificate.","Require an explicit complete checked domain for any saturation claim, and compare candidates by weakest-first exact support.","Have Mira perform the sole final validation; request no divisor_completion. Reject every near-square or numerically inferred cell."]}},{"round":41,"agentId":"solvi","privatePlan":{"objective":"Design a modular local sieve that prunes row palettes before exact validation and seeks candidates with minimum row and column support at least 3.","checks":["For the 411/438 and high-multiplier palettes, compute quadratic-residue masks for d²+4N modulo several small primes and combine them by CRT.","Discard residue classes that cannot permit at least five compatible differences for one N; retain only classes with a possible five-row support.","Use the same masks to branch-and-bound extensions of the existing 4×3 and 411/438 seeds, then send survivors for exact bigint checking by Mira.","Compare exact support against Bremner’s 4×4 certificate; treat modular emptiness only as a sieve result, not an impossibility proof."]}},{"round":41,"agentId":"tess","privatePlan":{"objective":"Run an exact-cover/SAT search on recomputed verified-cell incidence, targeting a strict 4×5 or 6×4 Pareto improvement.","checks":["Recompute every row-column edge from N=t(t+d), discarding stale support metadata.","Merge the 411/438 and 27720 incidence graphs only after exact deduplication.","Search subsets with five columns and at least four rows, then six rows and four columns; score weakest supports first.","Do not request divisor_completion or revisit registry-covered parameter regions."]}},{"round":42,"agentId":"mira","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery in fresh multiplier and gcd-diversified layers, then exact-bigint verify only support-balanced candidates.","checks":["Generate mutations outside all registry-covered and prior saturated palettes.","Optimize weakest row and column supports, retaining candidates that can reach a Pareto improvement.","Use exact square tests for every retained incidence and reject all numerical near-squares.","Prioritize five-column support and balanced 4×5 or 5×4 structures before larger isolated supports."]}},{"round":42,"agentId":"orum","privatePlan":{"objective":"Develop a genuinely new constructive rational/elliptic family for simultaneous factor differences, avoiding the covered fixed specializations and broad denominator search.","checks":["Rewrite simultaneous cells as N=t(t+d) and derive a low-parameter intersection of the resulting quadrics; identify whether the residual curve is genus 1 rather than another finite palette.","Compute a Weierstrass model for the first promising three- or four-row configuration, determine its rational torsion and generators, and test exact height-bounded multiples for a fifth compatible row.","Reject any candidate based only on numerical square proximity; send only exact bigint incidence data to Pip for validation, without requesting divisor_completion."]}},{"round":42,"agentId":"pip","privatePlan":{"objective":"Adversarially falsify the fresh 372/690 pivot branches, certify scoped saturation exactly, and adapt only from genuine Pareto improvements.","checks":["Recompute exact support masks and weakest-first column profiles independently for the completed seven-row baseline.","Run complete exact family scans for the 372 and 690 mutations; reject any apparent gain lacking a certified support mask and completion record.","Require any negative conclusion to be explicitly scoped to the enumerated family and check budget.","Treat Mira’s high-layer palettes as exploratory only; demand complete enumeration before using them as saturation evidence.","If a novel candidate is first produced by a non-divisor primary scan, use at most one divisor_completion job to verify its two defining differences; otherwise request none."]}},{"round":42,"agentId":"solvi","privatePlan":{"objective":"Use modular and local congruence sieves to construct fresh compatible row palettes, avoiding broad repeats and all divisor_completion requests.","checks":["For candidate row sets S, compute exact intersections R_S(p^e)={t(t+d) mod p^e:d in S} over small prime powers; discard palettes with an empty local intersection.","Use CRT lifts of surviving N residue classes to force compatibility among rows before any family_scan enumeration.","Seed fresh palettes around 372 and 690 but reject every previously scanned difference set and registry-covered regime.","Run only family_scan jobs with d1:null, d2:null, differences specified, limit:5000, maxChecks bounded, and startDifference/endDifference/stride:null.","Rank returned masks by weakest row and column support, then send only survivors to Pip for deterministic validation."]}},{"round":42,"agentId":"tess","privatePlan":{"objective":"Build an incidence-hypergraph exact-cover search from all verified cells in the 372/690 pivot branches and prior certified seeds, targeting five difference vertices and five N vertices with complete incidence.","checks":["Construct row and column support masks using only exact verified cells.","Enumerate 5-row candidate covers with branch-and-bound on weakest column support; retain only (5,5,5,5,5) or strict Pareto improvements.","Run an adversarial SAT-style obstruction check for the best (4,4,4,3,3) components before expanding to fresh verified palettes.","Deduplicate by normalized incidence signatures and record every falsification with exact missing cells."]}},{"round":43,"agentId":"mira","privatePlan":{"objective":"Run stochastic, optimization-guided family_scan seed discovery on fresh high-layer projective mutations and large-difference bridge mutations, then perform exact bigint verification through orum only.","checks":["Use family_scan only; no divisor_completion. Encode all eight parameter keys when submitting jobs, with null for irrelevant fields and a space-separated differences string.","Seed hash-random mutations beyond covered multiplier layers and around the 225/690 bridge, retaining candidates by lexicographically sorted weakest row/column support.","Reject registry-covered regions and retain only exact support-mask improvements or structurally diverse seeds.","Verify every finalist with bigint perfect-square tests, distinct N and d conditions, and explicit support masks; submit validation-only finalists to orum."]}},{"round":43,"agentId":"pip","privatePlan":{"objective":"Adversarially test the fresh large-difference and high-layer regimes while seeking an exact Pareto improvement, using only saturation certificates and adaptive scans.","checks":["Recompute and archive exact support masks and maxima for the 372/438 and 411/690 completed branches.","Test whether the bridge differences can share at least five columns at support four; reject isolated high-difference hits.","Audit every proposed high-layer palette against the registry and remove any covered multiplier overlap, especially 83.","Design fresh family_scan and boundary_scan jobs from surviving co-occurrence rows, with complete parameter objects and exact bigint checks only.","Send any genuine candidate exclusively to orum for validation; request no divisor_completion job."]}},{"round":44,"agentId":"mira","privatePlan":{"objective":"Run stochastic, optimization-guided family_scan seed discovery in uncovered high-difference and prime-layer regimes, then submit only Pareto-promising seeds for exact bigint verification.","checks":["Use family_scan only; do not request divisor_completion.","Reject registry-covered regions and any numerical near-square heuristic.","Score exact row and column supports lexicographically, prioritizing balanced 4×5 or 5×4 structures.","Verify every survivor with exact bigint square tests, distinct positive N values, and distinct differences.","Record incomplete scans as scoped results rather than impossibility claims."]}},{"round":44,"agentId":"orum","privatePlan":{"objective":"Construct rational parametrizations of multi-difference columns and seek a genus-one family whose rational points produce a fifth shared difference.","checks":["Use (u_p-u_q)(u_p+u_q)=p²−q² with u_d=2x+d to enumerate exact simultaneous cells and expose conic transitions.","For promising 3-row scaffolds, derive the remaining incidence condition as a quartic/genus-one curve and enumerate rational points by exact arithmetic.","Reject isolated trivial anchors and score only exact Pareto support vectors.","Validate every surviving column with deterministic bigint squares; request no divisor_completion job."]}},{"round":44,"agentId":"pip","privatePlan":{"objective":"Adversarially test whether the 51099/61599 boundary promotions or fresh >149 layers yield a genuine Pareto improvement, while producing scoped saturation certificates.","checks":["Recompute every claimed cell via exact bigint square tests and construct binary incidence masks with weakest-first row and column supports.","For each finite experiment, distinguish complete exhaustion from scan prefixes and certify only the searched parameter domain.","Test boundary palettes jointly and separately, recording whether any 4×5, 5×4, or stronger mask survives after requiring distinct N and differences.","Adapt experiments only from columns sharing at least four nontrivial exact rows; avoid registry-covered N, denominator, multiplier, and bounded-difference regions.","Keep Solvi as the sole final verifier and request no divisor_completion job."]}},{"round":44,"agentId":"solvi","privatePlan":{"objective":"Build a modular local sieve that forces compatible non-boundary rows against exact four-row seeds, while pruning covered and structurally degenerate regimes.","checks":["For each prospective difference set and small prime p, compute the exact residue set A_p={n mod p: d^2+4n is a quadratic residue mod p for every d}; discard empty intersections and rank surviving CRT classes by density.","Lift surviving N-residue classes p-adically, then enumerate only compatible parameter residues in the primary family search; require each proposed row to retain at least five possible columns before exact bigint testing.","Canonicalize row-versus-column support and reject d=N−1 boundary rows as candidates for a shared row.","Request no divisor_completion unless this modular primary search first produces a new exact candidate with at least four shared rows; then use at most one completion job solely for verification."]}},{"round":44,"agentId":"tess","privatePlan":{"objective":"Construct an incidence hypergraph from every exact verified cell in the current frontier and search for a strict 4×5 or 5×4 improvement via exact-cover/SAT constraints.","checks":["Use only deterministic verified (N,d) cells; deduplicate rows and columns.","Encode five distinct N-columns and five distinct d-rows with all selected incidences present.","Score feasible partial bicliques by weakest-first row and column supports.","Explicitly test whether the 51099/61599 promotions participate in balanced components, rather than counting isolated anchors.","Do not request divisor_completion or family_scan jobs this round."]}},{"round":45,"agentId":"mira","privatePlan":{"objective":"Run stochastic, optimization-guided family-seed discovery only in uncovered cross-core and high-layer regimes, then pass Pareto-promising seeds to Tess for exact bigint verification.","checks":["Use family_scan only, with d1=null, d2=null, startDifference=null, endDifference=null, stride=null, explicit differences, limit, and maxChecks; request no divisor_completion.","Generate and perturb pooled difference palettes while ranking by lexicographically sorted weakest-first row and column supports, prioritizing balanced 4×5 or 5×4 incidence.","Exclude the registry-covered domains and Pip’s finitely saturated boundary palette.","Verify every retained seed by exact tests d²+4N=m² and compute complete support masks; reject numerical near-squares.","Record only exact Pareto improvements over the current 4×3 scaffold, with Tess as the validation-only verifier."]}},{"round":45,"agentId":"orum","privatePlan":{"objective":"Construct a genuinely new rational/elliptic deformation of the square-additive condition X_i+Y_j=Z_{ij}^2, beginning from the exact 3-row, 4-column scaffold and targeting an additional shared row.","checks":["Normalize the scaffold algebraically, derive the conic intersections, and identify any nonsingular genus-one model.","Generate rational points by exact group-law arithmetic, avoiding fixed Bremner specializations and registry-covered regimes.","Require each X_i to be a rational square; clear denominators with parity checks so d_i and N_j are positive distinct integers.","Verify every cell with exact bigint arithmetic and retain only Pareto-improving support masks.","For each specialization, enumerate factor pairs to provide a scoped saturation certificate before escalation to Tess."]}},{"round":45,"agentId":"pip","privatePlan":{"objective":"Use adversarial falsification, scoped saturation certificates, and adaptive family experiments to seek a Pareto improvement without repeating covered regions.","checks":["Test Mira’s cross-core and high-layer palettes only in uncovered family-scan domains; reject any unsupported balance claim.","For each scan, record exact support masks, the maximum number of columns at support at least 4, and a finite saturation certificate.","Promote only fresh factor-pair rows that can change the weakest support; avoid redundant boundary rows.","Require Tess’s bigint validation for every surviving candidate; request no divisor_completion job."]}},{"round":45,"agentId":"solvi","privatePlan":{"objective":"Use modular, local congruence design to prune or force compatible difference rows before any broad search.","checks":["For small primes q, enumerate N mod q for which every d²+4N in a candidate palette is a quadratic residue; intersect these residue classes across q.","Use N=t(t+d) and m=2t+d to impose simultaneous local conditions on t for row pairs, prioritizing palettes with at least four locally compatible rows.","CRT-lift surviving residue classes, then apply exact bigint square tests and support-vector ranking; record only scoped congruence exclusions, never numerical near-square evidence."]}},{"round":45,"agentId":"tess","privatePlan":{"objective":"Construct an exact-cover/SAT incidence search using only verified cells from the 4×3 scaffold and Pip’s exact boundary columns, seeking a Pareto improvement in the weakest support vector.","checks":["Canonicalize each verified cell as (number,difference) and reject duplicate numbers or differences.","Search for 5-number/5-difference bicliques with support vectors prioritized weakest-first.","Use Mira’s and Pip’s verified objects only as seeds; do not treat proposed family-scan outputs as cells.","If a complete candidate emerges, independently bigint-verify every cell before any divisor_completion request."]}},{"round":46,"agentId":"mira","privatePlan":{"objective":"Adversarially test the lifted palettes, certify pairwise saturation, and adaptively generate only rows that improve the weakest exact support.","checks":["Recompute every proposed factorization and every d^2+4N square with deterministic bigint arithmetic; reject all near-square evidence.","For each strongest returned difference pair, enumerate every signed divisor factorization of |d^2-e^2| with parity constraints and compare the complete completion set against family output.","Run one fresh lift-palette experiment using Pip’s algebraically generated rows, retaining only complete, nontruncated support matrices and Pareto improvements.","If a fifth column reaches support at least four, independently verify all cells and distinctness before any further completion search; otherwise derive the next row layer from new admissible divisors rather than extending a saturated branch.","Do not request divisor_completion unless an independently generated family-scan candidate justifies it; use at most one such request."]}},{"round":46,"agentId":"orum","privatePlan":{"objective":"Design modular and local congruence filters for Pip-style lifted rows, forcing compatible difference intersections before exact family scanning.","checks":["For each candidate parameter, intersect parity and quadratic-residue conditions modulo selected small primes.","Screen parameter residues for simultaneous compatibility with at least five target differences and reject locally impossible row sets.","Use gcd and valuation constraints to eliminate duplicate or forced-incompatible factor pairs.","Require complete exact support matrices, truncation status, and Pareto sorting before accepting any improvement."]}},{"round":46,"agentId":"pip","privatePlan":{"objective":"Construct an iterated rational-point family from the two-row factor parametrization, targeting a genuinely new balanced support palette rather than another truncated boundary scan.","checks":["Derive symbolic compatibility conditions for three or more rows and identify conic or elliptic components.","Enumerate only exact rational/integral specializations with positivity, parity, distinctness, and bigint square checks.","Rank complete support matrices by weakest column and row support, rejecting every non-Pareto outcome.","Verify that all parameter regimes are outside the registry’s covered searches and avoid divisor_completion requests."]}},{"round":46,"agentId":"solvi","privatePlan":{"objective":"Build an incidence-hypergraph/SAT exact-cover search using only verified cells from the current reservoirs, targeting five difference vertices and five number vertices with every required edge present.","checks":["Canonicalize each verified (N,d) cell and reject any unverified or duplicate N or d.","Encode selected N-vertices and d-vertices with all 25 incidence constraints; optimize lexicographically for column support before seeking a 5-by-5 submatrix.","Run adversarial checks for unsupported edges, duplicate differences, and accidental reliance on truncated family output.","Do not request divisor_completion; leave numerical validation exclusively to Mira."]}},{"round":46,"agentId":"tess","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery in uncovered high-layer and cross-core family palettes, then send only Pareto-promising seeds for exact bigint verification.","checks":["Exclude all registry-covered multiplier, denominator, direct-N, and bounded-difference regions.","Mutate exact Pip/Mira seed rows stochastically, scoring weakest-first row and column support.","Prioritize candidates improving (4,4,5,5,5) toward (4,5,5,5,5) or a verified 5-by-5 incidence.","Verify every retained cell with exact integer-square tests and distinctness checks; reject numerical closeness and truncated-support claims.","Request no divisor_completion; use Mira only for validation."]}},{"round":47,"agentId":"mira","privatePlan":{"objective":"Adversarially test collision-generated support while building saturation certificates and adaptively selecting only exact Pareto-improving rows.","checks":["Enumerate admissible factor pairs for the new pair using C=120902^2-59999^2=3^2·7·43·67·101·601; certify its complete common-completion pool without requesting divisor_completion.","For every surviving pool, compute exact row and column supports, reject any candidate below the weakest-support frontier, and route only survivors to pip.","Conditionally issue a family_scan with d1=null, d2=null, differences equal to the certified high-support rows, limit=5000, maxChecks=2000000, startDifference=null, endDifference=null, stride=null.","If saturation yields a viable boundary palette, issue a boundary_scan with d1=null, d2=null, differences=null, limit=null, maxChecks=1500000, startDifference=120903, endDifference=2000000, stride=59999; otherwise terminate that branch as falsified."]}},{"round":47,"agentId":"orum","privatePlan":{"objective":"Use modular, local congruence filters to prune collision-derived candidates and force compatible additional rows before exact support testing.","checks":["Derive valuation and residue obstructions for simultaneous equations m_i^2=d_i^2+4N across candidate row sets.","Build CRT-compatible residue classes for columns sharing at least four prescribed differences, rejecting classes whose maximum possible support is below five.","Test only surviving local classes with exact bigint cells and weakest-first support accounting; request no divisor completion."]}},{"round":47,"agentId":"pip","privatePlan":{"objective":"Construct a new rational-point family by converting three simultaneous factor-difference conditions into an elliptic quartic, then seek five rational points whose scaled integer columns share additional rows.","checks":["For fixed d,e, set C=e^2-d^2, uv=C, m=(v-u)/2, and N=(m^2-d^2)/4; impose a third difference f through W^2=u^4+[4(f^2-d^2)-2C]u^2+C^2.","Search elliptic-curve specializations with a non-torsion rational point and test multiples for five distinct positive columns, clearing denominators only by exact scaling.","Impose fourth and fifth rows on the resulting columns using exact bigint square tests and weakest-first support scoring.","Do not request a divisor_completion job unless this algebraic construction first yields a concrete candidate; then request at most one job solely for independent verification."]}},{"round":47,"agentId":"solvi","privatePlan":{"objective":"Run an exact SAT/exact-cover search on the incidence hypergraph formed solely from pip-certified cells, prioritizing the new collision rows and the saturated-pair rows.","checks":["Discard every reported row or candidate lacking pip’s deterministic bigint certification.","Encode selected differences and N-columns with exact cell-incidence constraints; search first for five columns sharing five rows, then Pareto profiles beginning with (4,5,5,5,5).","Cross-check every selected cell via d^2+4N=m^2 with parity and positive-factor conditions.","Record exact witnesses or maximal support profiles; do not treat numerical closeness as evidence and do not request divisor completion."]}},{"round":47,"agentId":"tess","privatePlan":{"objective":"Perform stochastic, optimization-guided mutation of collision parameters and divisor-pair seeds, targeting Pareto improvements in weakest-column exact support before bigint verification.","checks":["Sample new (c,d,t,y,z) seeds beyond covered difference and N regimes; enforce collision identities exactly.","Construct candidate difference palettes from exact factor rows, scoring sorted column supports and retaining only nondominated seeds.","Reject numerical near-squares and any candidate lacking at least four exact rows across five columns.","Submit only the strongest seeds to pip for deterministic bigint verification; request no divisor_completion jobs."]}},{"round":48,"agentId":"mira","privatePlan":{"objective":"Adversarially test the new CRT and collision palettes while retaining only exact saturation certificates and adaptive Pareto improvements toward (4,5,5,5,5) or (5,5,5,5,5).","checks":["Audit all claimed residue intersections by exact modular enumeration; reject unsupported class-count assertions.","Compute exact row and column supports for Orum’s shell and Solvi’s roots, using intersection upper bounds to discard unsaturable branches early.","Treat truncated, timed-out, and zero-check jobs as unknown; adapt the next experiment only from surviving exact support structure.","Do not request divisor_completion; use only exact bigint checks and the existing certified completion information."]}},{"round":48,"agentId":"orum","privatePlan":{"objective":"Use a fresh c=2 odd-z shell and modular intersections to force compatible rows, then exact-test only the surviving CRT classes without repeating registry-covered searches.","checks":["Generate z=75,77,79,81,83 with r=(z²−1)/2, N=3(r⁴−z⁴), and d±=2(r²±2z²); verify distinctness and exclude all previously tested rows.","Recompute quadratic-residue admissibility independently modulo 32, 3, 5, 7, 11, 13, and 17; retain only CRT classes common to at least five rows.","Run exact bigint tests m²=d²+4N and report the complete weakest-first incidence matrix, rejecting numerical near-squares.","Cross-check Solvi’s P=25050025 roots and merge only rows whose local conditions survive independent validation.","Do not request divisor_completion unless a separate primary search first produces a genuine candidate; then use at most one completion job solely for verification."]}},{"round":48,"agentId":"pip","privatePlan":{"objective":"Construct a new elliptic or rational-point family of three-row common completions, then seek a fifth compatible row without repeating covered searches.","checks":["Derive an exact quartic/elliptic model from m_i^2=d_i^2+4N for three parametrized rows, retaining its birational maps and nonsingularity conditions.","Use known exact rectangles as rational base points; compute low-height rational multiples and impose distinct positive N and differences.","Test every generated incidence with bigint square identities, sorting weakest-first supports and rejecting numerical near-squares.","Avoid divisor_completion requests; submit only algebraically generated survivors to Orum for validation."]}},{"round":48,"agentId":"solvi","privatePlan":{"objective":"Build an incidence-hypergraph and exact-cover search exclusively from Orum-validated cells in the new z=63–73 palettes, Mira’s collision palettes, and the complete CRT-root palette.","checks":["Reject every cell unless bigint evaluation gives m^2=d^2+4N exactly with a valid positive factor pair.","For each saturated row pair, enumerate only verified common columns and construct the row-column incidence graph.","Solve for distinct 5-row/5-column bicliques, with lexicographic support optimization and explicit tests for (4,5,5,5,5).","Use SAT/exact-cover constraints forbidding every missing row-column incidence; report weakest-first supports and witnesses.","Do not request divisor_completion; route candidate validation exclusively through Orum and record nonexistence only within the enumerated verified universe."]}},{"round":48,"agentId":"tess","privatePlan":{"objective":"Use stochastic, optimization-guided sampling of unscanned CRT-root subsets and collision-layer seeds, then submit only Pareto-promising supports for exact bigint verification.","checks":["Sample row subsets outside registry-covered ranges and rank them by residue density, row diversity, and predicted balanced support.","Apply weakest-first Pareto filtering before any expensive exact test; reject seeds driven only by isolated or support-two columns.","Run exact checks using m^2=d^2+4N with bigint arithmetic and require distinct rows and columns.","Use Orum as the sole validation-only verifier; record exact support matrices and reject numerical near-square evidence.","Do not request divisor_completion; use only stochastic seed discovery followed by exact verification."]}},{"round":49,"agentId":"mira","privatePlan":{"objective":"Adversarially falsify or certify the fresh elliptic, divisor-lattice, and z-shell regimes using exact support and saturation criteria, without requesting divisor_completion.","checks":["Recompute every displayed factorization and square identity with deterministic bigint arithmetic; deduplicate rows and columns.","Require exhaustive or explicitly saturated family-scan output, rejecting result-limited runs as noncertificates.","Verify Orum’s mod-13 and mod-17 calculations and test the 221 divisibility filter against exact completions.","For every survivor, construct the full incidence matrix, sort column supports weakest-first, and retain only Pareto improvements toward (4,5,5,5,5) or (5,5,5,5,5).","Adapt the next experiment to uncovered rows and columns only after falsifying seed-only closures; send any final candidate to Solvi for validation."]}},{"round":49,"agentId":"orum","privatePlan":{"objective":"Use modular, local congruence design to prune compatible rows before any broad search.","checks":["Recompute every claimed factor product with deterministic bigint arithmetic and discard mismatched column assignments.","For the fresh ten-row shell, derive exact residue filters modulo 13, 17, and 221, then intersect them with parity and gcd constraints on N=t(t+d).","For Pip’s five-row palettes, compute row-pair local compatibility classes modulo small primes and retain only classes capable of five-column support.","Require an exact support table before sending any survivor to Solvi; request no divisor-completion job."]}},{"round":49,"agentId":"pip","privatePlan":{"objective":"Construct and test an exact rational-point family from the (204,225,411) genus-one intersection, targeting a balanced 4-by-5 incidence without divisor_completion requests.","checks":["Derive an exact Weierstrass/Jacobian model and verify smoothness and the maps from the displayed rational points.","Compute exact group relations and rational multiples; clear denominators primitively and impose positivity, parity, and duplicate-scaling filters.","For each resulting N, test all candidate differences by exact square identities and retain only distinct columns with support at least four.","Audit whether Orum’s 221 filter holds symbolically on the selected family, rejecting any claim based on numerical near-squares.","Send only surviving exact incidence certificates to Solvi for validation."]}},{"round":49,"agentId":"solvi","privatePlan":{"objective":"Run an exact-cover/SAT search on a bigint-verified incidence hypergraph, combining Pip’s genus-one palette with Orum’s audited shell while excluding Mira’s disputed cells.","checks":["Recompute every edge using d²+4N=m² and positive factor pairs; discard all arithmetic mismatches.","Audit Orum’s ten shell rows and use 221|N only as a necessary prefilter.","Encode distinct five-row/five-column incidence constraints and optimize first for (5,5,5,5,5), then Pareto targets (4,5,5,5,5).","Require deterministic support certificates for every selected cell; make no divisor_completion request unless the primary SAT search first produces a candidate."]}},{"round":49,"agentId":"tess","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery in uncovered elliptic, divisor-lattice, and shell boundary regimes, then submit only exact bigint survivors for validation.","checks":["Deduplicate against the registry and prior palettes; exclude covered Bremner specializations and z≤73 shell regimes.","Optimize lexicographically for sorted exact support vectors, prioritizing (4,5,5,5,5) and then (5,5,5,5,5).","For every candidate, verify d²+4N=m², parity, positivity, distinctness of N and d, and every claimed incidence with bigint arithmetic.","Recheck all modular filters and factor-pair identities independently before retaining a seed.","Use Solvi only for validation; do not request divisor_completion."]}},{"round":50,"agentId":"mira","privatePlan":{"objective":"Adversarially test the new elliptic and mixed-lattice palettes for a strict exact-support Pareto improvement, while seeking saturation certificates rather than isolated anchors.","checks":["Run only complete exact family scans on Pip’s two five-row palettes and the mixed six-row palette; reject every result-limited or check-limited output as inconclusive.","Require at least four columns of support five, or five columns of support four, before accepting any improvement; independently recompute every surviving cell with bigint square tests.","Compare the repaired boundary shell against the current frontier and record explicit maximum support profiles, including weakest-first column ordering.","Use Tess solely for final validation of survivors; do not request divisor_completion."]}},{"round":50,"agentId":"orum","privatePlan":{"objective":"Design modular and local congruence filters around the new anchors before any further exact family scan, seeking residue classes that force compatible multi-row columns.","checks":["Recompute every displayed factor pair and derive exact row signatures for 260820 and 647500.","For each candidate difference palette, enumerate N modulo small primes and retain only residue classes where every d satisfies d²+4N being a quadratic residue.","Intersect these conditions by CRT and measure local density; discard palettes with no compatible classes or only isolated signatures.","Use surviving classes to target collisions with at least five distinct columns and report exact support vectors; request no divisor_completion job."]}},{"round":50,"agentId":"pip","privatePlan":{"objective":"Construct rational-point families from the 260820 and 647500 factor-pair seeds, targeting multiple integral columns on one fixed five-row palette.","checks":["Derive the exact intersection-of-quadrics and a genus-one/Weierstrass model using U_i=2x_i+d_i.","Generate chord-tangent or Mordell-Weil multiples from each known point, retaining only positive integral N and x_i.","Test every survivor by exact bigint square identities, requiring support at least four and seeking 4×5 or 5×5.","Send only exact survivors to Tess; request no divisor_completion job."]}},{"round":50,"agentId":"solvi","privatePlan":{"objective":"Run an exact-cover search on the union of verified cells, targeting a strict 4x5 improvement before 5x5.","checks":["Recompute every listed cell with bigint arithmetic and reject any missing square or parity condition.","Construct the verified incidence hypergraph with distinct N and distinct differences enforced.","Use SAT or branch-and-bound exact cover with row and column degree constraints, branching on the rarest verified row.","Report canonical support vectors and a deterministic certificate, or an explicit infeasibility result for the searched verified-cell union."]}},{"round":50,"agentId":"tess","privatePlan":{"objective":"Use stochastic, optimization-guided mutations of exact factor-pair seeds from the new anchors, targeting a strict (4,5,5,5,5) improvement before attempting (5,5,5,5,5), while excluding registry-covered regions.","checks":["Generate reproducible random seed mutations from verified factor-pair tuples and retain Pareto-nondominated support profiles.","Reject duplicate or covered difference palettes and score only exact incidence support, never numerical square closeness.","Run deterministic bigint verification using integer square roots, parity, positivity, distinctness, and all required incidence cells.","Request no divisor_completion job unless a different primary method first produces a complete candidate."]}},{"round":51,"agentId":"mira","privatePlan":{"objective":"Construct elliptic and rational-point families from the exact support-four anchors, seeking multiple integral columns on one palette rather than isolated collisions.","checks":["Derive the exact pair-factor parametrization for variable difference pairs and impose third-row square conditions symbolically.","Form the genus-one intersection of quadrics for triples such as 411, 488, 513 through the N=260820 point; compute rational group-law multiples exactly.","Repeat with a 647500 triple and test whether fourth and fifth rows become integral on distinct columns.","Clear denominators only when the resulting differences and N remain integral; reject numerical near-squares.","Require the complete support matrix and a 4x5, 5x4, 5x5, or strict 6x4 improvement before reporting.","Request no divisor_completion job unless an independently derived exact candidate first appears."]}},{"round":51,"agentId":"orum","privatePlan":{"objective":"Run an exact incidence-hypergraph search over newly verified cells, targeting a certified 5x5 or 6x4 subgraph.","checks":["Represent each verified (N,d) cell as an edge and solve the balanced support problem by SAT or exact cover.","Require distinct N and d vertices and verify every selected edge with bigint square tests.","Reject numerical near-squares, duplicate known regions, and unsaturated partial components.","Emit a saturation certificate listing all selected cells and each column’s complete support."]}},{"round":51,"agentId":"pip","privatePlan":{"objective":"Design modular and local congruence sieves for replacement palettes around the 260820, 647500, and 70536 structures.","checks":["Compute exact 2-adic and prime-modulus residue filters for support at least four.","Intersect CRT classes and discard palettes with no locally admissible column classes.","Test surviving classes for compatible multi-column incidence using deterministic bigint square checks only.","Record obstruction moduli separately from merely check-limited search failures."]}},{"round":51,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided mutation of exact collision palettes to discover new balanced seeds, then submit survivors to deterministic bigint verification.","checks":["Exclude all registry-covered regions and previously scanned palettes; optimize exact-support incidence, not numerical square proximity.","Run family_scan only with params {d1:null,d2:null,differences:\"<generated space-separated palette>\",limit:5000,maxChecks:4000000,startDifference:null,endDifference:null,stride:null}.","Generate palettes by random row replacement, conic-parameter mutation, and modular-support filtering; retain only candidates with plausible 4x5, 5x4, or 6x4 incidence.","Send every survivor through exact bigint tests of d²+4N=m² and accept only a certified k=5 result or strict Pareto improvement."]}},{"round":51,"agentId":"tess","privatePlan":{"objective":"Adversarially falsify the new bridge and replacement palettes while designing exact, saturation-certified experiments toward a strict support improvement.","checks":["Use only exact bigint tests for every cell, with parity, positivity, distinctness, and support sorting enforced.","Deduplicate all outputs against known single-column anchors and reject palettes whose apparent progress is only one support-4 seed.","Require each bounded scan to report parameter coverage, CRT branches, and unresolved tails; never treat maxChecks exhaustion as a negative result.","Adapt the next family_scan or boundary_scan to the branch with the strongest independent multi-column support; do not request divisor_completion."]}},{"round":52,"agentId":"mira","privatePlan":{"objective":"Construct a rational-point/elliptic family from the 260820 six-difference factorization, then seek parameter values that retain four or more rows and extend to a balanced 5-by-5 configuration.","checks":["Derive the genus-one quartics for triples such as (57,107,201) and (57,107,216), including exact parity and positivity conditions.","Map each quartic to a Weierstrass model and enumerate rational multiples of the known integral point, retaining only exact integral N and m values.","Test whether the resulting parameter values simultaneously satisfy the extra rows 312 and 481, then cross-check compatibility with the 135,225,732 seed family.","Use deterministic bigint support counts only; reject candidates based on missing exact square identities, not numerical closeness."]}},{"round":52,"agentId":"orum","privatePlan":{"objective":"Run an exact incidence-hypergraph/SAT search over verified cells from the new palettes, targeting a 5x5 rectangle or the Pareto improvement (4,5,5,5,5).","checks":["Represent each exact cell (N,d) as an edge with independently recomputed m, parity, positivity, and distinctness checks.","Search row subsets and column subsets by exact-cover constraints; sort supports weakest-first and reject numerical near-squares.","Emit a saturation certificate listing every selected cell and all missing incidences, plus an independently reproducible bigint verification record.","If no improvement appears, retain only maximal support profiles and record the exhausted row/column universe without requesting divisor completion."]}},{"round":52,"agentId":"pip","privatePlan":{"objective":"Build a modular local sieve for the new mixed palettes, forcing compatible four-row residue classes before any exact search.","checks":["Enumerate N modulo compatible powers of 2 and primes 3, 5, 7, 11, and 13; retain only classes with at least four locally admissible rows.","Intersect residue classes across the 260820-derived and mixed elliptic palettes, rejecting classes that cannot support five distinct columns or a common fifth row.","Run exact bigint support scoring only on survivors; record weakest-first column supports and falsify any numerical near-square heuristic.","Reserve at most one divisor_completion verification, only if an independently produced candidate supplies two exact differences."]}},{"round":52,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided mutations of the exact seed palettes to discover novel balanced support patterns, then submit only Pareto-promising survivors for exact bigint verification.","checks":["Sample new difference and parameter seeds near the Mira elliptic anchors and Pip’s 260820 factor palette, rejecting registry-covered regions.","Score candidates by modularly predicted support and weakest-first Pareto support, rewarding novelty and balanced 4×5 or 5×5 potential.","Require exact square tests, parity, positivity, distinct N values, and distinct differences after generation; never use numerical closeness.","Use family-scan-style exploration only; submit survivors to Pip for validation and do not request divisor_completion."]}},{"round":52,"agentId":"tess","privatePlan":{"objective":"Adversarially test the mixed palettes while producing saturation-grade certificates and adapting toward the sparsest viable residue branches.","checks":["Recompute every seed factorization and square cell with bigint arithmetic, including positivity and parity.","Audit each family-scan enumeration domain and require an explicit exhaustive parameter-box or residue-class certificate.","Independently enumerate the CRT filters, add prime 11 and 13 constraints, and retain only exact support improvements.","Adapt subsequent scans to palettes maximizing shared admissible residue classes; reject any result based on numerical square proximity."]}},{"round":53,"agentId":"mira","privatePlan":{"objective":"Construct a rational-point parametrization linking the 260820 and 647500 anchor systems through their shared difference 411, then specialize to integral columns with support at least four.","checks":["Derive the shared-row elliptic intersection and its birational parameter map.","Generate independent rational points rather than rescanning covered bounded regions.","Test integrality and positivity symbolically before exact bigint square verification.","Reject every candidate lacking a balanced multi-column support improvement."]}},{"round":53,"agentId":"orum","privatePlan":{"objective":"Run a SAT/incidence-hypergraph search using only explicitly verified cells from the new Mira/Pip palettes, targeting a saturated 5-by-5 rectangle and then the Pareto fallback (4,5,5,5,5).","checks":["Create one incidence literal per exact bigint identity d^2+4N=m^2; reject every unverified or numerically close cell.","Enforce five distinct rows and five distinct columns with all 25 incidences, while excluding anchor-only solutions.","Sort every returned support vector weakest-first and emit the exact missing-cell certificate for failures.","Record the finite cell pool and blocked clauses so any negative result is reproducible; request no divisor_completion job unless an independent primary search first produces a candidate."]}},{"round":53,"agentId":"pip","privatePlan":{"objective":"Design a new modular sieve for the 647500-derived and 260820 bridge palettes, targeting columns with exact support at least four while avoiding all registry-covered regions.","checks":["For each difference and selected primes, compute the exact residue classes for which d²+4N is a quadratic residue; enumerate CRT combinations supporting at least four rows.","Add primes only after checking that the known seed residues survive, and record reproducible class counts and densities.","Run exact bigint square tests only on surviving classes; reject every numerical near-square.","Sort resulting columns by weakest support and test only genuine balanced Pareto improvements.","Have Orum perform validation-only checking; do not request divisor_completion."]}},{"round":53,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided mutations and crossovers of the new exact anchor palettes to discover distinct high-support columns, then submit only exact candidates for bigint validation.","checks":["Exclude registry-covered parameter and difference regions, while retaining the new 260820 and 647500 seed regimes.","Score candidate palettes by sorted exact-support vectors, prioritizing balanced Pareto improvements over isolated anchors.","Apply CRT filters only as necessary prefilters; verify every retained cell with exact bigint square roots and parity checks.","Require five distinct N values and five distinct differences before treating any result as a k=5 candidate.","Use Orum only for final validation; do not request divisor_completion."]}},{"round":54,"agentId":"mira","privatePlan":{"objective":"Construct and specialize the q=411 elliptic fiber product linking the 260820 and 647500 conic systems, seeking a genuinely multi-column rational-point family rather than isolated anchors.","checks":["Derive a birational elliptic model and record the map back to x, N, and every difference row.","Verify both known anchors symbolically and identify independent rational sections via chord-tangent operations.","Impose exact integrality, positivity, parity, and distinctness conditions before any bounded specialization.","Use deterministic bigint square tests to retain only balanced Pareto improvements, with no divisor_completion request."]}},{"round":54,"agentId":"orum","privatePlan":{"objective":"Construct an incidence hypergraph solely from exact verified cells and search it by SAT/exact cover for balanced K5,5 or strict Pareto improvements.","checks":["Recompute every cell with deterministic bigint square and parity tests; reject near-squares and duplicate columns.","Encode distinct rows and columns, then test K5,5, 5x4, 4x5, and stronger Pareto-support targets.","Require complete candidate-output and parameter-bound records before accepting any saturation claim.","Send only surviving exact finalists to Solvi for validation; request no divisor_completion job."]}},{"round":54,"agentId":"pip","privatePlan":{"objective":"Design a modular, local congruence sieve for a new d≡3 (mod 16) palette, forcing compatible four-row or five-row support before exact scanning.","checks":["Enumerate L_p support classes for p=3,5,7,11,13,17,31 and recompute all CRT intersections independently.","Reject residue classes lacking simultaneous local support for at least four selected differences; compare weakest-column support lexicographically.","Use only previously unscanned palettes and defer every survivor to deterministic bigint square tests; request no divisor_completion job."]}},{"round":54,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery in new difference regimes, then perform deterministic exact-bigint verification of the highest-support candidates.","checks":["Mutate conic-anchor and 2-adic difference seeds while penalizing duplicate or registry-covered palettes.","Optimize weakest-column exact-support score, retaining only Pareto improvements or balanced high-support candidates.","Verify every retained cell with d²+4N=m², enforce distinct N and distinct differences, and report no result from numerical closeness.","Request no divisor_completion job unless an independently generated seed first warrants exact completion verification."]}},{"round":54,"agentId":"tess","privatePlan":{"objective":"Adversarially test the q=411 and CRT proposals while constructing bounded saturation certificates and adapting the next palette to observed exact support.","checks":["Recompute every anchor cell with exact bigint square tests; reject duplicates, near-squares, and columns with support below four.","Run family_scan with parameters {d1:null,d2:null,differences:\"57 107 201 411 488 513\",limit:5000,maxChecks:4000000,startDifference:null,endDifference:null,stride:null}.","Run family_scan with parameters {d1:null,d2:null,differences:\"135 225 411 795 3525\",limit:5000,maxChecks:1200000,startDifference:null,endDifference:null,stride:null}.","Record complete bounded support histograms; if no balanced component appears, issue a bounded saturation certificate rather than a global nonexistence claim.","Adaptively add a new row only when at least five distinct columns retain support at least four; otherwise tighten local sieves and stop without divisor_completion."]}},{"round":55,"agentId":"mira","privatePlan":{"objective":"Construct an explicit parameterized q=411 rational-point family from the shared-t equations, targeting multiple integral columns with common rows.","checks":["Eliminate the t-coordinates for two-column configurations and identify the resulting conic, elliptic, or higher-genus model explicitly.","Use exact chord-tangent or elliptic transport only after verifying the birational maps and integrality conditions.","Generate distinct positive integer columns and apply deterministic exact square tests; submit only genuine support improvements to Tess.","Avoid divisor_completion and all registry-covered bounded regions."]}},{"round":55,"agentId":"orum","privatePlan":{"objective":"Run an exact-cover/SAT search on the verified-cell incidence hypergraph, using the q=411 and 1459 seed cells only as certified input.","checks":["Encode distinct columns and differences with exact cell witnesses.","Search specifically for support patterns (4,5,5,5,5) and (4,4,5,5,5).","Reject isolated one-column seeds and all unverified near-square evidence.","Use Tess solely for final deterministic validation."]}},{"round":55,"agentId":"pip","privatePlan":{"objective":"Strengthen the 1459 modular lane by intersecting local support constraints before any exact family scan.","checks":["Compute support-at-least-four residue classes modulo 13, 17, and 19 for all three palettes.","Intersect these with the existing moduli 3, 5, 7, and 11; discard empty classes and classes excluding the exact 260820 seed.","Retain only palettes with materially reduced CRT density, then perform exact bigint support scans and have Tess validate every survivor.","Report weakest-first column supports and reject isolated one-column anchors."]}},{"round":55,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery on unscanned difference palettes, then submit only exact bigint-verified support improvements.","checks":["Generate palettes by mutation/crossover around the 260820 and 647500 row signatures, penalizing repeated or registry-covered regions.","Optimize the weakest column supports, prioritizing balanced 4×5 and 5×4 candidates rather than isolated high-support seeds.","Apply modular feasibility filters before exact family scans; reject every candidate lacking exact square identities.","Request family_scan only, with Tess reserved for validation; do not request divisor_completion."]}},{"round":55,"agentId":"tess","privatePlan":{"objective":"Adversarially falsify the proposed q=411 and 1459 lanes, then design one adaptive exact scan only if a saturation certificate leaves an uncovered support regime.","checks":["Independently enumerate the claimed CRT classes and test whether support-four necessity is stated without hidden assumptions.","Construct exact divisor-coordinate saturation certificates for the displayed seed palettes, separating isolated columns from any repeatable family.","Prioritize the residue class or shared-h fibre with the largest surviving untested support potential; request no divisor_completion unless an unrelated primary method first produces a candidate."]}},{"round":56,"agentId":"mira","privatePlan":{"objective":"Design a new row palette by intersecting local quadratic-residue constraints modulo several small primes, centered on exact seed incidences but disjoint from covered palettes and difference regions.","checks":["Enumerate the allowed residue classes and verify the CRT density exactly before any bigint scan.","Require each proposed row to pass simultaneous local tests against the selected seed columns; treat this only as a necessary filter.","Run exact bigint verification and sort column supports weakest-first, accepting only distinct columns with the stated 4×5 or 5×5 support.","Use no divisor_completion unless this primary congruence method first produces a concrete candidate; then make at most one exact verification request."]}},{"round":56,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery over new difference palettes and progressions, followed only by exact family-scan bigint verification.","checks":["Optimize modular support density and predicted biclique potential while excluding all registry-covered regions.","Request no divisor_completion; use bounded-complete family scans with valid limits and maxChecks.","Verify every edge via exact square roots, parity, positive factor pairs, distinct columns, and weakest-first support vectors.","Reject anchors, near-squares, and finite-slice failures as non-certificates; prioritize [5,5,5,5,5] and then Pareto improvements such as [4,5,5,5,5]."]}},{"round":56,"agentId":"pip","privatePlan":{"objective":"Run an exact-cover search over the union of verified cells from the 1459 palette and CRT-flat family, targeting a 4x5 or 5x5 biclique without requesting divisor completion.","checks":["Canonicalize and deduplicate every verified (difference,N) edge.","Enumerate five-column exact covers, then four-row relaxations, using weakest-first support vectors.","Recheck every selected edge with bigint square roots, parity, positive factors, and distinct N values.","Record the finite tested domain and stop if no balanced cover appears."]}},{"round":56,"agentId":"solvi","privatePlan":{"objective":"Adversarially falsify the two new palettes, then use exact modular saturation to choose the next bounded experiment without dispatching divisor_completion.","checks":["Recompute every local residue table exhaustively, including Mira’s corrected mod-11 set, and certify the resulting CRT support graph.","For each palette, enumerate modular support patterns and record exact upper bounds for 5×5 and five-column 4-row bicliques before any large scan.","Treat 260820 only as a seed: test whether distinct additional columns exist, rejecting any one-column or near-square evidence.","Adaptively select the next row palette by maximizing modular extension capacity while remaining outside the registry-covered regions.","Require deterministic bigint square-root, parity, positivity, distinctness, and weakest-first support checks; route validation-only work to mira."]}},{"round":56,"agentId":"tess","privatePlan":{"objective":"Construct a rational-point family by extending the four-row square-difference equations on an intersection of quadrics, targeting a symbolic fifth row rather than an unbounded incidence scan.","checks":["Eliminate each N_j from m_{ij}^2-d_i^2=4N_j and identify genus-one or elliptic fibers for candidate fifth-row parameters.","Use exact rational arithmetic to derive identities, track denominators, and specialize only when all d_i and N_j are distinct positive integers.","Verify every specialized cell with exact bigint square roots and positive factor pairs, then record any finite fiber obstruction without making a global impossibility claim."]}},{"round":57,"agentId":"mira","privatePlan":{"objective":"Design a new modular row palette that prunes N classes while forcing shared local compatibility, without requesting divisor completion.","checks":["Enumerate CRT sign palettes modulo products of 8, 9, 25, 7, 11 and one additional small prime, retaining five distinct differences outside covered bounds with identical d² residues.","Compute the exact intersection of admissible N residue classes and rank palettes by sieve density and predicted shared support.","Use a bounded family_scan only on the best new palette; independently verify every returned cell by bigint square, parity, positivity, and product checks.","Reject palettes whose modular intersection is merely nonempty but does not improve the weakest-column support target."]}},{"round":57,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery to select new five-row CRT/sign palettes and seed N-columns, maximizing the Pareto score of weakest-column exact support; then perform deterministic bigint verification only on the best seeds.","checks":["Exclude every registry-covered difference region and previously tested fixed specialization.","Score seeds by exact incidence counts and support vector, prioritizing 5×5, then 4×5 and 4×4; never score numerical closeness to squares.","Require an auditable finite sampling domain, random-seed record, and saturation/stop certificate for each optimization batch.","Recompute every surviving cell with bigint square root, parity, positivity, and ab=N checks, then independently verify distinctness and the complete support matrix.","Do not request divisor_completion; use exact validation only after stochastic seed selection."]}},{"round":57,"agentId":"pip","privatePlan":{"objective":"Run an exact-cover incidence search on verified cells from the union of the 312 and 1459 palettes, requiring selected row sets to contain both differences and targeting 5×5 before 6×4 or 4×5.","checks":["Recompute every edge with exact bigint square-root, parity, positivity, and reconstructed-product tests.","Build the complete support matrix, enforce distinct rows and columns, and sort column supports weakest-first.","Apply the finite stop rule for the bounded verified-cell domain; report no global impossibility claim."]}},{"round":57,"agentId":"solvi","privatePlan":{"objective":"Adversarially falsify both proposed slices, certify their support saturation, and choose any follow-up experiment only from an uncovered high-support frontier.","checks":["Independently recompute Mira’s residue identity, CRT count, and registry disjointness.","Recompute every returned edge in the six-row union with exact bigint arithmetic and build the full deduplicated support histogram.","Test whether any apparent improvement is merely a shared anchor or an old-only rectangle; require distinct rows and columns.","If both slices saturate below support four, select the next palette by uncovered exact compatibility rather than numerical size."]}},{"round":57,"agentId":"tess","privatePlan":{"objective":"Construct a rational algebraic family by fixing three differences, parametrizing the two-row conic, and converting the third-row square condition into an explicit nonsingular quartic or elliptic curve; seek rank-positive specializations that support five common differences and multiple distinct N columns.","checks":["Derive the exact substitution u=m₂−m₁, v=(d₂²−d₁²)/u and verify the resulting N and third-row quartic symbolically.","Compute an exact Weierstrass model, torsion, and rational points; reject singular or merely numerical points.","Lift candidate rational points to fourth and fifth square conditions, clear denominators, and verify every cell by bigint square, parity, positivity, and product tests.","Require five distinct differences and five distinct N values, then recompute the complete support matrix and compare against the registry without claiming novelty from bounded failure."]}},{"round":58,"agentId":"mira","privatePlan":{"objective":"Design a new modular row palette using local square-residue branches and CRT intersections, with no divisor_completion request; route any candidate-cell validation solely through Orum.","checks":["Enumerate exact root and admissible-N tables for several fresh prime-power moduli, then intersect them by CRT and retain only palettes with a nonempty, explicitly counted compatible class set.","Vary sign and lift branches while enforcing distinct positive differences and recording registry overlap as bounded scope rather than novelty.","Rank palettes by surviving local density and predicted common-row compatibility; discard branches whose local intersection is empty or whose bounded support cannot reach the target.","For every retained cell, require Orum to verify exact squarehood, parity, positive factors, product, distinct columns, and weakest-first support ordering.","Treat every bounded failure as a finite exclusion only, and stop without claiming progress unless a common 4x5 or 5x5 biclique is returned."]}},{"round":58,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery to generate new factor-pair palettes outside the ledger, then apply deterministic bigint verification only to the highest-scoring candidates.","checks":["Sample and mutate divisor-pair seeds and scale parameters while excluding covered difference regimes and previously tested palettes; optimize the exact-support proxy for multiple distinct columns, not merely one high-support anchor.","Use modular sieves only for candidate ranking, then recompute every retained cell with bigint square-root, parity, positivity, and product checks; reject numerical near-squares.","Rank candidates by weakest-first column support and search explicitly for 5x5, common 4x5, and 6x4 bicliques with distinct rows and columns.","Require a reproducible seed, parameter record, complete support mask, and finite-domain saturation report before treating failure as informative.","Do not request divisor_completion unless stochastic discovery first produces a concrete promising candidate; if needed, use at most one completion job solely to verify that candidate, with no interpretation of its limit as an N bound."]}},{"round":58,"agentId":"pip","privatePlan":{"objective":"Build an exact-cover incidence hypergraph from independently verified cells in the new Mira and Pip palettes, prioritizing 5x5, then common 4x5 and 6x4 structures.","checks":["Recompute every cell with bigint square-root, parity, positive factors, and product checks.","Merge duplicate N columns only after verifying complete row-support masks and distinctness.","Enumerate row-subset intersections and sort candidate columns weakest-first; reject numerical closeness entirely.","Record finite domain, stop conditions, and exact witnesses so any negative result remains bounded-complete only."]}},{"round":58,"agentId":"solvi","privatePlan":{"objective":"Adversarially falsify both new families, then extract saturation certificates and adapt the next experiment to whichever row intersections survive.","checks":["Recompute Mira’s congruences and all local admissible-residue counts independently before scanning; discard the family if any residue or count fails.","Run exact bigint incidence scans for Mira’s 19-adic palette and Pip’s scale-3 palette, recording complete support masks and weakest-first column supports.","For each bounded negative result, certify the exact traversal domain, maximum support, row-intersection counts, and absence of 5x5, common 4x5, and 6x4 bicliques.","Send every surviving candidate cell and any proposed rectangle to Orum for sole validation; require squarehood, parity, positive factors, products, and distinctness.","Adaptively branch only from row subsets with repeated multi-column support; abandon branches capped at support 2 and avoid divisor_completion."]}},{"round":58,"agentId":"tess","privatePlan":{"objective":"Construct a genuine rational-point family by extending the Bremner four-row rectangle: eliminate each column N from the square equations, derive the resulting genus-one curves for a fifth difference, and search exact rational points whose denominator clearing yields multiple distinct positive columns.","checks":["Derive and verify the elliptic/Weierstrass model symbolically from the five equations; reject any parametrization that is only numerical.","Enumerate rational points by exact height and denominator bounds, then clear denominators and verify every cell with bigint squarehood, parity, positivity, and product identities.","Require at least five distinct columns and a common five-row support before claiming a candidate; otherwise record only the exact lower-support family and its finite domain."]}},{"round":59,"agentId":"mira","privatePlan":{"objective":"Design a new modular-local family_scan using CRT-compatible row palettes, selecting congruence branches that maximize common admissible N residue density while avoiding all recorded search regions.","checks":["Enumerate exact square roots and admissible N classes independently modulo each chosen prime power, then recompute CRT intersections and densities.","Require at least two genuinely distinct local branches and verify row distinctness, parity compatibility, and positivity conditions before scanning.","Use modular support upper bounds to prune columns unable to reach support four or five; send any surviving exact cells only through Solvi’s validation-only route.","Record a finite domain, exact stopping rule, and weakest-first support profile; reject numerical near-squares and any unverified candidate."]}},{"round":59,"agentId":"orum","privatePlan":{"objective":"Generate stochastic, optimization-guided row palettes outside covered regimes, then exact-bigint verify only Pareto-promising support structures; request no divisor_completion.","checks":["Sample and mutate prime-modulus branches, affine lifts, and mixed divisor palettes; score predicted common support, support-4/5 density, and registry overlap.","Maintain a Pareto archive and submit only bounded family scans with explicit domains, stop-losses, and completeness targets.","Recompute every accepted edge by exact bigint square testing, parity, positivity, reconstructed factors, distinct columns, and full support masks.","Use Solvi only for validation and reject any candidate based solely on numerical closeness or modular compatibility."]}},{"round":59,"agentId":"pip","privatePlan":{"objective":"Run an incidence-hypergraph/exact-cover search on the mixed scale-5 palette, using only independently verified exact cells and prioritizing 5×5, then common 4×5 support.","checks":["Recompute every edge with bigint square, parity, positivity, and product tests.","Build the complete bounded support matrix, deduplicate N columns, and search exact bicliques with weakest-column-first ranking.","Keep Mira’s CRT palette separate until its cells pass the same validation gates; do not request divisor completion."]}},{"round":59,"agentId":"solvi","privatePlan":{"objective":"Adversarially falsify both palettes, then choose the next experiment from exact support saturation rather than isolated dense seeds.","checks":["Independently recompute every reported edge with bigint identities, reconstructed factors, distinctness, and full support masks.","Require saturation certificates: enumerate all bounded columns at each support threshold and certify weakest-first Pareto ordering, including omitted-column counts.","Compare each domain against the supplied registry and prior scale-3 scan without upgrading finite exclusion to a global claim.","Adapt the next family experiment to the palette with the best verified support-4 density; request divisor_completion at most once, and only for an independently produced candidate."]}},{"round":59,"agentId":"tess","privatePlan":{"objective":"Construct a genuinely parametrized square-additive family by treating the cell equations d_i^2+4N_j=m_{ij}^2 as an intersection of quadrics, beginning with rational parametrization of a two-row/two-column seed and identifying the resulting elliptic curve for an added row or column.","checks":["Derive the birational parametrization symbolically, clear denominators, and verify every generated identity over exact integers after specialization.","Search rational points outside the registry-covered difference and N regimes; reject numerical square proximity and retain only exact bigint witnesses.","For every specialization, enforce positive reconstructed factors, parity, distinct rows and columns, and complete support masks.","Test whether the family yields a strict support improvement such as (4,5,5,5,5), while recording only bounded-complete conclusions when the parameter search is finite."]}},{"round":60,"agentId":"mira","privatePlan":{"objective":"Design a new modular-local row palette using CRT branches distinct from 23·29 and 31·37, maximizing forced common admissible N residues before exact scanning.","checks":["Enumerate and intersect exact square-residue branches modulo selected prime powers; count admissible N classes by CRT.","Require five distinct differences outside covered ranges and test whether local compatibility forces at least four common rows.","Use only bounded-complete family scans; never infer global exclusion from a nonhit.","Send every surviving cell exclusively to Tess for deterministic bigint validation; request no divisor_completion."]}},{"round":60,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery for new CRT-compatible, nonhomothetic difference palettes, then submit only the highest-scoring seeds to Tess for exact bigint incidence verification.","checks":["Generate randomized prime-pair CRT palettes outside the recorded registry domains, rejecting duplicates and purely homothetic rescalings.","Optimize a surrogate score combining predicted common-residue density, row distinctness, primitive-column potential, and Pareto support prospects; never use numerical closeness to squares.","Verify every selected cell exactly via m²=d²+4N, parity, positive reconstructed factors, and ab=N.","Rank exact columns weakest-first and test 5×5, then 4×5 and 5×4 support objectives, with all exclusions reported only for the tested bounded domain.","Use Tess as the sole validation-only verifier; request no divisor_completion and assign Solvi no validation task."]}},{"round":60,"agentId":"pip","privatePlan":{"objective":"Run an incidence-hypergraph/exact-cover search using only Tess-verified cells from the new prime-31 and 31·37 palettes, prioritizing a true 5×5 block and then exact 4×5 or 5×4 support.","checks":["Recompute every cell by bigint square equality, parity, positivity, and product reconstruction.","Deduplicate N columns, compute complete support masks, and rank weakest-first without counting homothetic duplicates as novelty.","Record finite-domain exclusions separately from any candidate certificate."]}},{"round":60,"agentId":"solvi","privatePlan":{"objective":"Apply adversarial falsification, saturation certificates, and adaptive branch selection to the two new palettes without requesting divisor completion.","checks":["Independently recompute every displayed seed cell and CRT congruence before treating either palette as admissible.","Test whether support saturates below four before accepting any bounded negative result; record a complete support profile when saturation occurs.","Treat any 6×4 output as diagnostic and search adaptively for a five-column Pareto-relevant substructure.","If a scan fails, distinguish traversal saturation from family-wide exclusion and select the next CRT branch from its observed common-row support."]}},{"round":60,"agentId":"tess","privatePlan":{"objective":"Construct a non-homothetic rational-point family from common square conditions, using the new CRT row palette only as a seed and seeking elliptic-curve or rational parametrizations rather than another raw scan.","checks":["Derive exact equations for simultaneous cells and determine whether the relevant component is rational or elliptic.","Generate only exact integral points outside the recorded registry domains; verify every support cell by bigint identities and reconstructed factors.","Require five distinct N-columns before treating a result as k=5 progress; request at most one divisor_completion job only after an independently produced candidate exists."]}},{"round":61,"agentId":"mira","privatePlan":{"objective":"Run an incidence-hypergraph/SAT exact-cover search on independently verified cells from the new CRT and mutation families, prioritizing balanced 5-by-5 support and then strict 4-by-6 or 5-by-4 improvements.","checks":["Recompute every cell with exact bigint square testing, parity, positivity, factor recovery, and product identity.","Deduplicate differences and columns, construct the full verified support matrix, and solve the stated biclique constraints with weakest-first column ordering.","Reject any claim based only on modular residue compatibility or numerical square proximity; request no divisor-completion job unless an independently generated candidate requires final verification."]}},{"round":61,"agentId":"orum","privatePlan":{"objective":"Adversarially audit the 41·43 construction and obtain a genuine saturation certificate with one exact family scan; request no divisor_completion.","checks":["Recompute M=138600·41·43, all five differences, pairwise distinctness, and d_i^2≡69814801 (mod 244351800); record only finite-domain conclusions.","Require this manifest: familyFingerprint='Five-root 138600 square-residue family lifted through auxiliary primes 41 and 43'; registryOverlap='new'; domain='first 5000000 deterministic exact pair-completion checks for the five listed differences, N unrestricted'; completenessTarget='bounded-complete'; targetShape='5×5, then 4×6, 5×4, or 4×5 exact support block'; successCriterion='return an exact target or the complete bounded support/intersection profile'; stopLoss='hard-stop at 5000000 checks and report the profile'; symbolicIdentity='d_i^2 ≡ 69814801 (mod 244351800) for i=1,...,5'; params={d1:null,d2:null,differences:'157781449 124115201 78799049 69814801 24498649',limit:null,maxChecks:5000000,startDifference:null,endDifference:null,stride:null}.","Demand bigint square equality, parity, positivity, recovered factors, products, distinct columns, exact support masks, and independent CRT recomputation before accepting any target.","If Pip’s cloud is reconsidered, first require explicit generating tuples and signs for every submitted difference; otherwise reject it as non-reproducible."]}},{"round":61,"agentId":"pip","privatePlan":{"objective":"Use stochastic and optimization-guided CRT/linear-mutation seed discovery to maximize balanced support floors, followed by exact bigint verification.","checks":["Reject seeds overlapping covered registry domains or lacking explicit provenance.","Rank candidates by exact-support potential, then run family_scan only; request no divisor_completion.","Verify every cell by bigint square equality, parity, positive factor recovery, and exact product identity.","Recompute the full incidence matrix and test 5×5, 4×5, and 5×4 support targets with bounded-complete claims stated precisely."]}},{"round":61,"agentId":"solvi","privatePlan":{"objective":"Develop a constructive rational-point family from the square-additive matrix model X_{ij}^2=A_i+B_j, with A_i=d_i^2 and B_j=4N_j, starting from the known 4-row rectangle and treating a fifth row as a genus-one lifting problem.","checks":["Eliminate the fifth-row variables symbolically, identify the resulting conic/genus-one model, and verify every birational substitution exactly.","Generate rational points by exact elliptic-curve group operations, then clear denominators and enforce positive distinct differences, positive distinct N, and parity.","For each lifted point, verify all claimed cells by bigint square equality and reconstructed factor products; reject numerical near-squares.","Compare the resulting family against the registry and discard homothetic or already-covered specializations."]}},{"round":61,"agentId":"tess","privatePlan":{"objective":"Design a modular-local row palette whose congruence conditions force compatible pair-completion classes before any broad incidence scan.","checks":["Enumerate CRT branches combining the common square-residue modulus with new small auxiliary primes, rejecting branches whose local roots cannot support at least four shared rows.","Derive exact necessary congruences for N and m in each branch, then intersect them across candidate row subsets to maximize forced compatibility.","Use only bounded family scans after the congruence sieve, recording exact support masks and testing whether a strict 4×5 or 5×4 block survives."]}},{"round":62,"agentId":"mira","privatePlan":{"objective":"Use only reconciled, bigint-verified cells from the CRT and corrected endpoint palettes to build an incidence hypergraph, then search exact-cover/SAT encodings for a 5×5 block and strict 4×6 or 5×4 frontiers.","checks":["Recompute every endpoint-projection difference from its source pair and coefficients; discard the entire inconsistent palette until reconciled.","Have Pip validate each retained cell by exact square equality, parity, positivity, factor recovery, and product.","Deduplicate verified row-column cells and encode support masks with distinct-row and distinct-column constraints.","Run deterministic SAT/exact-cover searches in descending target order, recording finite domain and weakest-first support profiles."]}},{"round":62,"agentId":"orum","privatePlan":{"objective":"Construct an adaptive, adversarially falsifiable family scan whose finite domain and saturation profile are fully reproducible, without requesting divisor_completion.","checks":["Require a deterministic generator specification: seed, parameter order, duplicate handling, emitted vertices, and exact check domain before selecting a branch.","Use checkpointed support profiles and common-row intersections; continue only when the support floor or balanced frontier improves, otherwise record a domain-limited exclusion.","Demand a complete incidence matrix and maximal-biclique certificate for the emitted domain, with no early termination based solely on low support.","Route every surviving cell and every claimed witness through Pip’s exact bigint validation: square equality, parity, reconstructed factors, products, and distinctness.","If both branches fail, mutate one auxiliary-prime choice or projection coefficient adaptively rather than broadening an under-specified stochastic cloud."]}},{"round":62,"agentId":"pip","privatePlan":{"objective":"Use stochastic, optimization-guided endpoint/CRT seed discovery to maximize balanced exact support, then submit only internally consistent palettes to deterministic bigint verification.","checks":["Regenerate every projected difference directly from its recorded factor pair and coefficient pair; reject any mismatch and avoid covered registry regimes.","Score candidate palettes by weakest-column support and balanced row intersections using exact modular filters, never numerical square closeness.","Run exact bigint square, parity, positivity, factor-recovery, distinctness, and support-intersection checks on the selected palette.","Request no divisor_completion unless a primary stochastic family method first produces a verified candidate; then use at most one such job solely for independent confirmation."]}},{"round":62,"agentId":"solvi","privatePlan":{"objective":"Develop a low-dimensional constructive family by rewriting simultaneous square completions as rational-point conditions, using CRT lifts only to select promising fibers.","checks":["Derive the exact conic/elliptic equations for shared factor-difference rows and identify any birational parametrization or elliptic curve.","Generate rational points symbolically, clear denominators, and enforce positivity, parity, distinctness, and exact support.","Test whether a rational family preserves at least four common rows before seeking a fifth; discard families with only accidental congruence compatibility.","Have Tess independently verify every resulting cell with deterministic bigint arithmetic."]}},{"round":62,"agentId":"tess","privatePlan":{"objective":"Design a new local CRT palette that forces compatible factor-pair endpoint residues and targets balanced four- or five-row support, without requesting divisor_completion.","checks":["Choose fresh small coprime moduli and solve exact square-residue constraints for five distinct differences, excluding all supplied palettes.","Enumerate only residue-compatible local endpoint pairs, tracking predicted row intersections before bigint expansion.","Verify every retained cell by exact square equality, parity, positivity, reconstructed factors, and product identity.","Report only bounded-complete support results and preserve finite-exclusion language for failed slices."]}},{"round":63,"agentId":"mira","privatePlan":{"objective":"Construct an exact-cover/SAT search over only independently bigint-verified incidence cells from the 59·61 and smooth-factor palettes, maximizing balanced biclique support.","checks":["Canonicalize cell enumeration and record its exact finite domain before solving.","Recompute every cell and support mask with deterministic bigint arithmetic.","Search 5×5 first, then 6×4 and 4×5, with distinct rows and columns.","Reject any witness lacking independent replay, parity, positivity, and product checks."]}},{"round":63,"agentId":"orum","privatePlan":{"objective":"Adversarially filter the proposed palettes and require a saturated, replayable incidence certificate before authorizing substantial computation.","checks":["Independently verify every CRT congruence, distinctness condition, and exact square-residue identity in Mira’s palette.","Demand a deterministic generator specification, seed, ordering, deduplication rule, and replay hash for any family scan.","Use support-floor and maximal-intersection certificates to choose the next experiment adaptively; do not infer global impossibility from bounded failure.","Request at most one divisor-completion job, and only if a separate primary search first produces an independently verified candidate column set."]}},{"round":63,"agentId":"pip","privatePlan":{"objective":"Run the rotating stochastic, optimization-guided seed-discovery lane on reproducible smooth/CRT hyperbola seeds, then submit only the best balanced palettes for exact bigint verification.","checks":["Generate disjoint candidate palettes with recorded seed, mutation, auxiliary-prime, and scoring metadata; exclude all covered registry and prior-round regions.","Optimize lexicographically for minimum column support, minimum row support, and balanced incidence before retaining a palette.","Use a canonical bounded family_scan traversal with all required manifest fields, never divisor_completion.","Require exact square, parity, positivity, factor-product, distinctness, and support-mask checks before accepting any witness.","Report only verified bicliques or finite bounded-complete exclusions; reject numerical near-squares."]}},{"round":63,"agentId":"solvi","privatePlan":{"objective":"Construct a rational-point family from the Bremner rectangle: for selected differences define C by y_i^2=4N+d_i^2, convert the resulting genus-one intersection to Weierstrass form, and use its group law to generate new common columns before imposing additional rows.","checks":["Verify the birational map, nonsingularity, and group-law identities symbolically over the rationals.","Generate multiples of known rational points, clear denominators exactly, and reject nonpositive or duplicate N values.","Impose fourth and fifth difference conditions by exact rational-square tests, not numerical proximity.","Submit only explicit resulting candidates to orum for independent bigint validation; request no divisor_completion job."]}},{"round":63,"agentId":"tess","privatePlan":{"objective":"Design a modular, local-congruence sieve that forces or prunes compatible rows for the 59·61 palette and a disjoint auxiliary-prime variant, without requesting divisor_completion.","checks":["Compute exact local square-residue sets for each difference modulo selected small primes and auxiliary-prime factors; intersect them by row subsets before any bigint completion search.","Use CRT to enumerate N residue classes compatible with at least four or five differences, rejecting classes that cannot support distinct positive factor pairs.","For every surviving class, perform exact parity, square-root, factor-recovery, distinctness, and support-mask checks; record finite exclusions only on the stated bounded residue slice.","Compare balanced support profiles against the registry and prior palettes, with no credit for numerical near-squares."]}},{"round":64,"agentId":"mira","privatePlan":{"objective":"Run an incidence-hypergraph/SAT exact-cover search using only independently verified cells from the admissible new palette, targeting 5x5 first and strict 6x4 or 4x5 support second.","checks":["Recompute every cell with deterministic bigint square tests, parity, positivity, and factor-product verification.","Construct the full row-column incidence masks and enforce distinct rows and columns in each biclique.","Use weakest-first support ordering and record the complete bounded support profile.","Reject any palette entry or witness failing decimal distinctness, family membership, or independent replay."]}},{"round":64,"agentId":"orum","privatePlan":{"objective":"Adversarially prefilter both palettes, then adaptively select at most one exact incidence experiment with a reportable saturation profile.","checks":["Recompute all listed distinctness, CRT congruences, common residues, and family exclusions using deterministic bigint arithmetic.","Reject any scan whose traversal, support profile, or stop condition cannot certify the entire declared prefix.","Prioritize palettes only after finding support-3-or-better columns; require weakest-first row and column profiles and independent Solvi validation.","Have Tess audit every accepted incidence matrix and distinguish bounded exclusion from any genuine frontier improvement."]}},{"round":64,"agentId":"pip","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery to generate one genuinely new balanced-incidence family, then perform exact bigint verification through a family scan only.","checks":["Exclude all registry-covered regions and prior palettes before scoring seeds.","Optimize weakest-first row and column support, prioritizing balanced 5×5 and strict 6×4 or 4×5 structures.","Replay every returned cell with exact square, parity, positivity, factor-product, distinctness, and support-intersection checks.","Have Solvi perform the independent validation; do not request divisor_completion."]}},{"round":64,"agentId":"solvi","privatePlan":{"objective":"Construct a reproducible low-dimensional algebraic family by reparameterizing shared factor-difference cells as rational points on conics, then derive and inspect the resulting elliptic fibers for five-row incidence.","checks":["Normalize each cell with N=xy and d=y-x, and eliminate scale symmetries before searching.","Parameterize a four-row seed exactly, compute the remaining square conditions as a genus-one curve or intersection of quadrics, and enumerate rational points by bounded height.","Clear denominators, enforce positivity and pairwise distinctness, and independently verify every incidence with bigint arithmetic.","Search only outside the supplied registry regions; use at most one divisor-completion check solely after a candidate column is produced by this primary algebraic construction."]}},{"round":64,"agentId":"tess","privatePlan":{"objective":"Develop a modular local-congruence palette that prunes incompatible rows and preferentially forces shared square-completion residues, without requesting divisor_completion.","checks":["Choose coprime auxiliary moduli and enumerate admissible d-residue classes with exact conditions on d^2+4N modulo each modulus.","Intersect local N-residue masks across five rows, retaining only residue patterns capable of producing common columns with support at least four.","Use CRT lifts of surviving residue patterns to form a small exact family, then audit distinctness, parity, and positivity before any scan.","Send only exact surviving incidence cells to Solvi for validation; compare the resulting weakest-first support profile against the current frontier."]}},{"round":65,"agentId":"mira","privatePlan":{"objective":"Run an incidence-hypergraph/SAT exact-cover search on Pip’s Markov palette using only independently verified exact cells, targeting 5×5 first and then 6×4 or 4×5.","checks":["Recompute every cell with exact bigint square equality, parity, positivity, reconstructed factors, and product.","Deduplicate and verify all rows and columns before support scoring.","Require direct verification of every selected biclique and report only bounded-complete results.","Investigate the Tess rejection discrepancy before reusing any 73·79 data."]}},{"round":65,"agentId":"orum","privatePlan":{"objective":"Adversarially replay both palettes, then adapt computation only after obtaining a parser-clean input and a saturation certificate.","checks":["Re-encode Mira’s five differences as canonical decimal tokens; verify pairwise distinctness, both CRT congruences, and the common square residue with bigint arithmetic.","Recompute Pip’s Markov terms and all 20 derived differences, checking recurrence, distinctness, and overlap against the ledger.","For any family_scan, demand a complete support histogram and maximal-intersection certificate over the declared prefix; escalate only if support reaches at least 3.","Reserve Tess for independent validation of every returned cell and biclique; make no divisor_completion request."]}},{"round":65,"agentId":"pip","privatePlan":{"objective":"Generate stochastic, optimization-guided seeds from a hybrid Markov/CRT incidence model, then submit one family_scan for exact bigint verification only.","checks":["Exclude every registry-covered and previously tested palette.","Optimize weakest-first support for 5×5, then 6×4 and 4×5.","Verify generated identities, distinctness, parity, reconstructed factors, and every square by deterministic bigint arithmetic.","Use maxChecks no greater than 5000000 and request no divisor_completion."]}},{"round":65,"agentId":"solvi","privatePlan":{"objective":"Construct a reproducible algebraic-family lift by fixing the Markov seed M_0=3524578, M_1=9227465 and deriving exact rational parametrizations of shared-cell intersections, including any genus-one subfamilies.","checks":["Verify the recurrence-generated differences symbolically and test pair completions through (2x+d)^2-(2y+e)^2=d^2-e^2 with exact parity and positivity conditions.","Search rational-point specializations for distinct positive N and differences, excluding the registry’s covered palettes and Bremner specializations.","Require exact bigint support matrices and Tess validation; request no divisor_completion job."]}},{"round":65,"agentId":"tess","privatePlan":{"objective":"Design a fresh modular palette that locally forces compatible row completions before any exact search, staying outside covered regions.","checks":["Use new auxiliary primes and CRT classes; verify pairwise distinct differences, all residue identities, and exact CRT moduli.","For each candidate N residue class, test every listed discriminant d_i^2+4N for quadratic-residue compatibility modulo several independent primes, retaining only shared local classes.","Lift surviving local classes to bounded integer candidates and perform exact square, parity, positivity, and factor-product checks; claim only verified support.","Request no divisor_completion job unless this primary congruence construction first yields a concrete candidate column."]}},{"round":66,"agentId":"mira","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery in a new hybrid incidence space, then perform deterministic bigint verification of retained seeds.","checks":["Generate seeds from independently selected Markov-CRT parameter combinations, excluding all registry-listed and previously scanned palettes.","Score seeds by Pareto support profile, prioritizing balanced row and column support before magnitude.","Verify every retained cell by exact square-root, parity, positivity, factor-product, and distinctness checks.","Request no divisor_completion job unless this primary method first produces a concrete candidate; then use at most one completion job solely for verification."]}},{"round":66,"agentId":"orum","privatePlan":{"objective":"Construct a verified rational/elliptic family of simultaneous factor-difference cells, using Markov mutations only as algebraic seeds rather than as unverified numeric palettes.","checks":["Derive pair completions from (2t+d)^2-(2u+e)^2=d^2-e^2 and retain only exact positive integral solutions.","Verify Markov identities, mutation preservation, distinct gaps, parity, reconstructed factors, and products by bigint arithmetic.","Search rational intersections imposing three or four difference incidences on one N; test discriminants and specialize rational points to integral columns.","Build the complete incidence matrix and require a certified 4×5, 5×4, or 5×5 support pattern before claiming progress."]}},{"round":66,"agentId":"pip","privatePlan":{"objective":"Adversarially falsify the CRT and off-axis Markov proposals while extracting exact saturation certificates, then adapt the next experiment only toward branches showing shared support at least three.","checks":["Recompute every proposed CRT difference, Markov identity, gap, parity condition, and exact square cell with bigint arithmetic.","Construct the complete returned incidence matrix; record weakest-first column supports, all maximal 4-row and 5-row intersections, and exact witnesses for every support threshold.","Reject any branch with no shared support-3 structure before extending it; retain only branches that improve the balanced support profile.","State every negative result as finite-exclusion over its tested prefix, and route any nonempty candidate set exclusively through Mira’s validation gate without requesting divisor_completion."]}},{"round":66,"agentId":"solvi","privatePlan":{"objective":"Use modular and local congruence signatures to force compatible four- or five-row subsets before any exact incidence search, without requesting divisor_completion.","checks":["For each proposed difference palette, compute S_i(q)={n mod q: d_i^2+4n is a quadratic residue mod q} for q=2 and selected small odd primes.","Intersect S_i(q) across row subsets of size four and five; discard subsets with an empty local intersection and CRT-combine surviving residue classes.","Rank CRT classes by simultaneous row support and add congruence filters that separate repeated or weak support patterns.","Exact-check only the resulting bounded residue classes, verifying square equality, parity, positivity, reconstructed factors, and distinct columns."]}},{"round":66,"agentId":"tess","privatePlan":{"objective":"Build an incidence-hypergraph exact-cover/SAT search using only independently verified cells, targeting 5×5 first and strict Pareto shapes second.","checks":["Recompute every cell by exact bigint square-root, parity, positivity, and factor-product tests.","Deduplicate rows and columns, then solve for 5-column support on five rows, followed by 4×5 and 6×4.","Audit every returned biclique by reconstructing the full incidence matrix and weakest-first support ordering.","Exclude all unverified CRT or Markov-generated cells from the constraint set until independently certified."]}},{"round":67,"agentId":"mira","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery on a fresh modular/CRT family, maximizing balanced incidence support before exact bigint verification.","checks":["Use a deterministic seeded Pareto beam and family_scan only; do not request divisor_completion.","Rank seeds by weakest-first column support, prioritizing 4×5 and 5×5 structures rather than isolated columns.","Verify every retained incidence by exact square, parity, positivity, and reconstructed-factor checks.","Recompute complete support matrices, distinctness, and frontier improvement; reject all numerical near-squares.","Submit finalists to pip for independent exact validation."]}},{"round":67,"agentId":"orum","privatePlan":{"objective":"Construct a genuine fifth-row algebraic family from a known 4×6 square-additive rectangle: use three columns to derive the rational-point/elliptic structure, then test the remaining columns exactly.","checks":["Derive an explicit birational elliptic model from r²+4N_j=s_j², and verify its discriminant and inverse maps symbolically.","Separate certified rational-point enumeration from exploratory height search; record all denominators and exceptional points.","Clear denominators with explicit parity and positivity conditions so every retained point yields integral d, N, and factor pairs.","Check all six columns with exact bigint arithmetic, distinctness, support ordering, and registry non-overlap; then send only surviving candidates to pip for validation.","Do not request divisor_completion."]}},{"round":67,"agentId":"pip","privatePlan":{"objective":"Adversarially falsify both proposed families, extract exact saturation certificates, and adapt only toward incidence patterns with support at least three.","checks":["Recompute Mira’s five lifts, residues, parity, and every retained incidence using deterministic bigint arithmetic; reject all near-squares.","Enumerate all divisor pairs for each power-of-two row pair, verify positivity and parity, deduplicate resulting N, and compare the complete shell profile with any scan output.","Require a repeated support-3-or-higher pattern before adapting the search; otherwise record only bounded finite exclusion and shell statistics.","Request no divisor_completion job unless an independent primary method first produces a concrete candidate requiring verification."]}},{"round":67,"agentId":"solvi","privatePlan":{"objective":"Design a deterministic local-congruence palette that forces compatible pair shells, using moduli and residue classes distinct from the registry, the 83·89 and 97·101 lifts, and the powers-of-two lane; retain only patterns capable of support at least three before extension toward 4×5 or 5×5.","checks":["For each selected modulus, enumerate admissible d and m residue classes and derive exact N residue classes; discard a class only with a proved congruence obstruction.","For every retained row pair, solve (m−n)(m+n)=d²−e² by exact divisor enumeration, then verify parity, positivity, reconstructed factors, and N.","Use deterministic support masks and weakest-first column ordering; report only bounded-complete support statistics, never a global exclusion.","Request no divisor_completion; require pip’s independent bigint audit for every retained incidence or claimed improvement."]}},{"round":67,"agentId":"tess","privatePlan":{"objective":"Use only exact verified cells to build an incidence hypergraph and run an exact-cover/SAT search for a strict Pareto improvement over the 4×4 frontier.","checks":["Recompute every cell with bigint square, parity, positivity, and product checks.","Encode distinct selected differences and N-columns, enforcing 5×5 first and then 4×5 or 6×4 objectives.","Verify every SAT/exact-cover solution directly and reject numerical near-squares.","Compare weakest-first column supports against the known Bremner certificate."]}},{"round":68,"agentId":"mira","privatePlan":{"objective":"Use stochastic, optimization-guided pair-shell seed discovery on a fresh high-index Pell–Lucas block, then perform exact bigint incidence verification and submit only retained records to orum.","checks":["Generate q_40 through q_59 exactly from q_{n+2}=2q_{n+1}+q_n, then use deterministic SplitMix64 seed 0x88568A to sample 5,000,000 pair-shell trials; for each row pair factor |d_i^2-d_j^2| and enforce parity before constructing completion seeds.","For every retained seed compute N=(m_i^2-d_i^2)/4 with bigint arithmetic, test d^2+4N for every selected row, verify positivity, parity, reconstructed factors, products, distinctness, and complete support masks.","Rank candidates lexicographically by biclique deficit, number of columns of support at least four, total support, and weakest column support; retain Pareto-undominated candidates only, never using floating-point closeness.","Run family_scan with params d1=null; d2=null; differences=\"1023286908188737 2470433131948081 5964153172084899 14398739476117879 34761632124320657 83922003724759193 202605639573839043 489133282872437279 1180872205318713601 2850877493509864481 6882627192338442563 16616131878186749607 40114890948711941777 96845913775610633161 233806718499933208099 564459350775477049359 1362725420050887306817 3289910190877251662993 7942545801805390632803 19175001794488032928599\"; limit=5000; maxChecks=5000000; startDifference=null; endDifference=null; stride=null.","Manifest: familyFingerprint=\"Pell–Lucas norm family q_n with stochastic pair-shell seed discovery\"; registryOverlap=\"new\"; domain=\"5,000,000 fixed-seed pair-shell trials on the listed q_40,...,q_59 differences, with unrestricted positive N\"; completenessTarget=\"exploratory\"; targetShape=\"5×5 primary, then 4×5 or 6×4\"; successCriterion=\"Return exact distinct N and rows proving one target shape, otherwise the best Pareto support profile\"; stopLoss=\"Stop at 5,000,000 trials, or earlier after 1,000,000 trials if no support-3 column occurs\"; symbolicIdentity=\"q_{n+2}=2q_{n+1}+q_n and q_n^2-2p_n^2=(-1)^n\"; proofObligations=\"independent bigint cell checks; support-mask and distinctness recomputation; recurrence/norm verification; exact parity and factor reconstruction\".","Send only exact retained candidates and complete incidence tables to orum for validation; request no divisor_completion job."]}},{"round":68,"agentId":"orum","privatePlan":{"objective":"Construct a new exact algebraic family by eliminating N from simultaneous equations m_i^2-d_i^2=4N, prioritizing rational-point and elliptic-curve parametrizations rather than stochastic scans.","checks":["Derive the resulting conic intersections or elliptic curve explicitly and test rational points symbolically.","Clear denominators exactly, enforcing parity, positivity, distinct differences, and distinct N values.","Use divisor-pair saturation to certify every claimed shared completion; request at most one divisor_completion job only after a candidate is produced algebraically.","Reject numerical near-squares and record only exact support matrices with weakest-first column ordering."]}},{"round":68,"agentId":"pip","privatePlan":{"objective":"Adversarially test whether the CRT and Pell families escape support-2 saturation, using exact pair-completion certificates and adaptive family scans only; no divisor_completion request.","checks":["Recompute every CRT congruence, Pell recurrence/norm identity, parity condition, and candidate support mask with bigint arithmetic; reject residue-only evidence.","For every support-bearing row pair, enumerate admissible divisor pairs of Δ=d_j²-d_i², reconstruct all positive common completions, and label the saturation domain exactly.","Run a family_scan with d1=null, d2=null, differences=\"22619537 54608393 131836323 318281039 768398401 1855077841 4478554083 10812186007 26102926097 63018038201 152139002499 367296043199 886731088897 2140758220993 5168247530883 12477253282759 30122754096401 72722761475561 175568277047523 423859315570607\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Manifest: familyFingerprint=Pell–Lucas q_n for n=20..39; registryOverlap=new; domain=first 5000000 exact canonical traversal checks with N unrestricted; completenessTarget=bounded-complete; targetShape=5x5, then 6x4 or 4x5; successCriterion=exact certificate or complete support profile; stopLoss=5000000 checks; symbolicIdentity=q_(n+2)=2q_(n+1)+q_n and q_n²−2p_n²=(−1)^n.","Require proof obligations for exact cell reconstruction, complete support intersections, distinctness, and finite-versus-global claim labeling; send only accepted outputs to Orum for validation."]}},{"round":68,"agentId":"solvi","privatePlan":{"objective":"Design a row-selective modular family in which local congruence classes force at least five differences to remain simultaneously admissible for the same N residue, then run only a deterministic family_scan.","checks":["Search small-prime residue patterns for d_i and N where m^2 ≡ d_i^2+4N has a common solution for a targeted five-row subset, avoiding palettes with identical d_i^2 residues.","Lift surviving local patterns by CRT to differences outside the covered registry ranges and record the exact finite domain and seed.","Prune candidate N classes locally before bigint generation; request no divisor_completion job.","Require exact parity, square, factor reconstruction, distinctness, and support-matrix checks, with orum as validation-only verifier."]}},{"round":68,"agentId":"tess","privatePlan":{"objective":"Use only deterministically verified cells to construct an incidence hypergraph and run exact-cover/SAT searches for 5x5, then strict Pareto improvements such as 4x5 or 6x4.","checks":["Recompute every cell with bigint square testing, parity, positivity, reconstructed factors, and product equality.","Deduplicate differences and N-columns, then build exact row-column incidence masks.","Enumerate all candidate row subsets and common-column intersections, checking distinctness and weakest-first support ordering.","Reject numerical near-squares and classify any zero result only over the explicitly enumerated verified-cell domain.","Require Orum’s independent exact audit for every reported biclique."]}},{"round":69,"agentId":"mira","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery on the certified six-root CRT family, followed only by exact bigint verification.","checks":["Use family_scan only with d1=null, d2=null, differences=\"112518847 152684933 192850916 235700669 286997862 327163948\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null; request no divisor_completion.","Recompute M=1113121, every residue, and every congruence d_i^2≡1 mod M before sampling.","Use deterministic stochastic seeds and a Pareto beam maximizing weakest row and column supports; reject isolated anchors.","Manifest: familyFingerprint=\"six signed CRT-root lifts modulo 101·103·107\"; registryOverlap=\"new\"; domain=\"5,000,000 checks with x in [10^12,10^24]\"; completenessTarget=\"exploratory\"; targetShape=\"balanced 5×5, then 6×4 or 4×5\"; successCriterion=\"exactly verified target support\"; stopLoss=\"stop at 5,000,000 checks or earlier if the stated finite threshold fails\"; symbolicIdentity=\"d_i^2≡1 (mod 1113121)\".","Verify every retained cell by exact square, parity, positivity, reconstructed factors, product, distinctness, and complete support-mask recomputation; send survivors to Solvi for independent validation."]}},{"round":69,"agentId":"orum","privatePlan":{"objective":"Construct a non-specialized rational square-additive rectangle by parametrizing the conic fibers q^2-d^2=4N and studying the resulting elliptic-curve fiber product for a fifth row or column.","checks":["Perform exact symbolic elimination over Q and retain only identities valid for free parameters, not numerical near-squares or fixed Bremner specializations.","Compute and document the associated Weierstrass model, rational-point map, nonsingularity, and denominator saturation exactly.","Enforce positivity and pairwise distinctness after integer scaling, then verify all claimed cells by bigint identities and exact support masks.","Use Solvi only for independent validation; reject any family that yields merely bi-maximal or support-2 specializations."]}},{"round":69,"agentId":"pip","privatePlan":{"objective":"Use adversarial falsification, exact saturation certificates, and adaptive experiment design to compare the CRT and odd-Lucas palettes without requesting divisor_completion.","checks":["Replay every returned incidence with exact bigint square, parity, positivity, reconstructed factors, and product checks; reject numerical near-squares.","For every support-bearing row pair, enumerate positive same-parity divisor pairs of Δ=d_j^2-d_i^2 and reconstruct all admissible completion shells, certifying saturation or finite traversal limits.","Compute complete incidence matrices, weakest-first column supports, and all 3×3, 4×5, 5×4, and 5×5 substructures before judging progress.","Select any subsequent family_scan solely from these exact shell statistics; retain maxChecks at or below 5000000 and require a full manifest, domain, stop-loss, and at least two proof obligations before execution."]}},{"round":69,"agentId":"solvi","privatePlan":{"objective":"Build a fresh CRT palette in which all selected differences share prescribed square classes modulo several small primes, then use these local conditions to prune pair-shell candidates and seek balanced exact support.","checks":["Choose odd differences with d_i^2 congruent to a common c modulo a squarefree product M; verify every congruence and parity condition exactly.","Enumerate admissible N residue classes modulo M for which c+4N is a square modulo every selected prime, then generate candidates from a primary shell N=x(x+d_0) outside covered bounds.","Apply exact bigint tests d_i^2+4N=m_i^2, parity, positivity, reconstructed factors, distinctness, and complete support-mask/Pareto checks; reject all near-squares.","Retune the prime palette only when local filters produce support concentration; do not request divisor_completion unless an independently generated candidate already has support at least four."]}},{"round":69,"agentId":"tess","privatePlan":{"objective":"Construct an incidence-hypergraph/SAT exact-cover search using only bigint-verified cells, targeting a 5x5 biclique and strict 4x5 or 5x4 improvements.","checks":["Reverify every cell by exact square, parity, factor, and product tests before insertion.","Build row-difference/column-N incidence masks and enumerate maximal bicliques with weakest-first support ordering.","Reject numerical near-squares and report only finite results for the verified-cell domain.","Send any surviving certificate to Solvi for independent validation; request no divisor_completion job."]}},{"round":70,"agentId":"mira","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery on a fresh CRT-lift family, then perform exact bigint support verification; request no divisor_completion.","checks":["Generate deterministic SplitMix64 perturbations of uneven lift vectors and rank them only by exact shell-factor and support-mask objectives; retain balanced candidates, not numerical near-squares.","Submit family_scan with params {d1:null,d2:null,differences:\"1607313 3140836 4748148 9373236 14022960 21856572\",limit:5000,maxChecks:5000000,startDifference:null,endDifference:null,stride:null}. Manifest: familyFingerprint=\"six selected nontrivial roots of 1 modulo 1564259=109·113·127 with CRT lifts r_i+1564259k_i\"; registryOverlap=\"new\"; domain=\"the first 5000000 deterministic SplitMix64 states of this fixed six-difference traversal, with unrestricted N and exact bigint incidence tests\"; completenessTarget=\"bounded-complete for the declared state stream\"; targetShape=\"5x5 first, then 6x4 or 4x5, retaining weakest-first Pareto support\"; successCriterion=\"return distinct N values with exact masks certifying one target shape, otherwise the strongest balanced support profile\"; stopLoss=\"stop after 5000000 checks unless a certified target appears\"; symbolicIdentity=\"r_i^2≡1 (mod 1564259) and d_i=r_i+1564259k_i\"; proofObligations=\"recompute every square test, parity, factor product, row/column masks, distinctness, and all claimed intersections exactly; verify every CRT root modulo 109, 113, and 127.\"","After completion, send only the returned candidates and full support matrix to Tess for validation; reject all unverified or support-1 claims."]}},{"round":70,"agentId":"orum","privatePlan":{"objective":"Construct a non-CRT algebraic family by parametrizing simultaneous difference-of-squares equations and imposing a third or fourth incidence condition as an explicit elliptic-curve or rational-point problem.","checks":["Derive the rational parametrization of u^2-v^2=d_i^2-d_j^2, then identify the exact curve imposed by additional differences; record its genus, singularities, and rational base points.","Search only rational points generated by certified group-law or factorization steps, and lift denominators to positive integral N with distinct columns and differences.","For every candidate, recompute all cells, support masks, and complete divisor factorizations of each relevant pair shell; reject numerical approximations and incomplete saturation.","Compare only against the registry and known frontier, with a finite, replayable domain and no divisor_completion request."]}},{"round":70,"agentId":"pip","privatePlan":{"objective":"Adversarially falsify both proposed palettes, then adapt toward the family with the strongest exact Pareto support while producing saturation certificates rather than heuristic density claims.","checks":["Recompute every CRT root, lift, ternary power, shell identity, and 728 divisibility with bigint arithmetic.","Before any experiment, bind the deterministic seed, candidate-generation rule, pair-shell ordering, x-to-N map, and exact finite domain.","For every support-bearing pair, exhaustively factor the shell and enumerate admissible same-parity divisor pairs; never label a check-limited sample saturated.","Maintain weakest-first row and column support profiles, rejecting isolated anchors and requiring distinct positive N values.","Use the first reproducible support-three intersection to choose the next adaptive palette; otherwise terminate that family as a finite exclusion and switch lanes.","Keep Tess limited to validation and independently replay all accepted masks before any frontier claim."]}},{"round":70,"agentId":"solvi","privatePlan":{"objective":"Construct a new modular/local CRT palette that forces compatible row pairs and prunes impossible five-row incidence before any global scan.","checks":["Enumerate exact roots modulo products of small primes and select difference residues with explicitly compatible N residue classes.","Derive and verify pair-shell congruences, including parity and divisibility conditions, for a prospective 5-row support graph.","Use local residue compatibility to eliminate palettes lacking a possible K5,5 pattern before requesting a family scan.","Keep the domain outside the supplied registry and require deterministic bigint support masks; Tess performs validation only."]}},{"round":70,"agentId":"tess","privatePlan":{"objective":"Run an incidence-hypergraph/exact-cover search over independently verified cells from the strongest nonzero family output, targeting a strict Pareto improvement before K5.","checks":["Recompute every cell with exact bigint arithmetic and reject numerical near-squares.","Construct row and column support masks, deduplicate N values, and solve exact-cover instances for 5x5, 6x4, and 4x5 patterns.","Report only finite-domain conclusions and require an independent replay hash before any frontier claim."]}},{"round":71,"agentId":"mira","privatePlan":{"objective":"Adversarially falsify both proposed families, certify exact support saturation where applicable, and adapt the next experiment only from balanced-support evidence.","checks":["Recompute every displayed CRT residue and quadratic-unit invariant with exact integers before accepting any scan output.","Audit complete incidence masks, weakest-first column supports, maximal intersections, and distinctness; label all zero-support conclusions as finite-domain exclusions.","Compare shell-factor and parity structure of any surviving columns, rejecting numerical near-squares and isolated supports.","If an independent primary method produces a genuine candidate, use at most one divisor_completion job solely for exact verification; otherwise request none."]}},{"round":71,"agentId":"orum","privatePlan":{"objective":"Develop a modular local sieve that prunes or forces compatible rows before any expensive search, using the CRT-root and quadratic-unit identities only as sources of residue constraints; do not request divisor_completion.","checks":["For selected primes p, compute exact residue masks S_d(p)={n mod p: d^2+4n is a quadratic residue mod p} and intersect masks across candidate rows.","Derive and test local compatibility conditions for d_i^2−d_j^2 and the two square variables, rejecting families with empty intersections before bigint enumeration.","Require a finite-domain saturation count for every retained residue class, then independently verify parity, positivity, reconstructed factors, distinct N values, and full support masks.","Keep every negative conclusion restricted to the explicitly enumerated modular and traversal domain."]}},{"round":71,"agentId":"pip","privatePlan":{"objective":"Construct a distinct simultaneous-square family from a tunable norm identity or rational-point/elliptic parametrization, avoiding the CRT-root and quadratic-unit palettes.","checks":["Derive and record a literal identity forcing d^2+4N to be a square, with positivity and distinctness conditions.","Define an exact finite family_scan domain and deterministic traversal, explicitly excluding covered registry regions.","Use symbolic pair-intersection conditions to select balanced-support rows before any bounded scan.","Require independent bigint incidence, parity, product, support-mask, and decimal-format audits; report only finite-sample conclusions."]}},{"round":71,"agentId":"solvi","privatePlan":{"objective":"Use only verified incidence cells to build an N–difference hypergraph, then solve exact-cover/SAT instances for balanced 5-by-5 and strict Pareto-improving supports.","checks":["Recompute every cell with bigint square, parity, positivity, and factor-product tests.","Encode distinct-N and distinct-difference constraints explicitly; reject unsupported or near-square edges.","Enumerate maximal support masks and solve 5-column exact-cover/SAT targets before weaker profiles.","Audit completeness against the declared finite verified-cell domain; request no divisor_completion."]}},{"round":71,"agentId":"tess","privatePlan":{"objective":"Run stochastic, optimization-guided discovery of fresh CRT-lift and quadratic-unit seed palettes, then apply exact bigint verification only to Pareto-leading candidates; request no divisor_completion.","checks":["Optimize exact structural proxies for shared factor shells and balanced row diversity, never numerical closeness to squares.","Use deterministic replay of each selected seed and verify every incidence by bigint square, parity, positivity, reconstructed factors, and product checks.","Reject support-1 and support-2 columns as non-progress; recompute complete support masks and biclique claims before retaining any result.","Record finite-exclusion domains and stop conditions separately from any global conjecture."]}},{"round":72,"agentId":"mira","privatePlan":{"objective":"Adversarially falsify both proposed families, then build exact finite saturation certificates and adapt only from balanced-support evidence.","checks":["Independently recompute every displayed difference, factorization identity, positivity, distinctness, and registry-bound separation.","Require a deterministic candidate enumeration specification, exact bigint incidence matrix, deduplicated N columns, and weakest-first support ordering.","Reject isolated and support-2 columns; certify maximal 3x3-or-larger intersections and distinguish check-limit truncation from domain saturation.","For the cofactor shell, compute gcd/residue statistics before further expansion; for the quadratic family, test whether its two gap factors actually recur across multiple rows.","If either family yields balanced support, refine its parameter neighborhood; if both saturate negatively, alter only the algebraic gcd pattern rather than launching an unrelated broad scan."]}},{"round":72,"agentId":"orum","privatePlan":{"objective":"Design a modular sign-code family: choose a modulus M and distinct CRT sign patterns so every selected difference satisfies d_i^2 congruent to d_0^2 modulo M, forcing identical local square-residue conditions for N.","checks":["Enumerate small-prime moduli and sign codes, retaining only five or six distinct positive differences outside covered bounds with an explicit CRT domain.","Certify for each retained N residue that d_i^2+4N is a quadratic residue modulo every prime divisor of M, with exact residue tables.","Define a deterministic bounded lift enumeration and independently recompute every exact bigint square, parity, positivity, and support mask.","Reject any slice lacking five columns with support at least four before considering weaker profiles."]}},{"round":72,"agentId":"pip","privatePlan":{"objective":"Develop a constructive rational-point or elliptic-curve family extending a square-additive four-row seed, varying parameters rather than repeating fixed Bremner specializations.","checks":["Derive and symbolically verify the rational identities defining the family and its square-additive cells.","Establish a nonconstant rational parameter and determine exact denominator-clearing conditions yielding distinct positive integer differences and N-values.","Compare every specialization against the supplied registry and reject covered fixed examples.","Require exact bigint square, parity, positivity, product, and weakest-first support checks before considering any candidate; request at most one divisor_completion job only after an independently derived candidate exists."]}},{"round":72,"agentId":"solvi","privatePlan":{"objective":"Construct the incidence hypergraph from already verified exact cells and solve for K5,5 first, then Pareto-improving 4x5 or 5x4 bicliques.","checks":["Recompute every edge with bigint square, parity, positivity, and product tests.","Encode exact-cover/SAT variables for distinct selected differences and N-columns, with all required incidences enforced.","Apply symmetry breaking and enumerate maximal support profiles, rejecting support-one and support-two anchors.","Independently verify every reported biclique and compare its weakest-first support vector against the ledger."]}},{"round":72,"agentId":"tess","privatePlan":{"objective":"Use stochastic, optimization-guided sampling of new algebraic seed families, then exact bigint verification of only the highest-scoring balanced-support seeds; request no divisor_completion job.","checks":["Sample coprime quadratic, cofactor-shell, and norm-identity parameters outside the registry domains.","Optimize lexicographically for maximum exact row support, number of columns with support at least three, and balanced four- or five-row intersections.","Deduplicate differences and candidate N values before exact testing.","Verify every retained cell by bigint square root, parity, positivity, and reconstructed-product checks.","Recompute complete support masks and record finite domain, stop-loss, and registry overlap for every seed."]}},{"round":73,"agentId":"mira","privatePlan":{"objective":"Adversarially test Pip’s quartic family first, certify only the exact finite traversal actually completed, and adapt toward a cofactor permutation only if balanced-support data—not raw candidate counts—justifies it. Use Orum only for validation; request no divisor_completion job.","checks":["Request one family_scan with params {d1:null,d2:null,differences:\"1001 2004 17049 82244 257769 627876 1300889 2409204 4109289 6581684 10031001 14685924 20799209 28647684 38532249 50777876 65733609 83772564 105291929 130712964\",limit:5000,maxChecks:5000000,startDifference:null,endDifference:null,stride:null}.","Manifest: familyFingerprint=\"quartic-index family A+B t^4, specialized to A=1001 and B=1003\"; registryOverlap=\"new\"; domain=\"t=0,...,19 and the deterministic first 5,000,000 exact pair-completion checks, with distinct positive N\"; completenessTarget=\"bounded-complete slice\"; targetShape=\"5x5 first, then 6x4, 5x4, or 4x5; balanced 4x4 only diagnostically\"; successCriterion=\"Return an independently verified target biclique, or a replayable complete support profile for the tested prefix\"; stopLoss=\"Stop at 5,000,000 checks; if no support-4 column or balanced 4x4 occurs, record finite exclusion only for this slice\"; symbolicIdentity=\"d(s)^2-d(r)^2=1003(s^4-r^4)(2002+1003(s^4+r^4))\".","Recompute all 20 differences, positivity, distinctness, the quartic identity, every square test, parity, reconstructed factors, and products with bigint arithmetic.","Require independent replay of the traversal count/hash and recomputation of every row/column support mask before calling the tested prefix saturated.","Reject numerical near-squares, duplicate N values, isolated anchors, and unbalanced support profiles.","If the quartic slice is negative, select any subsequent primorial permutation solely by maximizing certified balanced intersections; do not broaden bounds without a new manifest."]}},{"round":73,"agentId":"orum","privatePlan":{"objective":"Apply a complete local congruence sieve to quartic and reversed-primorial row subsets before exact lifting.","checks":["Use M=64·3·5·7·11·13=960960 and enumerate, exactly, all residue classes n mod M for which d²+4n is a quadratic residue modulo every factor.","Certify empty intersections and counts for every tested five-row subset, distinguishing modular exclusion from global impossibility.","Lift only surviving CRT classes through deterministic exact bigint tests, enforcing distinct N, parity, positivity, and weakest-first support.","Do not request divisor_completion this round; only consider one later if this primary modular lift first produces a concrete candidate pair."]}},{"round":73,"agentId":"pip","privatePlan":{"objective":"Construct a rational-point or elliptic-curve family extending a verified four-row square-additive rectangle, with the fifth row generated algebraically rather than by an isolated quartic scan.","checks":["Write the simultaneous equations m_{ij}^2-d_i^2=4N_j and eliminate selected N_j to obtain an explicit genus-one quartic or equivalent Weierstrass model.","Generate rational points and specialize only to distinct positive integer differences and columns, outside the registry-covered bounds.","Verify every specialization by exact bigint square tests, parity, factor reconstruction, distinctness, and weakest-first support profiles.","Retain only balanced 4x4-or-better structures and prepare a family_scan manifest; request no divisor-completion job."]}},{"round":73,"agentId":"solvi","privatePlan":{"objective":"Run an exact-cover/SAT search on the union of all already verified incidence cells, seeking a balanced 5-by-5 biclique and Pareto improvements without generating unverified numerical candidates.","checks":["Recompute every imported cell by exact bigint square, parity, positivity, and product tests.","Deduplicate rows and columns, build row and column incidence masks, and enumerate maximal bicliques.","Require five distinct differences and five distinct N values for any claimed witness.","Report complete finite exclusion only for the imported verified-cell hypergraph, with weakest-first support profiles."]}},{"round":73,"agentId":"tess","privatePlan":{"objective":"Use stochastic, optimization-guided sampling of uncovered quartic and primorial-cofactor seeds, rank them by balanced-support proxies, then exact-bigint verify only the best seed through a family scan; request no divisor_completion.","checks":["Sample distinct integer parameter seeds and cofactor permutations, excluding all supplied registry regions and canonical relabeling duplicates.","Rank seeds using integer gcd/divisor structure and modular square-residue filters, optimizing balanced support rather than raw candidate count; use no floating-point closeness.","Before submission, construct a complete family_scan manifest with d1=null, d2=null, a concrete differences string, limit=null, maxChecks=5000000, startDifference=null, endDifference=null, and stride=null.","Recompute every retained cell by exact bigint square testing, parity, positivity, reconstructed factors, and product equality.","Accept only distinct-column 5x5, 6x4, 5x4, or 4x5 incidence; otherwise record the finite bounded result and weakest-first support profile."]}},{"round":74,"agentId":"mira","privatePlan":{"objective":"Adversarially falsify the cubic slice, obtain a deterministic saturation certificate, and adapt only from its balanced-support profile.","checks":["Use a corrected family_scan manifest: d1=null, d2=null, differences=\"1009 3035 11139 31399 69893 132699 225895 355559 527769 748603 1024139 1360455 1763629 2239739 2794863 3435079 4166465 4995099 5927059 6968423\", limit=5000, maxChecks=5000000, startDifference=null, endDifference=null, stride=null.","Verify every generated difference, pair completion, parity condition, reconstructed factor pair, and product using exact bigint arithmetic and decimal-string parsing.","Produce weakest-first row/column supports, all support-at-least-3 columns, and exact four-row/five-row intersections; label failure only as finite exclusion of this slice.","If saturation is below support 4, adapt one parameter family at a time toward homogeneous forms t(t^2+u^2), retaining only choices justified by measured support balance.","Require Solvi’s validation of any claimed rectangle and record a replayable saturation hash; request no divisor_completion job."]}},{"round":74,"agentId":"orum","privatePlan":{"objective":"Build a modular, local congruence sieve for the cubic family before any large scan.","checks":["Compute residue classes of d(t)^2 modulo selected small prime powers.","Intersect local conditions for shared N across four- and five-row subsets.","Discard row subsets with incompatible N residues before exact lifting.","Require a finite saturation certificate for the tested modulus box, then send only survivors to Solvi for exact validation."]}},{"round":74,"agentId":"pip","privatePlan":{"objective":"Construct a genuinely parametrized algebraic family by deforming the common-N equations for three or four difference rows into genus-one curves, then use rational points to generate new columns and test whether two additional rows can be imposed without repeating fixed Bremner specializations.","checks":["Write the exact intersection-of-quadrics system m_i^2-d_i^2=4N and derive its pair-completion factorization symbolically.","Obtain a birational elliptic model for a non-fixed rational deformation, and verify the map in both directions over Q.","Generate rational points, clear denominators, and retain only distinct positive integer differences and N-values outside covered registry regions.","Test every incidence by exact bigint arithmetic, recompute weakest-first supports, and require a verified 4x5, 5x4, 6x4, or 5x5 structure before claiming progress.","Record rank, specialization, exclusions, and failed rational points separately; do not infer impossibility from a finite search."]}},{"round":74,"agentId":"solvi","privatePlan":{"objective":"Build an incidence-hypergraph/SAT exact-cover search from all currently verified cells, targeting a balanced 5x5 biclique and strict 4x5 improvements.","checks":["Recompute every imported cell with bigint square, parity, positivity, and product tests.","Encode distinct differences and distinct N-columns as exact incidence constraints; rank solutions weakest-column-first.","Check all 5-row/5-column and 4-row/5-column witnesses independently, reporting only finite conclusions for the verified-cell universe.","Request no divisor_completion job unless an independently generated candidate requires verification."]}},{"round":74,"agentId":"tess","privatePlan":{"objective":"Run stochastic, optimization-guided discovery over registry-separated cubic and primorial-shell seeds, then perform exact bigint verification of the highest-balanced-support candidates.","checks":["Generate deterministic-seed mutations of scales, cofactor permutations, and cubic index parameters while excluding covered N and difference regions.","Optimize lexicographically for weakest-column support, then total support and maximal 4-row intersections.","Submit only valid family_scan jobs with all parameter keys populated, integer limit, and maxChecks within bounds.","Verify every incidence by exact square, parity, positivity, reconstructed-factor, and product tests.","Report only verified bicliques or bounded-complete finite exclusions."]}},{"round":75,"agentId":"mira","privatePlan":{"objective":"Adversarially falsify the elliptic scaling claim, obtain exact saturation certificates for the active fixed families, and adapt the next bounded experiment only from its balanced-support profile.","checks":["Normalize and independently recompute all elliptic coordinates, numerators, gcds, and parity; classify every returned biclique as homothetic or genuinely new.","Require a deterministic complete support histogram, maximal biclique dimensions, and check-count certificate for each bounded slice.","Use Tess’s exact bigint validation for every surviving cell and reject numerical near-squares or representation artifacts.","If support saturates below four, change the algebraic family or cofactor assignment rather than merely enlarging the budget; if support reaches four, target the missing fifth row or column adaptively.","Do not request divisor_completion."]}},{"round":75,"agentId":"orum","privatePlan":{"objective":"Design a modular local sieve that forces retained candidate columns into residue classes simultaneously admissible for at least five selected differences, then test exact support only inside those classes.","checks":["For small odd primes and powers of two, enumerate R_l(d)={n mod l: d^2+4n is a quadratic residue mod l} exactly.","Use CRT to enumerate common residue classes for five-row difference sets; record nonemptiness, density, and complete local saturation.","Apply the sieve before exact bigint square tests and report all surviving support masks, with no divisor_completion request.","Independently verify every surviving cell by exact square, parity, positivity, reconstructed factors, and product checks.","Treat the result as bounded-complete only over the explicitly enumerated residue classes and parameter range."]}},{"round":75,"agentId":"pip","privatePlan":{"objective":"Develop a genuinely varying elliptic or genus-one parametrization for simultaneous factor differences, rather than another fixed scaled five-row specialization.","checks":["Eliminate N from three- and four-row equations and identify the resulting rational curve or elliptic curve exactly.","Generate rational points by varying the point or multiple, then canonicalize all numerators and denominators before imposing parity and integrality.","Test distinctness and exact square identities symbolically and with bigint arithmetic; submit only a family_scan, with Tess as validation-only verifier."]}},{"round":75,"agentId":"solvi","privatePlan":{"objective":"Using only Tess-verified cells from the accepted family scans, construct the incidence hypergraph and search for exact 5×5, 4×5, or 5×4 bicliques via SAT/exact-cover methods.","checks":["Normalize every verified cell into row and column bitmasks, rejecting duplicate N or difference labels.","Enumerate row-subset intersections and solve the corresponding column exact-cover instances, ordered by weakest-first support.","Independently recompute the claimed rectangle dimensions and distinctness constraints before recording any candidate.","Report complete search coverage, maximal biclique dimensions, and finite exclusions without requesting divisor_completion."]}},{"round":75,"agentId":"tess","privatePlan":{"objective":"Run the rotating stochastic lane: generate algebraically specified five-row seeds by annealed or cross-entropy mutation of square-cofactor and elliptic parameters, optimize exact modular-support surrogates, then bigint-verify the strongest survivors.","checks":["Canonicalize all elliptic fractions and reject arbitrary unreduced-numerator or common-scaling duplicates.","Use modular tests only as necessary filters; verify every claimed cell with exact square, parity, positivity, factor, and product checks.","Recompute distinct differences, distinct N columns, support masks, and strict 5x5/4x5/5x4 objectives.","Record registry overlap and finite tested domains explicitly; request no divisor_completion unless a concrete candidate is first produced by this primary method, and then use at most one verification job."]}},{"round":76,"agentId":"mira","privatePlan":{"objective":"Construct a genuinely varying rational-point family of square-additive rectangles: derive an elliptic/genus-one fiber for adding a fifth difference, then vary the base parameters rather than repeating fixed Bremner specializations.","checks":["Derive and symbolically verify the rational parametrization and its elliptic-curve group law.","Use exact rational arithmetic, clear denominators primitively, and reject coincident rows or columns.","Verify every resulting cell by bigint square, parity, positivity, and factor-product tests, with complete support masks.","Do not request divisor_completion yet; reserve at most one such job only after an independently derived candidate exists."]}},{"round":76,"agentId":"orum","privatePlan":{"objective":"Build an incidence hypergraph from independently verified exact cells returned by the new family scans, then search it by SAT or exact cover for 5x5, 6x4, and 4x5 bicliques.","checks":["Recompute every cell with bigint squarehood, parity, positivity, and factor-product tests before inserting it into the hypergraph.","Encode each candidate rectangle with distinct-row and distinct-column constraints; require all selected incidences, not merely modular compatibility.","Use deterministic weakest-first support certificates and independently recheck every reported biclique.","Reject any result whose domain or traversal overlaps the supplied registry without an explicit exact-overlap audit."]}},{"round":76,"agentId":"pip","privatePlan":{"objective":"Design a genuinely varying modular sieve that forces compatible local square roots across at least five difference rows, while excluding scaled Bremner and the tested CRT ±1 slice.","checks":["For selected prime powers Q, compute exact sets S_i={n mod Q: d_i^2+4n is a quadratic residue mod Q} and their five-row intersections.","Enumerate compatible residue templates for d_i and m_i, including sign patterns, then Hensel-lift them across prime powers.","Require distinct positive rows and nondegenerate residue classes; treat local compatibility only as a pruning condition, never as an exact-square claim.","Lift surviving N residues to bounded integers and verify every cell by exact bigint square and factor-product tests.","Record the finite domain, registry exclusions, support profile, and any failure without making a global impossibility claim."]}},{"round":76,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery to generate new balanced-support difference families, then submit only high-scoring seeds for exact bigint verification; request no divisor_completion job.","checks":["Score seeds by predicted weakest-column and weakest-row support, penalizing automatic 4×6 constructions and registry duplicates.","Use randomized modular/affine perturbations outside covered bounds while retaining exact symbolic provenance.","Verify every surviving candidate with bigint square, parity, positivity, factor-product, distinctness, and complete support-mask checks.","Stop only on a certified 5×5 improvement or a bounded-complete support profile."]}},{"round":76,"agentId":"tess","privatePlan":{"objective":"Adversarially falsify both proposed mechanisms, obtain exact saturation certificates for their finite incidence structures, and adapt the next experiment only from weakest-first biclique support.","checks":["Enumerate the six affine translated columns symbolically, then recompute every cell, row mask, column mask, and maximal biclique exactly; distinguish forced four-cell layer support from accidental cross-layer cells.","Certify separately the embedded scaled Bremner incidences and any genuinely new incidences; record finite exclusion only for the explicitly enumerated affine orbit.","For the CRT family, apply the exact quadratic-residue sieve modulo 120120 before interpreting any returned support, and test whether paired rows add support beyond the common-residue baseline.","Choose the next algebraic parameter regime conditionally: vary a translation parameter if layer separation saturates, or alter the CRT modulus/residue pattern only if the exact support profile improves.","Require bigint square, parity, positivity, factor-product, distinctness, and deterministic-traversal checks; request no divisor-completion job."]}},{"round":77,"agentId":"mira","privatePlan":{"objective":"Construct a new rational-point/elliptic family of square-additive rectangles, with parameters varying independently rather than another fixed row list.","checks":["Derive and symbolically verify the elliptic or genus-one identities, including integrality, parity, positivity, and distinctness conditions.","Enumerate bounded rational specializations with exact bigint arithmetic, recording every incidence and rejecting numerical near-squares.","Search the resulting exact incidence hypergraph for 5×5, 6×4, and 4×5 support without requesting divisor_completion."]}},{"round":77,"agentId":"orum","privatePlan":{"objective":"Construct an incidence hypergraph from independently verified exact cells in Pip’s eight-branch CRT family, then search it for exact 5×5, 6×4, and 4×5 bicliques.","checks":["Recompute every listed difference and every accepted cell with bigint perfect-square tests and reconstructed positive factor pairs.","Deduplicate N columns and publish row-support and column-support vectors before search.","Run support-prioritized SAT/exact-cover constraints for the three target shapes, requiring distinct rows and columns.","Emit a machine-checkable certificate or a bounded-complete support profile with the exact checked-domain and truncation status."]}},{"round":77,"agentId":"pip","privatePlan":{"objective":"Design a modular local sieve for shifted-Pell and CRT-derived rows: enumerate prime-power residue classes of N and retain only classes supporting at least five selected differences simultaneously.","checks":["For each modulus, verify every retained condition by exact square-root enumeration rather than numerical approximation; combine compatible classes with CRT.","Prioritize residue classes forcing structured roots m (including divisibility conditions), then generate a candidate row set and N class before any factor completion.","Exclude all registry-covered difference and fixed-family regions, and record finite-domain completeness separately from exploratory lifting.","Request at most one divisor_completion job only after the modular sieve produces a concrete candidate column; otherwise perform no completion job."]}},{"round":77,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery for new affine-recursive and modular-hybrid difference families, then send only the best non-overlapping family_scan candidates for exact bigint verification by Pip.","checks":["Sample reproducible seeds outside the covered small-difference, fixed-Bremner, and exhausted CRT ±1 regimes.","Rank seeds by modular incidence upper bounds and predicted balanced support; never use numerical closeness to squares.","Deduplicate rows and prospective columns, then retain Pareto improvements in weakest-first support.","Before any claim, verify d²+4N exactly, parity, positive reconstructed factors, distinct rows and columns, and the complete support profile.","Do not request divisor_completion; issue a family_scan manifest only after seed selection, with all irrelevant parameters explicitly null."]}},{"round":77,"agentId":"tess","privatePlan":{"objective":"Adversarially falsify the CRT and affine-Pell proposals, then adapt experiment design from verified saturation data toward an uncovered high-support boundary.","checks":["Recompute every proposed row, recurrence identity, parity condition, and exact bigint square incidence.","Audit row-family collisions against the supplied registry before accepting the claimed novelty.","Require support histograms, maximal 5×5/6×4/4×5 intersections, and explicit truncation certificates.","If saturation is weak, select the next family or modulus from observed high-support residue classes rather than expanding an arbitrary range.","Do not request divisor_completion; use only exact family-scan evidence and never infer results from numerical closeness."]}},{"round":78,"agentId":"mira","privatePlan":{"objective":"Develop a constructive rational-point family of square-additive rectangles, using the c=10054 CRT branch as a congruence sieve rather than as a standalone search; prioritize a parameterized extension from a verified lower-dimensional rectangle toward five common differences.","checks":["Write the shared-cell equations as an intersection of quadrics and identify an explicit rational parametrization or elliptic curve, with its group-law map stated symbolically.","Impose the CRT branch and derive exact integrality, positivity, distinctness, and parity conditions after denominator clearing.","Verify every generated cell by bigint square identities and reconstructed factor pairs; reject numerical near-squares.","Compare parameters against the registry and fixed Bremner specializations, recording only genuinely new families.","Use Orum for validation only; do not request divisor_completion."]}},{"round":78,"agentId":"orum","privatePlan":{"objective":"Construct an incidence hypergraph from independently verified cells in the fresh c=10054 CRT family and search it by exact-cover/SAT constraints for 5×5, 6×4, or 4×5 support.","checks":["Recompute every bigint square root and factor witness before inserting a cell.","Enforce distinct positive N and distinct differences, then recompute weakest-first column supports.","Use exhaustive exact-cover/SAT enumeration on the verified finite incidence set; record a finite-exclusion certificate if no target exists.","Request no divisor_completion job unless an independently generated candidate requires final verification."]}},{"round":78,"agentId":"pip","privatePlan":{"objective":"Construct a modular local sieve for CRT-derived difference rows, retaining residue classes of N that can support at least five selected differences simultaneously.","checks":["Enumerate exact square classes d² modulo several coprime prime powers and intersect compatible N residues.","Separate forced divisibility of m from actual simultaneous solvability and reject unsupported branches early.","Test new sign-pattern and non-global-square residue families without repeating the c=1 or registry-covered domains.","Use exact bigint verification only after modular pruning; reserve Orum for validation."]}},{"round":78,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery on fresh CRT/local-residue and affine-parameter mutations, then submit only the highest exact-support seeds for bigint verification.","checks":["Sample novel prime-power residue patterns and parameter mutations while rejecting registry-overlapping difference regimes.","Optimize the exact support vector, prioritizing 5×5 and then strict Pareto improvements such as 4×5.","Verify every cell with bigint square tests, parity, reconstructed positive factors, distinctness, and complete support recomputation.","Record bounded failures precisely; use family_scan discovery only and route validation-only work to Orum, with no divisor_completion request."]}},{"round":78,"agentId":"tess","privatePlan":{"objective":"Adversarially falsify the new CRT family, seek exact saturation certificates, and adapt the next family experiment from observed support patterns without requesting divisor_completion.","checks":["Recompute the complete CRT root set and independently test every displayed difference for distinctness and registry non-overlap.","Search exact incidences by residue branch, then test maximal five-row and five-column intersections; record weakest-first supports and certify saturation or failure.","Use Orum for validation-only replay of any candidate; distinguish check-limit truncation from finite-domain completeness.","If support concentrates in a residue class, design the next family by varying the local square class or modulus rather than repeating the c=1 construction."]}},{"round":79,"agentId":"mira","privatePlan":{"objective":"Construct new parametrized square-additive families for a fifth row, emphasizing elliptic curves and rational points outside the registry’s covered regimes.","checks":["Derive exact identities before enumerating rational parameters.","Use deterministic bigint verification of every support cell and reject covered specializations.","Test whether the family yields a genuine 5-by-5 support pattern, recording finite-domain completeness and falsifications."]}},{"round":79,"agentId":"orum","privatePlan":{"objective":"Run an incidence-hypergraph/exact-cover search using only independently verified square-additive cells, seeking a 5-by-5 all-support certificate outside the covered registry regions.","checks":["Regenerate every cell with deterministic bigint arithmetic and verify d^2+4N=m^2 with valid positive factors.","Reject registry-covered parameter regions and duplicate N or d values before search.","Require an exact 5-column, 5-row cover and emit the complete incidence certificate for independent checking.","If no cover exists, record only finite-domain exclusion with explicit domain bounds."]}},{"round":79,"agentId":"pip","privatePlan":{"objective":"Develop modular, local congruence filters that prune or force compatible fifth rows without repeating registry-covered regions.","checks":["For each tested row, verify d^2+4N=m^2 exactly with bigint arithmetic and parity conditions.","Intersect quadratic-residue constraints modulo selected small primes before any large-integer test.","Enforce distinct positive N and d values and record the exact finite tested domain.","Compare every surviving parameter slice against the 2026 registry before interpreting it as new evidence."]}},{"round":79,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery in an uncovered parameter regime, then perform exact bigint verification of support.","checks":["Exclude all registry-covered regions before seeding.","Rank seeds by exact or cheaply certifiable support potential, never numerical closeness.","Verify every proposed N,d incidence via exact perfect-square tests and distinctness checks.","Do not request divisor completion unless another primary method first produces a candidate; if requested, use it only as independent verification."]}},{"round":79,"agentId":"tess","privatePlan":{"objective":"Run adversarial falsification and saturation checks on uncovered algebraic or boundary regimes, without divisor_completion.","checks":["Require exact bigint support counts and perfect-square identities for every candidate.","Seek saturation certificates showing each tested family or adaptive slice is exhausted, and distinguish bounded exclusion from global impossibility.","Adapt the next experiment toward the weakest-supported boundary regime revealed by exact results."]}},{"round":80,"agentId":"mira","privatePlan":{"objective":"Construct a genuine rational-point family by parametrizing the three-column intersection y_i²=x²+4A_i, then use elliptic group-law multiples and denominator clearing to generate integral rows and columns outside covered bounds.","checks":["Derive the birational elliptic model and verify the inverse map symbolically on every generated rational point.","For each multiple, clear denominators with exact gcd normalization and test d²+4N=m², parity, positivity, distinctness, and support masks by bigint arithmetic.","Intersect the resulting family with paired CRT residue classes only as a local pruning step, never as evidence of an integer square.","Record finite exclusions separately from any certified biclique, with Tess performing the final independent validation."]}},{"round":80,"agentId":"orum","privatePlan":{"objective":"Build an incidence-hypergraph and exact-cover search over only exact verified cells returned by the new scans, targeting Pareto-optimal 5-row and 5-column supports.","checks":["Canonicalize every verified cell with bigint square, parity, positivity, and factor-product checks.","Construct row and column bitmasks; search SAT/exact-cover instances for 5x5, 4x5, and 5x4 bicliques.","Emit saturation certificates: complete candidate intersections, weakest-first support ordering, and deterministic domain counts.","Have Tess perform validation-only recomputation of any surviving certificate."]}},{"round":80,"agentId":"pip","privatePlan":{"objective":"Design a genuinely new paired-CRT row family whose local square classes force a common N residue while avoiding all fixed Bremner specializations.","checks":["Enumerate small prime-square moduli and solve CRT systems for two or more distinct d² residue classes with one common positive N residue.","Reject shells with repeated differences, covered difference ranges, or residue classes equivalent to prior scans.","Prove the local congruence identities and parity conditions symbolically before any bounded exact family scan.","Use only a family_scan request, with an explicit finite difference list and deterministic N-residue filter; reserve Tess for validation."]}},{"round":80,"agentId":"solvi","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery across perturbed height and CRT shells outside covered regions, then perform exact bigint verification of the highest-support seeds.","checks":["Optimize weakest-first support potential using modular admissibility and distinctness, never numerical square closeness.","Exclude registry-covered ranges and enforce differences above 50000 and columns above 3000000 where applicable.","For retained seeds, verify every d²+4N square, parity, positivity, factor product, distinctness, and complete support matrix exactly.","Record finite domains and deterministic seeds; request no divisor_completion, with Tess restricted to validation."]}},{"round":80,"agentId":"tess","privatePlan":{"objective":"Adversarially falsify the claimed family mechanisms, then adapt the next exact experiment to the weakest uncovered support while seeking saturation certificates.","checks":["Recompute every displayed CRT residue, parity condition, and exact square incidence with deterministic bigint arithmetic.","Test each of Mira’s displayed differences directly against all three fixed A_i; reject the genus-one label unless the claimed common-support structure is present.","Enumerate complete row masks, weakest-first column supports, and all maximal 4-row and 5-row intersections for each proposed finite family.","Select any follow-up shell only from uncovered support deficits; request divisor_completion at most once, and only to verify a candidate first generated by a primary scan."]}},{"round":81,"agentId":"mira","privatePlan":{"objective":"Develop a modular sign-allocation filter that forces compatible witness residues across several rows before any global square search.","checks":["Verify the paired CRT residues and derive all admissible N classes prime-by-prime.","Enumerate compatible choices of m_i−m_j versus m_i+m_j modulo selected prime powers, rejecting inconsistent row tuples.","Use CRT lifts to generate only row families with prescribed local compatibility, then apply exact bigint square tests.","Request no divisor_completion job unless a candidate incidence is first produced by this modular primary method."]}},{"round":81,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery to search fresh paired-CRT and algebraic parameter seeds, then perform exact bigint verification for Pareto-improving support patterns.","checks":["Reject all registry-covered differences and N domains before scoring seeds.","Optimize weakest-first row and column support, prioritizing 4×5 and 5×5 incidence potential.","For every retained seed, verify exact square identities, parity, positive factor witnesses, distinctness, and deterministic traversal counts.","Record bounded exclusions separately from verified bicliques; do not request divisor_completion."]}},{"round":81,"agentId":"pip","privatePlan":{"objective":"Use only verified cells to build an incidence hypergraph and search exact-cover/biclique structures in the new CRT shell.","checks":["Revalidate every cell by exact square, parity, positivity, and factor-product checks.","Deduplicate rows and columns, compute incidence bitsets, and enumerate maximal 5×5, 4×5, 5×4, and 6×4 bicliques.","Sort supports weakest-first and record a complete certificate or bounded-complete finite exclusion for the checked cell domain.","Require independent recomputation of distinctness, masks, and all claimed dimensions before escalation."]}},{"round":81,"agentId":"solvi","privatePlan":{"objective":"Use adversarial falsification, saturation certificates, and adaptive experiment design on the two proposed CRT shells without requesting divisor_completion.","checks":["Require deterministic enumeration metadata and independently recompute every CRT identity, parity condition, exact square, factor witness, and support mask.","Extract maximal four-row and five-row intersections, column-support histograms, and stopping-time saturation data; treat truncation as non-exclusion.","If both shells saturate below target, adapt the next family by selecting residues that maximize exact pair-completion density rather than adding primes solely for local admissibility.","Track Tess as validation-only and require an independent audit before accepting any claimed rectangle."]}},{"round":81,"agentId":"tess","privatePlan":{"objective":"Develop a genuine rational-point family: parametrize common completions for three differences by an elliptic quartic, then seek group-law-generated columns and an algebraic fourth- or fifth-row extension outside covered regimes.","checks":["For fixed distinct d,e,f, derive and verify the exact quartic y²=(t²+Δ)²+4t²(f²−d²), with Δ=d²−e², and its birational elliptic model.","Enumerate and independently verify rational points, excluding singular, zero, duplicate, and sign-equivalent cases.","Clear denominators and check positivity, integrality, parity, distinctness, and every cell by exact bigint arithmetic.","Require a verified 4×5 or 5×5 support profile before any bounded family scan; do not request divisor_completion."]}},{"round":82,"agentId":"mira","privatePlan":{"objective":"Design a new modular shell whose local root/sign assignments are simultaneously compatible across at least five rows, without requesting divisor_completion.","checks":["Factor and verify the modulus and all row residues exactly, including the 2-adic case.","Enumerate local square roots for candidate column residues and discard incompatible sign vectors before bigint testing.","Use CRT to derive explicit column congruence classes, then test only distinct positive columns in a new finite slice.","Require exact square witnesses, reconstructed factor products, and weakest-first support profiles; treat any negative result as finite-slice exclusion."]}},{"round":82,"agentId":"orum","privatePlan":{"objective":"Generate stochastic, optimization-ranked seeds from fresh affine/AP–CRT hybrid families outside the registry, then route the highest-support seeds to Pip for deterministic exact bigint verification; request no divisor_completion job.","checks":["Reject registry-overlapping parameter regions and rank seeds lexicographically by estimated weakest-column support and Pareto improvement.","Use modular filters only for pruning; require exact d²+4N=m², parity, positivity, reconstructed factors, distinctness, and complete incidence masks.","Have Pip certify every retained candidate and distinguish verified bicliques from bounded-complete failures or truncation."]}},{"round":82,"agentId":"pip","privatePlan":{"objective":"Run an exact incidence-hypergraph/SAT search over independently verified cells, prioritizing Mira’s p-adic shell and recording local sign-allocation masks before biclique search.","checks":["Recompute every cell by exact bigint square testing, parity, positivity, and factor-product reconstruction.","Deduplicate verified (d,N) edges, then enumerate complete 5×5, 4×5, 5×4, and 6×4 bicliques over the declared finite edge set.","Use branch-and-bound or SAT constraints on row and column supports; reject numerical near-squares and report weakest-first supports.","Do not request divisor_completion unless a candidate is first produced independently; then use at most one verification-only request."]}},{"round":82,"agentId":"solvi","privatePlan":{"objective":"Run an adversarial, saturation-focused round that tests whether either proposed family has genuine high-support structure rather than merely useful necessary conditions.","checks":["Require an explicit finite candidate generator for Mira’s shell, then independently verify every retained cell and all local sign allocations.","For Pip’s progression, enumerate every positive factor pair of each square gap, enforce parity and positivity, and certify that no pair completion is omitted.","Compute complete row and column support masks, weakest-first profiles, and all maximal 4- and 5-row intersections in the declared domains.","Adapt the next family choice only from these support ceilings and modular incompatibilities; use Pip solely for validation and request no divisor_completion job."]}},{"round":82,"agentId":"tess","privatePlan":{"objective":"Construct a genuinely new rational-point family of square-additive rectangles, using the projective equations m_{ij}^2=d_i^2+4N_j and an elliptic-curve model for extending a variable 4-row or 4-column configuration. Target a verified 5×4 improvement first, then seek a fifth column.","checks":["Derive and verify birational maps between the rectangle equations and the proposed genus-one model using exact rational arithmetic.","Verify every rational point by substitution, positivity, distinctness, and exact square identities; clear denominators uniformly via d' = Ld, m' = Lm, N' = L^2N.","Compute rank, torsion, and saturation or an explicitly bounded rational-point search; do not infer results from numerical closeness.","Recompute all incidence masks and weakest-first support profiles with deterministic bigint arithmetic, excluding fixed registry examples and their specializations.","If the curve is singular or higher genus, record that obstruction and retain only rigorously verified rational parametrizations or finite exclusions."]}},{"round":83,"agentId":"mira","privatePlan":{"objective":"Design a rotating-lane modular sieve that couples several proposed rows through simultaneous local root and sign constraints, seeking compatible five-row residue patterns before exact validation by Orum.","checks":["Enumerate admissible roots of x²≡d²+4N modulo selected prime powers for each row and intersect column residue classes.","Test CRT compatibility of sign assignments across at least five rows, rejecting patterns with empty local intersections.","Choose a new difference family whose residues impose shared column classes while remaining outside the supplied registry.","Use only modular/local pruning and congruence design; defer all divisor completion and send any survivors solely to Orum for exact validation.","Record the exact finite domain, support objective, and falsifiable stopping condition before any scan."]}},{"round":83,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided sampling of untested p-adic/CRT sign states and parameter seeds to discover high-support candidates, then certify every surviving cell with exact bigint arithmetic.","checks":["Exclude all registry-covered parameter regions and score seeds by weakest-first exact-support potential before verification.","For each selected seed, recompute d^2+4N, perfect-square status, parity, positive factor reconstruction, distinctness, and complete incidence masks.","Maintain a bounded-complete seed-domain certificate with deterministic replay data and distinguish truncation from exclusion.","Request at most one divisor_completion job only after an independently generated candidate exists, solely for exact verification."]}},{"round":83,"agentId":"pip","privatePlan":{"objective":"Run a deterministic incidence-hypergraph/SAT or exact-cover search over independently verified cells from the accepted shells, without requesting divisor_completion.","checks":["Admit only cells with exact bigint square identities and reconstructed positive factor pairs.","Deduplicate N-columns and difference rows, enforcing all five-by-five distinctness conditions.","Enumerate maximal 5×5, 6×4, and 4×5 bicliques with weakest-first support profiles.","Compare any frontier improvement against the registry and label negative results only for the tested finite slice.","Reject modular-only witnesses, numerical near-squares, and unsupported columns."]}},{"round":83,"agentId":"solvi","privatePlan":{"objective":"Adversarially falsify both shells, derive saturation certificates for their attainable row and column supports, and adapt the next experiment only if an exact support frontier survives.","checks":["Independently derive the Mira shell’s sign-state equations and test whether simultaneous support across five distinct t-values is arithmetically possible.","For Pip’s shell, bound exact completion multiplicities after imposing M|N and identify whether any column can support four or five rows.","Require deterministic bigint verification of every surviving incidence, factor reconstruction, support mask, and distinctness condition; reject all near-square evidence.","Compare the resulting maximal support profiles and select the next finite slice only when it improves the current Pareto frontier without overlapping the registry."]}},{"round":83,"agentId":"tess","privatePlan":{"objective":"Construct an exact algebraic bridge from Mira’s p-adic shell to a rational-point or elliptic-curve family capable of extending a four-row rectangle by a fifth row, while avoiding the covered denominator regime.","checks":["Normalize simultaneous equations as x_i^2-t_i^2=n and derive the genus-one model for a fifth-row extension.","Compute rational maps, singularities, torsion, and rank using exact arithmetic; do not infer rank or solutions from numerical proximity.","Enumerate rational points with denominator greater than 10000 or genuinely new height, clear denominators, and enforce distinct positive N and differences.","Verify every resulting incidence by exact bigint square tests, factor reconstruction, and weakest-first support scoring.","Record any failure only as a finite exclusion of the explicitly parameterized slice."]}},{"round":84,"agentId":"mira","privatePlan":{"objective":"Construct a modular local sieve that preselects five-row difference tuples with simultaneous witness residues before any large exact scan.","checks":["For candidate rows, solve m_i²≡d_i²+4N modulo 8, 9, 5, 7, and 11, recording compatible sign branches.","Use CRT to require one shared N residue and reject row tuples with incompatible local branches.","Derive exact parity and divisibility conditions for the resulting parameterized witnesses, then retain only tuples admitting at least five common local states.","Request no divisor_completion; send only the surviving family to a bounded family_scan, with Solvi used solely for validation."]}},{"round":84,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery across the Q=729 shell and smooth-step progression, then perform exact bigint verification for maximal support.","checks":["Generate randomized seeds emphasizing shared five-row intersections and Pareto improvements over balanced 4×5 structures.","Reject all registry-covered differences and N regimes before scoring.","Verify every retained cell by d²+4N=m², parity, positive factor reconstruction, and exact products.","Recompute weakest-first row and column supports and certify any 5×5, 6×4, or 4×5 result deterministically.","Do not request divisor_completion; use only family-level seed discovery followed by exact validation."]}},{"round":84,"agentId":"pip","privatePlan":{"objective":"Use an incidence-hypergraph/SAT exact-cover search over independently bigint-verified cells from the accepted modular and progression families, targeting 5×5 first and strict 6×4 or 4×5 alternatives.","checks":["Recompute every cell via d²+4N=m², parity, positive factor reconstruction, and exact products.","Canonicalize distinct rows and columns, build row and column support masks, and prune by weakest-first support.","Run deterministic SAT/exact-cover branching for 5×5, 6×4, and 4×5 bicliques.","Log check counts and report only bounded-complete conclusions for the declared verified-cell domain."]}},{"round":84,"agentId":"solvi","privatePlan":{"objective":"Adversarially compare the 3^6 shell and the 27720-step progression using exact support saturation and adaptive allocation, while reserving divisor completion only for a candidate independently produced by a primary search.","checks":["Re-derive each family’s admissible witness parametrization, then test parity, positivity, distinctness, and exact square identities with deterministic bigint arithmetic.","Construct saturation certificates: canonical traversal counts, complete row and column masks, weakest-first support profiles, maximal bicliques, and explicit obstruction records for failed targets.","Use early 4-row/5-row and 4-column/5-column support statistics to allocate the next experiment toward the family with the strongest Pareto frontier.","If an independent primary method produces a candidate, request at most one targeted divisor_completion job solely to verify it; otherwise request none."]}},{"round":84,"agentId":"tess","privatePlan":{"objective":"Construct a new algebraic-family route by parametrizing simultaneous square conditions through rational points on conics and their elliptic-curve intersections, targeting five-row shared columns outside the registry.","checks":["Derive literal identities for the proposed rational/integer parametrization and impose positivity, parity, and distinctness symbolically.","Compute exact local compatibility conditions across at least five differences before any search.","Convert admissible rational points to integer columns and verify every incidence via d^2+4N=m^2 and reconstructed factors.","Compare the resulting domain against the registry and record only deterministic finite exclusions or verified bicliques."]}},{"round":85,"agentId":"mira","privatePlan":{"objective":"Design a modular CRT sieve that forces compatible five-row residue classes while explicitly retaining the divisibility and parity branches needed for genuine completions.","checks":["Enumerate exact square classes for d^2+4N modulo 3^r, 5, 7, 11, and selected coprime moduli; retain only N classes supporting all five rows.","For d=2Qv+ε and m=2Qu+δ, solve the parity and divisibility conditions symbolically and reject branches where the resulting N is not divisible by the intended modulus.","Compare lower and higher 3-adic shells by exact residue counts, prioritizing classes with simultaneous local witnesses rather than merely equal d^2 residues.","Prepare a bounded family_scan only after the residue table is complete; use Tess solely for deterministic validation and do not request divisor_completion."]}},{"round":85,"agentId":"orum","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery over new parameterized difference families, ranking seeds by predicted multi-row and multi-column support, then perform exact bigint verification only on the highest-scoring seeds.","checks":["Generate non-covered seeds by perturbing smooth quadratic shells, congruence-aligned shells, and mixed local-residue templates; reject registry overlap.","Optimize a surrogate support score using exact modular filters and pairwise gap factorization, targeting balanced 5x5 before 4x5 or 6x4.","Run family_scan only, with all eight parameter keys explicit and maxChecks≤5000000; do not request divisor_completion.","For every retained incidence, verify d^2+4N=m^2, parity, positive factor reconstruction, distinctness, row/column masks, and weakest-first support exactly.","Treat failures strictly as finite exclusions and pass any surviving candidates to Tess for validation."]}},{"round":85,"agentId":"pip","privatePlan":{"objective":"Construct an incidence hypergraph from independently verified cells in the two new families and solve exact-cover/SAT instances for 5×5, then strict 6×4 and 4×5 support.","checks":["Have Tess independently verify every retained cell using exact bigint square tests, parity, positivity, and reconstructed factor pairs.","Deduplicate all rows and columns; compute complete incidence masks and weakest-first support profiles.","Reject near-squares and require SAT/exact-cover selections to use distinct differences and distinct N values.","Emit a bounded, replayable certificate or a finite-exclusion report for the tested verified-cell hypergraph."]}},{"round":85,"agentId":"solvi","privatePlan":{"objective":"Adversarially falsify both proposed families, then build saturation certificates and choose the next experiment from exact support density rather than residue plausibility.","checks":["Recompute every displayed difference, symbolic identity, parity condition, and residue claim independently.","For each family, audit the first bounded slice for duplicate columns, unsupported anchors, witness parity, and exact row/column masks; certify the maximal balanced biclique or finite exclusion.","Compare support saturation per exact completion and test whether local alignment or quadratic gap factorization actually improves the Pareto frontier.","Reject any family whose observed support remains below the best verified frontier, then adapt the next family_scan only to the densest surviving parameter regime."]}},{"round":85,"agentId":"tess","privatePlan":{"objective":"Construct a new rational/elliptic parametrization of simultaneous cells, seeking a reusable family with five prescribed differences rather than scanning residue shells.","checks":["Derive an explicit identity for N and all m_i from a common rational-point parameter, then impose integrality, positivity, parity, and distinctness symbolically.","Search rational points on the resulting intersection of quadrics for a strict support improvement, with exact bigint verification of every cell and no inference from numerical near-squares.","Use at most one divisor_completion job only after an independently derived candidate exists, solely to verify its remaining completion."]}},{"round":86,"agentId":"mira","privatePlan":{"objective":"Run an incidence-hypergraph/exact-cover search using only deterministically verified cells, targeting five differences shared by five distinct N values.","checks":["Recompute every cell via d^2+4N=m^2 with parity and positivity checks.","Reject duplicate differences or N values and verify the complete 5-by-5 incidence matrix before claiming success.","Avoid all registry-covered parameter regions."]}},{"round":86,"agentId":"orum","privatePlan":{"objective":"Recover with an adversarial, non-validation experiment targeting an uncovered algebraic family or parameter regime, with an exact saturation certificate.","checks":["Confirm the proposed domain is disjoint from the 2026 registry, including all listed bounded searches and fixed Bremner specializations.","Require deterministic bigint perfect-square tests for every support cell and report the complete support vector, sorted weakest-first.","Use adaptive allocation: first test a cheap falsification slice; expand only if it yields a certified near-saturated rectangle.","Reject numerical near-squares, duplicate differences or columns, and any claim lacking finite-domain completeness and explicit failure counts."]}},{"round":86,"agentId":"pip","privatePlan":{"objective":"Run optimization-guided stochastic seed discovery in an uncovered family or parameter regime, followed by deterministic bigint support verification.","checks":["Exclude all registry-covered regions before sampling.","Score only exact square identities and Pareto column supports.","Verify every survivor with exact integer arithmetic and distinctness checks.","Do not request divisor_completion; reserve validation-only work for Mira."]}},{"round":86,"agentId":"solvi","privatePlan":{"objective":"Construct a new rational-point parametrization for simultaneous cells N=x(x+d), seeking a five-row common-support family outside the covered searches.","checks":["Derive the parametrization symbolically and verify every claimed cell by exact square identities.","Prove distinctness and positivity of all differences and N-values.","Compare the resulting parameter domain against the registry to exclude overlap.","Run exact bigint support scoring; reject any candidate lacking five-by-five support."]}},{"round":86,"agentId":"tess","privatePlan":{"objective":"Design a modular, local congruence filter for compatible rows without requesting divisor_completion.","checks":["Enumerate residue conditions making d^2+4N a quadratic residue modulo selected small primes.","Intersect locally admissible residues for several differences and identify forced shared N classes.","Require exact bigint square tests after pruning; distinguish bounded-complete domains from exploratory samples.","Use Mira only for validation of any resulting finite candidate set."]}},{"round":87,"agentId":"mira","privatePlan":{"objective":"Use an exact-cover search over only verified cells returned by the family scans, prioritizing balanced 5-row/5-column incidence and strict Pareto improvements.","checks":["Recompute every cell with exact square, parity, positivity, and product tests.","Build row and column incidence masks and enumerate maximal 5×5, 6×4, and 4×5 supports.","Deduplicate N and differences, and report truncation separately from finite exclusion."]}},{"round":87,"agentId":"orum","privatePlan":{"objective":"Adversarially eliminate unsupported shells before any new scan, then select only a regime with a certifiable path to balanced support.","checks":["For each shell, enforce the full m residue condition, positivity, parity, distinctness, and exact bigint square identity; reject any manifest whose domain omits these constraints.","Derive cheap modular and pair-compatibility upper bounds on common columns; discard a family if its surviving support cannot reach five rows and five columns.","For the first surviving bounded slice, require a deterministic traversal, complete incidence masks, weakest-first support profile, and enumeration of all maximal bicliques as the saturation certificate.","Use the certificate to adaptively choose one family_scan only; keep the stop-loss at the allowed hard ceiling and request no divisor_completion job."]}},{"round":87,"agentId":"pip","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery in uncovered high-difference algebraic families, then perform deterministic exact-bigint incidence verification; use no divisor_completion job unless another primary method first produces a concrete candidate.","checks":["Generate and anneal seed slates while excluding registry-covered regimes; score only as a heuristic using shared factor and pair-gap compatibility.","Freeze the best slate and verify every cell by exact square, parity, positivity, and reconstructed-factor product tests.","Construct complete incidence masks, reject duplicate N or differences, and enumerate weakest-first support for 5×5, 6×4, and 4×5 targets.","Record the exact finite domain, traversal count, truncation status, and any negative result strictly as finite exclusion."]}},{"round":87,"agentId":"solvi","privatePlan":{"objective":"Construct a new rational genus-one family by normalizing a square-additive rectangle, eliminating the column parameters, and deriving an elliptic or quartic curve whose rational points generate an additional row or column.","checks":["Verify the elimination identities and every birational map symbolically.","Enumerate rational points using exact arithmetic and verify the curve equation without numerical approximation.","Clear denominators, enforce positivity, distinctness, parity, and exact bigint identities d²+4N=m².","Compute complete support masks, test the 5×5 objective, and exclude all registry-covered parameter regions."]}},{"round":87,"agentId":"tess","privatePlan":{"objective":"Design a new family_scan whose rows share deliberately engineered local square conditions modulo a composite modulus, while using nonlinear offsets to avoid covered arithmetic-progressions and shells.","checks":["Choose a squarefree modulus and prescribed N residue, then verify every selected d has d²+4N locally quadratic-residue compatible at each prime.","Use CRT sign branches to prune incompatible row pairs before exact completion enumeration.","Require all retained candidates to pass exact bigint square, parity, positivity, distinctness, and complete incidence-mask checks.","Report only bounded-complete exclusions for the declared congruence slice; request no divisor_completion job."]}},{"round":88,"agentId":"mira","privatePlan":{"objective":"Build an incidence-hypergraph/exact-cover search using only cells independently verified by Orum from the new slates, targeting balanced 5×5 and strict 6×4 or 4×5 bicliques.","checks":["Reverify every retained cell by exact square, parity, positive-factor, and product tests before inserting it into the incidence matrix.","Deduplicate N and differences, recompute row and column support masks, and enumerate maximal bicliques with exact-cover/SAT constraints.","Record the precise finite domain, replay data, and whether any result is complete or merely prefix-bounded."]}},{"round":88,"agentId":"orum","privatePlan":{"objective":"Adversarially select the better primary family by cheap compatibility tests, then demand a deterministic saturation certificate for its finite incidence domain.","checks":["Independently regenerate both difference slates, verify distinctness and registry separation, and audit every claimed symbolic identity.","Before any large scan, compute exact pair-compatibility and attainable row/column support bounds; reject families with no plausible five-row intersection.","For the selected primary method, recompute every cell with bigint arithmetic, enumerate maximal bicliques, and distinguish complete traversal from check-limit truncation.","Request at most one divisor_completion job only if the primary method first produces a concrete candidate requiring independent completion verification."]}},{"round":88,"agentId":"pip","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery across uncovered high-difference offset and square-anchor parameter tuples, scoring only exact incidence support, then submit the strongest masks for deterministic bigint verification.","checks":["Use a reproducible seeded search with registry-disjoint differences and explicit finite domains.","Optimize lexicographically for weakest column support, prioritizing 5x5 and strict 6x4 or 4x5 shapes.","Verify every cell by exact integer square root, parity, positive factors, and ab=N; reject near-squares.","Independently replay, deduplicate N and differences, recompute incidence masks, and send only exact candidates to orum; request no divisor_completion.","Record truncation and finite-domain exclusions separately from any impossibility claim."]}},{"round":88,"agentId":"solvi","privatePlan":{"objective":"Develop a constructive rational/elliptic family from the intersection y_i^2=x+d_i^2, with x=4N, then glue exact rational points from overlapping difference triples toward balanced support.","checks":["Derive and record a birational Weierstrass model for each smooth triple intersection, treating singular cases separately.","Generate rational points by exact chord-tangent arithmetic and map them back to x,y_i; use no numerical near-square evidence.","Clear denominators only under a globally consistent row normalization, then verify positive integer factors and ab=N with bigint arithmetic.","Assemble triple overlaps into incidence masks and retain only distinct N and differences outside covered registry regions.","Report rank/height truncation and any negative result strictly as bounded-complete or exploratory, never global exclusion."]}},{"round":88,"agentId":"tess","privatePlan":{"objective":"Construct a modular local-congruence sieve for equal-offset shells before requesting any exact scan, targeting compatible five-row signatures and excluding the previously tested parameter window.","checks":["Re-derive N=625[625(u²−v²)+c(δu−εv)] and reject every implementation output failing exact divisibility by 625.","For c=3,5,7, enumerate common B=N/625 residues modulo 5,25,125,625; handle c=5 separately because it is not invertible modulo 5-powers.","Use CRT compatibility modulo 3,7,8 and selected odd primes to retain only row signatures admitting a common B residue and positive factor ordering.","Choose a new v-range and offset set outside the submitted window, then define an explicit finite u-domain and canonical traversal before any family_scan.","Require exact bigint square, parity, positivity, product, distinctness, and incidence-mask checks; request no divisor_completion, with Orum reserved for validation."]}},{"round":89,"agentId":"mira","privatePlan":{"objective":"Run an incidence-hypergraph/SAT exact-cover search using only independently verified cells, seeking K_{5,5}, K_{6,4}, and K_{4,5} without requesting divisor completion.","checks":["Replay every accepted square witness, parity condition, positive factor pair, and product before admitting an edge.","Deduplicate verified difference rows and N columns, then construct exact incidence masks.","Encode distinct-row and distinct-column constraints for each target biclique and enumerate all maximal solutions in weakest-first support order.","Report only exact bicliques; label absent targets as finite exclusion over the verified-cell ledger."]}},{"round":89,"agentId":"orum","privatePlan":{"objective":"Adversarially test the two new family scans for false support and seek a saturation certificate through adaptive exact incidence analysis.","checks":["Recompute every proposed difference from its defining formula and reject malformed or duplicate rows.","For each high-support column, independently verify every square witness, parity condition, positivity, and factor product.","Construct complete row/column support masks for the declared checked prefix and enumerate maximal bicliques, distinguishing prefix failure from finite exclusion.","Use observed partial masks to choose the next uncovered parameter slice only if it can improve the weakest support floor; otherwise falsify the family and stop."]}},{"round":89,"agentId":"pip","privatePlan":{"objective":"Run stochastic, optimization-guided seed discovery over uncovered high-difference offset and quartic parameter slates, optimizing weakest row/column exact-support floors, then submit only the strongest slates for deterministic bigint verification.","checks":["Exclude every registry-covered regime and deduplicate 2–80 positive differences.","Score candidates only by exact perfect-square incidence and Pareto biclique support; reject numerical near-squares.","Use family_scan only, with limit in 1–5000 and maxChecks at most 5000000; request no divisor_completion.","Replay every selected cell with exact square, parity, positivity, reconstructed-factor, and product checks, then label negative results as finite exclusion for the declared domain."]}},{"round":89,"agentId":"solvi","privatePlan":{"objective":"Construct an algebraic completion family by fixing three common differences and reducing the equations m_i^2-4N=d_i^2 to an explicit genus-one curve, then use rational-point generation to seek five distinct integral N-columns and two additional common differences.","checks":["Form the projective intersection of quadrics m_i^2-m_1^2=(d_i^2-d_1^2)z^2 and verify its birational map to a Weierstrass model, including singular-case parametrizations.","Generate rational points from an exact known point, clear denominators, and retain only positive integral N with correct parity and distinctness.","Test every proposed extra row by exact bigint square checks and enumerate the complete support matrix; reject all numerical near-square evidence.","If a 5x5 candidate first emerges from this rational-point construction, use at most one divisor-completion verification job; do not use that job for discovery."]}},{"round":89,"agentId":"tess","privatePlan":{"objective":"Construct a modular and local compatibility sieve for shared rows, prioritizing five-row support without numerical near-square scoring or divisor_completion.","checks":["For small primes and the 2-adic modulus, enumerate residues n and m satisfying m² ≡ d²+4n for each candidate difference.","Build compatibility graphs for row sets whose local N residues agree simultaneously, then combine surviving patterns by CRT.","Require distinct positive reconstructed factors and columns before promoting any pattern to exact bigint validation by Solvi.","Record only bounded-complete exclusions or independently verified biclique certificates."]}},{"round":90,"agentId":"mira","privatePlan":{"objective":"Run an incidence-hypergraph/SAT search using only independently verified cells from the cross-block support seeds, targeting balanced bicliques.","checks":["Recompute every admitted edge with exact bigint square, parity, positivity, and product checks.","Deduplicate all rows and columns, then construct the complete verified incidence matrix.","Search exact-cover/SAT profiles K_{5,5}, K_{6,4}, and K_{4,5}, scoring weakest supports first.","Have Tess perform validation-only replay of any reported certificate; treat failures as finite exclusions only."]}},{"round":90,"agentId":"orum","privatePlan":{"objective":"Adversarially falsify both proposed scans before accepting any support improvement, then design the smallest adaptive exact experiment with a saturation certificate.","checks":["Demand explicit t-to-difference reconstruction for the quintic slate and an exact canonical enumeration rule for cross-block completions.","Reject any claimed frontier unless every generated column has a complete incidence mask and maximal 5×5, 6×4, and 4×5 biclique enumeration.","Use the weakest column-support profile to choose the next uncovered row slate; do not request divisor completion.","Independently bigint-recompute every surviving edge, parity condition, factor pair, and distinctness constraint."]}},{"round":90,"agentId":"pip","privatePlan":{"objective":"Use stochastic, optimization-guided generation of uncovered high-difference shell slates, optimizing weakest row and column exact-support floors, followed by deterministic exact-bigint verification.","checks":["Reject registry-overlapping or duplicate differences and require explicit parameter-to-difference mappings.","Use only exact modular and algebraic compatibility filters during seed scoring; never rank numerical near-squares.","Verify every retained incidence by exact square, parity, positivity, and product checks; enumerate maximal target bicliques.","Do not request divisor_completion; have Tess perform validation-only checking after bounded scans with maxChecks at most 5000000."]}},{"round":90,"agentId":"solvi","privatePlan":{"objective":"Construct a new difference-of-squares family by changing to U=m+d and V=m-d coordinates, quotienting the scaling symmetry, and reducing the remaining simultaneous-incidence conditions to rational conics and genus-one fibers; submit only a family_scan, never divisor_completion.","checks":["Derive the parametrization symbolically and identify the residual quartic or elliptic curve after imposing at least three shared-incidence conditions.","Generate rational points by exact chord-tangent or conic parametrization, then clear denominators while enforcing distinct positive differences and distinct positive N.","Search an uncovered high-difference parameter slice for balanced support, with exact bigint square, parity, factor, and Cartesian-cell checks.","Treat every negative result as finite exclusion for the stated family and bounds, and compare weakest-first support against the registry without claiming novelty from computation alone."]}},{"round":90,"agentId":"tess","privatePlan":{"objective":"Design a modular local filter that forces at least five differences to remain simultaneously compatible with candidate N columns before exact bigint verification.","checks":["Derive exact residue conditions for d^2+4N to be a square modulo selected small-prime moduli.","Use CRT intersections to prune row sets and retain only five-distinct-row compatibility classes.","Reject residue classes with empty local square support or duplicate reconstructed parameters.","Run exact square, parity, positivity, and product checks on surviving candidates; request no divisor-completion job unless another primary method first produces a candidate."]}},{"round":91,"agentId":"mira","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery in uncovered algebraic families, maximizing lexicographic minimum row and column exact support, then perform deterministic bigint verification.","checks":["Generate only registry-disjoint parameter slates and rank seeds by balanced support potential, not numerical near-squares.","Verify every retained incidence exactly, reconstruct positive factor pairs, and independently recheck every Cartesian cell of any proposed biclique.","Record bounded-domain exclusions separately from structural conclusions; request no divisor-completion job unless another primary method first produces a candidate."]}},{"round":91,"agentId":"orum","privatePlan":{"objective":"Derive an explicit rational-point family for square-additive rectangles, preferably by extending a known 4-row configuration through an elliptic-curve or conic-intersection parametrization, before requesting any scan.","checks":["Prove the symbolic identity for every intended cell and isolate all nondegeneracy conditions.","Clear denominators exactly, then check positivity, distinctness, and disjointness from the registry.","Use deterministic bigint verification for every Cartesian cell and emit row/column support certificates.","Request only a fully specified family_scan after the parametrization and finite domain are fixed."]}},{"round":91,"agentId":"pip","privatePlan":{"objective":"Adversarially falsify both proposed saturation mechanisms, certify maximal support profiles, and adapt the next experiment only from independently verified residual neighborhoods.","checks":["Recompute every shell and Pell identity, difference list, deduplication rule, and exact incidence mask with deterministic bigint arithmetic.","Require independent saturation certificates: weakest-first row and column supports, all maximal target bicliques, and explicit finite-domain exclusions.","Use residual common-neighbor structure to select the next uncovered family or offset slate; do not request divisor_completion."]}},{"round":91,"agentId":"solvi","privatePlan":{"objective":"Design a new local congruence filter for shell parameters that prunes incompatible rows before exact biclique extraction, without requesting divisor completion.","checks":["Derive exact residue conditions for d^2+4N being a square modulo 8, 3, 5, 7, and 11.","Group offset and parameter choices by compatible residue classes, seeking five rows that can coexist with one column parameter class.","Reject any filter that removes a known exact support-3 witness.","Run only the surviving finite parameter slice, then perform deterministic bigint square, parity, factor, and full-incidence checks."]}},{"round":91,"agentId":"tess","privatePlan":{"objective":"Run an incidence-hypergraph/SAT search over the union of independently verified shell cells, targeting K_{5,5}, then strict K_{4,5} and K_{6,4} support profiles.","checks":["Deduplicate positive N and differences before constructing the incidence matrix.","Recompute every retained edge with exact bigint square and factor tests.","Require every claimed biclique Cartesian cell to be present; never infer edges from near-squares or symbolic proximity.","Report weakest-first row and column supports and state the exact finite verified domain for any exclusion."]}},{"round":92,"agentId":"mira","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery across new affine-shell parameter combinations, then exact bigint verification of the strongest support neighborhoods.","checks":["Anneal lexicographically on minimum row support, minimum column support, and biclique score; reject numerical near-squares.","Verify positivity, distinctness, parity, exact square witnesses, and reconstructed factor products for every incidence.","Independently rebuild the incidence matrix and test every Cartesian cell of each 5x5, 6x4, or 4x5 candidate.","Record only bounded-complete finite exclusions with explicit domains and check limits; request no divisor_completion."]}},{"round":92,"agentId":"orum","privatePlan":{"objective":"Construct a genuinely shared-N algebraic family by intersecting the affine square-root shell with rational-point or elliptic-curve constraints, rather than scanning unrelated difference slates.","checks":["Derive an explicit rational parametrization or elliptic model whose integral specializations produce at least five shared differences.","Prove positivity, parity, distinctness, and exact square identities symbolically before any scan.","Define a finite, reproducible family_scan domain with saturation and support certificates; request no divisor_completion.","Reject any family whose identity does not force multi-row common columns, and independently bigint-check every surviving biclique."]}},{"round":92,"agentId":"pip","privatePlan":{"objective":"Adversarially compare the T=1009 affine shell with the offset-Fibonacci shell through deterministic saturation certificates and adaptive support-neighborhood experiments.","checks":["Recompute every proposed incidence by exact bigint square, parity, positivity, and factor-product tests.","Build complete deduplicated row/column support masks and enumerate maximal 5x5, 6x4, and 4x5 bicliques.","Record deterministic seeds, traversal order, truncation status, and weakest-first support profiles.","Adapt only from independently verified residual neighborhoods; request no divisor_completion unless another primary method first produces a candidate."]}},{"round":92,"agentId":"solvi","privatePlan":{"objective":"Develop a modular, local congruence filter for affine-shell rows that forces shared column neighborhoods before exact validation; issue no divisor_completion request.","checks":["Derive parity and positivity conditions symbolically for each affine-shell branch.","For small pairwise-coprime moduli, retain only residue classes where every prospective d^2+4N is a quadratic residue with the required parity.","Use CRT intersections on row pairs and triples to identify residue classes capable of supporting at least five rows in one column.","Reject any residue pattern that cannot produce five distinct positive N columns, then pass only surviving exact candidates to Pip for bigint verification.","Record finite domains, support masks, and falsification boundaries separately from any conjectural family claim."]}},{"round":92,"agentId":"tess","privatePlan":{"objective":"Use only independently verified cells from the T=1009 shell and any validated residuals to build an incidence hypergraph, then run exact-cover/biclique search for strict support improvements.","checks":["Recompute every cell by exact square testing, parity, positivity, and factor reconstruction before graph insertion.","Canonicalize duplicate N values and enumerate exact row and column support masks.","Search deterministically for K_{5,5}, K_{6,4}, and K_{4,5}, then independently verify every Cartesian cell of each candidate.","If unsuccessful, record maximal verified neighborhoods and a finite-domain exclusion without extrapolation."]}},{"round":93,"agentId":"mira","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery over mixed affine-shell parameters, then perform exact bigint verification of the highest-support seeds.","checks":["Exclude all registry-covered and previously failed parameter regions before sampling.","Anneal toward balanced row and column support, retaining only distinct positive differences and promising support masks.","Verify every incidence by exact square, parity, reconstructed factors, and product tests.","Request no divisor_completion; send only any surviving candidate certificate to Orum for validation."]}},{"round":93,"agentId":"orum","privatePlan":{"objective":"Construct a lower-dimensional rational or elliptic parametrization of simultaneous square conditions, prioritizing shared five-row neighborhoods rather than broad shell scans.","checks":["Eliminate factor variables symbolically and identify the resulting conic intersections or elliptic curve; certify every birational map and integrality condition.","Enumerate rational points in a bounded height range with exact arithmetic, then test distinct positive N and differences using bigint square identities and full support matrices.","Request no divisor_completion job unless this primary parametrization first produces a concrete candidate; if needed, use at most one completion job solely for exact verification."]}},{"round":93,"agentId":"pip","privatePlan":{"objective":"Use adversarial falsification, saturation certificates, and adaptive family design to test whether the new affine-shell incidence geometry genuinely improves the support frontier.","checks":["Recompute every proposed difference and shell identity independently.","Require reproducible weakest-first row/column support masks and explicit truncation certificates.","Adapt the next family parameters only from exact residual support patterns, while excluding all covered regions.","Reject any conclusion based on near-squares, stochastic absence, or incomplete literature comparison."]}},{"round":93,"agentId":"solvi","privatePlan":{"objective":"Develop a deterministic modular sieve for the new affine and quadratic-unit shells, pruning or forcing compatible row neighborhoods before any family scan; send only surviving witnesses to Orum for validation.","checks":["Derive exact residue conditions for d^2+4N to be a square modulo 8, 3, 5, 7, 13, and selected CRT products.","Construct compatible residue masks for five-row and five-column support, rejecting edges with provable local nonresidue obstructions.","Run a deterministic bounded family scan only on survivors, recording weakest-first support and finite-domain completeness.","Recheck every survivor by exact bigint square, parity, positivity, and Cartesian biclique tests."]}},{"round":93,"agentId":"tess","privatePlan":{"objective":"Use only Orum-verified cells from the two proposed regimes to construct a deterministic incidence hypergraph, then solve exact-cover/SAT instances for K_{5,5}, K_{6,4}, and K_{4,5}.","checks":["Reject every unverified or duplicate (N,d) cell; keep canonical row and column order.","Encode distinct-column and distinct-difference constraints, requiring every selected Cartesian cell.","Compare weakest-first row and column supports and report only exact certificates or finite-domain exclusions.","Do not request divisor_completion; route validation exclusively through Orum."]}},{"round":94,"agentId":"mira","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery over previously untested mixed-gap affine-shell parameters, then exact bigint verification of the highest-support seeds; request only family_scan jobs.","checks":["Generate distinct positive differences outside covered low-difference regions and rank seeds by balanced row/column support, not square proximity.","Apply deterministic bigint tests for every incidence, including square witness, parity, positivity, reconstructed factors, and product equality.","Rebuild support masks independently and report only finite-prefix exclusions or fully verified bicliques.","Reject any family whose registry overlap or symbolic identity is not explicitly documented."]}},{"round":94,"agentId":"orum","privatePlan":{"objective":"Construct a genuinely coupled rational-point family by expressing five shared differences and five N-columns as an intersection of exact quadrics, then identify and exploit any genus-one or elliptic quotient rather than scanning an arbitrary difference slate.","checks":["Use factor coordinates x_{ij} with x_{ij}(x_{ij}+d_i)=N_j, eliminate the N_j, and derive a symbolic low-dimensional ansatz whose genus and Jacobian are explicitly certified.","Generate rational points by exact algebraic transformations or elliptic-curve group operations; clear denominators primitively and reject registry-covered specializations.","Verify all 25 cells with bigint square tests, positivity, parity, distinctness, and independent support reconstruction; report saturation or finite truncation separately.","Do not request divisor_completion; reserve validation-only checking for Solvi."]}},{"round":94,"agentId":"pip","privatePlan":{"objective":"Run an adaptive family-scan-only round centered on adversarial falsification and exact saturation certificates, without requesting divisor_completion.","checks":["Recompute each proposed generator, difference list, parity, positivity, and deduplication independently.","Record complete exact row/column support masks and distinguish capped-domain completeness from check-limit truncation.","Enumerate maximal bicliques and certify every accepted cell with bigint square, parity, positive-factor, and product checks.","Use residual support masks to alter one algebraic parameter at a time, avoiding all registry-covered regions."]}},{"round":94,"agentId":"solvi","privatePlan":{"objective":"Develop a modular and local CRT sieve that forces compatible difference rows before any large search, using a primary algebraic construction rather than another blind family scan.","checks":["For small prime powers, enumerate exactly the residue classes of N for which d^2+4N is a quadratic residue, then intersect these classes across proposed rows.","Lift nonempty local patterns by CRT and impose the factor parametrization N=a(a+d), rejecting duplicate columns and incompatible parity.","Require exact bigint verification of every surviving incidence and support mask; treat modular survivors only as candidates, never as certificates.","Use the single permitted divisor_completion job only if this primary congruence construction first yields a concrete candidate."]}},{"round":94,"agentId":"tess","privatePlan":{"objective":"Run an incidence-hypergraph exact-cover search over only independently verified cells from the new family scans, targeting K_{5,5} first and strict K_{4,5} or K_{6,4} second.","checks":["Require Solvi to validate every edge with bigint square, parity, positivity, and reconstructed-factor checks.","Deduplicate N-columns and difference-rows, then enumerate all target bicliques from the complete verified edge set.","Record row and column support histograms and certify that any negative result is restricted to the stated verified-cell domain and truncation.","Use SAT or exact-cover branching on weakest-support rows first; do not request divisor_completion."]}},{"round":95,"agentId":"mira","privatePlan":{"objective":"Use stochastic, optimization-guided seed discovery in an untested affine-shell/Pell-phase family, then apply deterministic bigint verification; request no divisor_completion and keep Tess as the sole validation-only verifier.","checks":["Exclude all registry-covered low-difference, small-N, bounded-denominator, and repeated fixed-specialization regions.","Use annealing or beam search scored first by minimum row and column support, then total exact incidence.","For top seeds, verify d²+4N=m² exactly, parity, positive reconstructed factors, distinctness, and every claimed biclique cell.","Require independent support-mask reconstruction and classify any negative result only as finite-prefix exclusion.","Keep any family_scan manifest within the eight-key schema and maxChecks≤5,000,000."]}},{"round":95,"agentId":"orum","privatePlan":{"objective":"Develop a genuinely shared-N parametrization by intersecting the affine-shell equations and reducing the resulting rational-point conditions to elliptic-curve or higher-genus algebraic families.","checks":["Derive the common-N equations symbolically and certify every claimed identity exactly.","Enumerate rational points in a stated height slice, convert only parity-valid positive points to integer factor pairs, and deduplicate N values.","Build the exact incidence matrix and test support floors and all target bicliques; use Tess only for validation, with no divisor_completion request."]}},{"round":95,"agentId":"pip","privatePlan":{"objective":"Run one adaptive family_scan-only round centered on adversarial falsification and exact saturation certificates, using the strongest residual support pattern rather than another blind parameter slate.","checks":["Compare both proposed families against prior parameterizations and reject reparameterization duplicates before selecting the scan.","Require exact bigint square, parity, positivity, factor-product, and distinctness checks for every incidence; use no numerical tolerance.","Produce complete weakest-first row and column support histograms, maximal 5x5/6x4/4x5 biclique enumeration, and explicit check-limit versus domain-cap accounting.","Use Tess only as the validation-only verifier of the resulting witness or negative saturation log; request no divisor_completion job.","Require a manifest with family fingerprint, registry overlap, exact finite domain, completeness target, target shape, falsifiable success/stop criteria, symbolic identity, and at least two proof obligations."]}},{"round":95,"agentId":"solvi","privatePlan":{"objective":"Design a modular and local congruence prefilter that identifies difference sets capable of sharing candidate N residues, then use those signatures to prioritize a deterministic family scan; request no divisor_completion.","checks":["For small prime powers q, compute exactly A_q(d)={n mod q: d^2+4n is a quadratic residue mod q} and intersect A_q(d) across candidate row sets.","Generate distinct differences with d_i congruent to ±d_0 modulo selected moduli, and retain only tuples whose CRT-compatible N residue classes survive every local test.","Apply the same residue-signature test to candidate column groups, recording empty intersections as finite modular obstructions rather than global impossibility claims.","Recheck all survivors with exact bigint square tests, reconstructed positive factors, distinctness, and an independently rebuilt support matrix."]}},{"round":95,"agentId":"tess","privatePlan":{"objective":"Use the rotating incidence-hypergraph/SAT/exact-cover lane on deterministically verified cells only, seeking a strict support-profile improvement.","checks":["Recompute every cell with exact bigint square, parity, positivity, and factor-product checks.","Deduplicate N and difference vertices, construct the finite incidence hypergraph, and audit all row/column supports.","Solve exact K_{5,5}, K_{6,4}, and K_{4,5} constraints; report finite unsatisfiability only for the stated domain.","Request one targeted divisor_completion job only if the primary solver first produces a concrete candidate."]}},{"round":96,"agentId":"mira","privatePlan":{"objective":"Run an adversarial, saturation-first comparison of new shell families, using adaptive experiment design rather than raw candidate volume.","checks":["Reconcile completion limits and independently replay every generated incidence, including self-incidence.","Require row/column support histograms, pair multiplicities, and truncation certificates before accepting any negative result.","Adapt the next family only from weakest support floors; request at most one divisor_completion job, and only to verify a candidate independently produced by the primary scan."]}},{"round":96,"agentId":"orum","privatePlan":{"objective":"Develop a modular, local congruence sieve that forces compatible shared-N rows before exact support search.","checks":["Derive necessary square-residue conditions for candidate rows modulo several small coprime moduli.","Use CRT intersections to discard row sets with empty admissible N residue classes.","Rank surviving residue patterns by predicted common pair support, not by size or numerical closeness.","Require exact bigint reconstruction, parity, positivity, distinctness, and independent support-mask saturation before any claim."]}},{"round":96,"agentId":"pip","privatePlan":{"objective":"Develop a corrected constructive affine-shell parametrization and reduce shared five-row support to explicit rational-point conditions on intersections of quadrics, preferably an elliptic fibration.","checks":["Recompute every shell row from d=10006v±c and enforce distinctness, parity, and positivity.","Eliminate N between three or more shell equations and identify genus-one/elliptic components with explicit rational parameters.","Impose integral m-values and positive distinct N, then independently verify every claimed incidence by exact square tests.","Retain only parametrizations that can be extended symbolically toward five common differences; do not request divisor_completion."]}},{"round":96,"agentId":"solvi","privatePlan":{"objective":"Build an exact incidence-hypergraph/SAT search over independently verified cells from corrected affine-shell and CRT-shell families, targeting K_{5,5}, then K_{6,4} or K_{4,5}.","checks":["Canonicalize and independently recompute every (d,N) cell using bigint square and parity tests.","Reject any affine-shell row inconsistent with its declared offset set before constructing the hypergraph.","Deduplicate N columns and encode exact row/column support constraints; report weakest-first supports.","Replay every claimed biclique from the raw verified-cell table and distinguish complete finite search from truncation."]}},{"round":96,"agentId":"tess","privatePlan":{"objective":"Use stochastic, optimization-guided sampling of new affine-shell parameter seeds, scoring predicted row/column support floors, then submit only the strongest non-registry seeds to exact bigint family scans.","checks":["Reject seeds overlapping the supplied low-difference, fixed-specialization, and bounded-denominator regions.","Optimize balanced support rather than raw completion count, prioritizing minimum row and column supports.","For each retained seed, deterministically verify every incidence, factor reconstruction, deduplication, and claimed biclique.","Record truncation and treat every negative result as finite-domain exclusion only; request no divisor_completion."]}},{"round":97,"agentId":"mira","privatePlan":{"objective":"Run the next round in the adversarial-falsification lane: demand saturation certificates before interpreting any negative shell result, then adaptively narrow toward families with demonstrably high support.","checks":["Audit every retained pair-generated completion for both generating incidences independently.","Require exact row/column support histograms and maximal-biclique reconstruction, with no near-square inference.","Compare Pell-shell and affine-shell outputs by verified support density, then allocate the next family scan only to the strongest unsaturated slice.","Do not request divisor_completion; reserve validation-only checking for Pip."]}},{"round":97,"agentId":"orum","privatePlan":{"objective":"Construct a modular row-compatibility filter that prunes or forces candidate five-row intersections before exact bigint square testing, without requesting divisor_completion.","checks":["For each candidate difference d and moduli 8,3,5,7,11,13,17,19, enumerate exactly the residue classes N for which d²+4N is a quadratic residue.","Use CRT intersections to rank difference tuples by surviving compatible N classes; reject tuples with empty five-row intersections and retain only locally viable tuples.","Verify parity and factor-residue constraints, then send only surviving exact candidates to Pip for validation; audit every generating pair for self-incidence.","Treat modular survival as necessary only, never as evidence of an exact square or a global obstruction."]}},{"round":97,"agentId":"pip","privatePlan":{"objective":"Construct a rational elliptic-family lift from a four-row square-additive configuration to a fifth common difference and five distinct positive columns.","checks":["Normalize the four-row equations, derive the intersection-of-quadrics model, and verify an explicit birational map to a Weierstrass elliptic curve in both directions.","Generate rational points by exact elliptic group operations and impose the fifth-row condition as an exact rational equation, not a numerical approximation.","Clear denominators and deterministically verify all 25 square identities, parity conditions, positivity, distinctness, and the complete 5-by-5 support matrix.","Use no divisor-completion job unless this construction first produces a concrete candidate requiring independent factor-pair verification."]}},{"round":97,"agentId":"solvi","privatePlan":{"objective":"Use only independently verified incidence cells to build an incidence-hypergraph and run exact-cover/SAT searches for a (5,5), (6,4), or (4,5) biclique; do not request divisor_completion.","checks":["Recompute each cell as an exact square identity with parity and positive-factor checks before insertion.","Reject any candidate column lacking independently verified support for every selected row.","Canonicalize rows and columns, remove dominated supports, and enumerate exact target bicliques deterministically.","Audit self-incidence and distinguish truncation from finite-domain completeness.","Have Pip perform validation-only checking of any surviving certificate."]}},{"round":97,"agentId":"tess","privatePlan":{"objective":"Run stochastic, optimization-guided discovery of undercovered affine/Pell shell seeds, then submit only the highest Pareto-support seeds to exact bigint verification.","checks":["Vary recurrence coefficient, affine phase, scale, parity mix, and shell spacing while excluding registry-covered regions.","Score seeds by exact modular support proxies and predicted row/column balance, never by numerical closeness to squares.","Verify every retained incidence with exact bigint squares, parity, positivity, reconstructed factors, and self-incidence.","Keep only independently verified bicliques; report truncation as finite-slice evidence, not exclusion.","Do not request divisor_completion."]}},{"round":98,"agentId":"mira","privatePlan":{"objective":"Run the next round in the adversarial-falsification lane: stress-test both proposed shell families, seek saturation certificates, and adaptively retain only slices showing strong exact support.","checks":["Use Orum as the sole validation-only verifier; request no divisor_completion job.","For each shell, independently recompute every difference, pair completion, factor reconstruction, and self-incidence before interpreting support.","Require complete row/column masks, weakest-first support vectors, distinct-N checks, and explicit truncation status.","If no target appears, narrow adaptively toward parameter slices with demonstrated cross-row support rather than merely larger numbers."]}},{"round":98,"agentId":"orum","privatePlan":{"objective":"Develop a modular and local congruence sieve for the two shell families before any broad computation; request no divisor-completion job unless an independently generated exact candidate appears.","checks":["For each shell-generated N=t(t+d), compute exact residue support masks for every proposed row modulo a fixed prime set, rejecting masks that cannot reach five rows or five columns.","Exploit parity and small-modulus square constraints, then combine compatible local masks by CRT to identify residue classes capable of saturation.","Apply exact bigint square tests and factor reconstruction only to survivors; record finite-domain saturation separately from exploratory pruning.","If a different primary method produces a complete candidate, use at most one divisor_completion job solely for independent verification."]}},{"round":98,"agentId":"pip","privatePlan":{"objective":"Construct a generic rational elliptic lift from a four-row square-additive rectangle to a fifth common difference, avoiding fixed Bremner specializations and requesting no divisor-completion job.","checks":["Parameterize two rows by m_1=(t+Delta/t)/2 and m_2=(t-Delta/t)/2, then derive the exact quartic condition for a third row.","Identify a nonsingular genus-one model with a rational base point and compute rational group-law multiples symbolically.","Seek five distinct rational parameters producing positive, distinct columns and then impose the fifth-row square conditions.","Clear denominators and verify every one of the 25 identities by exact bigint arithmetic, including parity, positivity, factor reconstruction, and distinctness.","Have Orum validate only the resulting explicit certificate and its complete support matrix."]}},{"round":98,"agentId":"solvi","privatePlan":{"objective":"Use only independently verified cells to construct an incidence hypergraph and run exact-cover/SAT searches for balanced 5×5 and Pareto-improving 6×4 or 4×5 supports; request no divisor completion.","checks":["Recompute every cell with exact bigint square, parity, positivity, and factor-product checks.","Deduplicate N columns and verify every claimed Cartesian biclique cell, including self-incidence records.","Enumerate maximal support masks exhaustively within the supplied verified-cell domain and separate complete slices from truncated search.","Test whether either shell contributes genuinely new row or column support before expanding its parameter range."]}},{"round":98,"agentId":"tess","privatePlan":{"objective":"Use stochastic mutation and crossover over nonlinear-shell and Pell-shell parameters, ranking seeds by exact support potential, then submit only the highest-scoring family_scan seeds for deterministic bigint verification.","checks":["Use modular square filters only for ranking; never treat numerical closeness as evidence.","Recompute every d²+4N as an exact bigint square and reconstruct positive factor pairs with parity checks.","Deduplicate N columns, build the complete incidence matrix, sort support vectors, and test balanced 5×5 plus strict 6×4 and 4×5 bicliques.","Require self-incidence auditing and explicit truncation status; use no divisor_completion request, with Orum reserved for validation-only checking."]}},{"round":99,"agentId":"mira","privatePlan":{"objective":"Adversarially test shell support while building saturation certificates and adaptively retaining only exact self-incidence survivors.","checks":["Recompute every row, parity condition, modular claim, and exact square incidence independently.","For each generated N, test all proposed differences and record self-incidence failures before support scoring.","Compare completion multiplicities across pair classes, deduplicate N columns, and certify weakest-first support masks.","Adapt the next bounded experiment toward pair classes with the strongest verified support, without requesting divisor_completion."]}},{"round":99,"agentId":"orum","privatePlan":{"objective":"Design a modular local sieve that retains only row pairs and completions compatible with five-way support, without requesting divisor completion.","checks":["Compute exact 2-adic and odd-prime residue intersections for both shell families.","Classify same-sign and opposite-sign pairs by gcd and square-residue compatibility modulo selected prime powers.","Require surviving local patterns to support at least five distinct rows before any broader exact scan.","Record explicit finite exclusions and reject any inference from numerical near-squares or incomplete completion lists."]}},{"round":99,"agentId":"pip","privatePlan":{"objective":"Construct a rational-point/algebraic-family extension of the known 4x4 Bremner rectangle, then derive a fifth row or fifth column rather than broad-scanning shell differences.","checks":["Write the simultaneous equations m_{ij}^2-d_i^2=4N_j and eliminate the N_j to identify the relevant elliptic or higher-genus fibers through the Bremner rational point.","Use chord-tangent or elliptic-curve rational-point generation to obtain parameterized candidate rectangles; clear denominators and retain only positive integral differences and columns.","Test every retained cell with exact bigint square and factor-pair reconstruction, then independently recompute the full incidence matrix and weakest-first supports.","Before any bounded family_scan, prepare a manifest naming the exact parametrized slice, registry overlap, completeness target, and falsifiable stopping conditions; do not request divisor_completion."]}},{"round":99,"agentId":"solvi","privatePlan":{"objective":"Use only independently verified incidence cells to build a row-column hypergraph, then run an exact-cover/SAT search for K_{5,5}, K_{6,4}, and K_{4,5} support across the cubic and corrected 4-adic slices.","checks":["Recompute every cell with exact bigint square, parity, positivity, and factor-pair tests before insertion.","Deduplicate N columns and record complete incidence masks; reject any pair-generated completion lacking independent self-incidence verification.","Encode distinct-row and distinct-column constraints in SAT/exact cover, then replay every satisfying model deterministically and report weakest-first supports.","Make no divisor-completion request unless the primary hypergraph search first produces a concrete candidate requiring independent verification."]}},{"round":99,"agentId":"tess","privatePlan":{"objective":"Use stochastic, optimization-guided mutations of the cubic-shell and 4-adic-shell parameters to discover seeds with improved exact weakest-first support, followed by deterministic bigint verification; do not request divisor_completion.","checks":["Reject seeds with duplicate, nonpositive, or parity-incompatible differences, and exclude all registry-covered regions.","Use modular filters only as candidate-ranking tools, never as evidence of incidence.","For retained seeds, enumerate family-scan completions, test d^2+4N as an exact bigint square, reconstruct positive factor pairs, deduplicate N, and recompute the full incidence matrix.","Optimize exact support lexicographically toward (5,5,5,5,5), then (4,5,5,5,5), while recording truncation and completion limits separately.","Route validation-only confirmation to Solvi and retain only independently reproducible candidates."]}},{"round":100,"agentId":"mira","privatePlan":{"objective":"Adversarially audit the two degree-five shells, obtain saturation diagnostics, and adapt the next bounded experiment only from exact support data.","checks":["Normalize every proposed row family and reject formula/list mismatches, duplicate rows, parity failures, or unverified self-incidences.","Require complete weakest-first row/column support histograms, completion multiplicities, and explicit truncation certificates before interpreting a negative result.","Use surviving support masks to select the next parameter slice; falsify any apparent improvement by independent bigint replay, with Tess reserved as the validation-only verifier."]}},{"round":100,"agentId":"orum","privatePlan":{"objective":"Design a modular, local congruence-pruned pentic shell before any broad scan; no divisor_completion request.","checks":["Choose several small coprime moduli and compute the exact allowed residue classes for N from d^2+4N being a square modulo each modulus.","Construct rows whose residue signatures have dense common compatibility while excluding incompatible row pairs locally.","Use CRT intersection filters to retain only pair completions compatible with every intended row, then require exact bigint square and factor-pair checks.","Record a saturation certificate: tested parameter domain, residue-pruned domain, surviving masks, and weakest-first supports."]}},{"round":100,"agentId":"pip","privatePlan":{"objective":"Construct a parity-controlled translated quintic family whose repeated pair-completions are constrained by rational-point or elliptic-curve equations, then target exact five-row common support.","checks":["Correct the quintic offset parametrization before any bounded scan, including explicit treatment of the zero offset.","Derive symbolic coincidence and self-incidence conditions and test their integrality, positivity, and parity exactly.","Compare rational-point parameter slices rather than numerical near-squares; retain only columns passing full bigint incidence replay.","Use Tess solely for independent validation and do not request divisor_completion."]}},{"round":100,"agentId":"solvi","privatePlan":{"objective":"Build an exact incidence hypergraph from independently replayed verified cells and search it with SAT/exact-cover methods.","checks":["Recompute every cell with bigint square, parity, positivity, and factor-pair checks.","Canonicalize N columns and sort support vectors weakest-first.","Search distinct-row/distinct-column K5,5, K6,4, and K4,5 patterns exactly.","Independently replay every claimed pattern and report only finite-domain conclusions."]}},{"round":100,"agentId":"tess","privatePlan":{"objective":"Use stochastic, optimization-guided discovery of new fifth-degree translated-shell seeds outside the covered registry, then perform exact bigint verification of the highest-support seeds.","checks":["Reject parameter seeds with duplicate differences or overlap with covered domains; use exact support upper bounds rather than numerical closeness.","For each retained seed, verify every incidence through d^2+4N=m^2, parity, positivity, and reconstructed factor pairs.","Deduplicate N columns, sort weakest-first supports, test all claimed bicliques, and independently replay any candidate.","Request no divisor_completion job unless a separate primary method first produces a candidate requiring completion verification."]}}],"researchRetrieval":[{"requestId":"524cdc3fce8cad2ed8de6a88a4205f64","agentId":"mira","round":1,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Map global invariants and full algebraic families; convert isolated hits into parameterized structure.","costUsd":0.007,"at":"2026-09-03T17:53:34.339Z","sourceCount":5,"ledgerOffset":0},{"requestId":"7e56b74ce90452d9a49de067c4260113","agentId":"orum","round":1,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Falsify cheaply, demand saturation certificates, and attack hidden assumptions before spending compute.","costUsd":0.007,"at":"2026-09-03T17:53:34.339Z","sourceCount":5,"ledgerOffset":1},{"requestId":"7e88cc58bbc91988093c71986cf5e7bc","agentId":"pip","round":1,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Cultivate smooth factorizations and divisor-pair completions with unusually dense exact branching.","costUsd":0.007,"at":"2026-09-03T17:53:34.339Z","sourceCount":5,"ledgerOffset":2},{"requestId":"2128db5f63da88d65d4be83f1ac04048","agentId":"solvi","round":1,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Change coordinates, steal symmetries from adjacent fields, and reduce the true dimension of the search.","costUsd":0.007,"at":"2026-09-03T17:53:34.339Z","sourceCount":5,"ledgerOffset":3},{"requestId":"7da1fff0af2389e4a1ed76e6e40b05d5","agentId":"tess","round":1,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Target uncovered boundary regimes and optimize strict Pareto improvements over the published frontier.","costUsd":0.007,"at":"2026-09-03T17:53:34.339Z","sourceCount":5,"ledgerOffset":4},{"requestId":"d1d4af7beea0eefb6ad3b2ab6ea1add5","agentId":"mira","round":2,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Prioritize exact intersections of the high-difference divisor rays, seeking a strict 6x4 or 4x5 support improvement without duplicating registry regions.","costUsd":0.007,"at":"2026-09-03T18:03:42.299Z","sourceCount":5,"ledgerOffset":5},{"requestId":"7ada05bf3ed9ca6dbb674dfe75f7b7e0","agentId":"orum","round":2,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Falsify or certify additional common rows in the two high-difference completion families, prioritizing exact support improvements.","costUsd":0.007,"at":"2026-09-03T18:03:42.299Z","sourceCount":5,"ledgerOffset":6},{"requestId":"de380ec215e1734a666a935b12b2a0c2","agentId":"pip","round":2,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Prioritize exact intersection tests for the high-difference divisor ray and the translated five-row family, while retaining saturation certificates.","costUsd":0.007,"at":"2026-09-03T18:03:42.299Z","sourceCount":5,"ledgerOffset":7},{"requestId":"7e3290da11f77bf5b4aed3b672d927af","agentId":"solvi","round":2,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Prioritize exact searches most likely to turn the new six-column seeds into a 6x4 or 5x5 support improvement.","costUsd":0.007,"at":"2026-09-03T18:03:42.299Z","sourceCount":5,"ledgerOffset":8},{"requestId":"87bf11e3c0d5097ba31d19c893d655e0","agentId":"tess","round":2,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Prioritize exact cross-intersections among the new high-difference completion families, seeking a verified 4×5 or 6×4 support improvement.","costUsd":0.007,"at":"2026-09-03T18:03:42.299Z","sourceCount":5,"ledgerOffset":9},{"requestId":"4fa034c89b3259b38d9342b2f4804fc4","agentId":"mira","round":3,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Prioritize exact saturated completion intersections in new high-difference rays before launching broad scans.","costUsd":0.007,"at":"2026-09-03T18:13:47.093Z","sourceCount":5,"ledgerOffset":10},{"requestId":"c55682d4343a161009b76bf14c9a8ae2","agentId":"orum","round":3,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Prioritize exact, nonduplicative t=38 and high-difference ray intersections while rejecting incomplete evidence.","costUsd":0.007,"at":"2026-09-03T18:13:47.093Z","sourceCount":5,"ledgerOffset":11},{"requestId":"7cbe9fb8aa568d89a7d34ce2c763f595","agentId":"pip","round":3,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Find an exact 4×5 or 6×4 improvement by intersecting fresh high-difference completion rays, beginning with the t=38 translated tuple.","costUsd":0.007,"at":"2026-09-03T18:13:47.093Z","sourceCount":5,"ledgerOffset":12},{"requestId":"61363b556c3cc6b73e8b5c67827a49a1","agentId":"solvi","round":3,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Test the strongest new rays and translated tuples by exact completion-set intersection, while eliminating duplicated registry regions.","costUsd":0.007,"at":"2026-09-03T18:13:47.093Z","sourceCount":5,"ledgerOffset":13},{"requestId":"3c257c84f58f3db96c06fd4b68ee99e2","agentId":"tess","round":3,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Prioritize exact intersections in the new high-difference regimes, seeking the first verified 4×5 or 6×4 support.","costUsd":0.007,"at":"2026-09-03T18:13:47.093Z","sourceCount":5,"ledgerOffset":14},{"requestId":"cd882a30e6c7d57f4f1074c8be7d242f","agentId":"mira","round":4,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Prioritize exact completion intersections capable of producing a strict support improvement, beginning with the new 3600/4200 and centered 600000 families.","costUsd":0.007,"at":"2026-09-03T18:23:48.110Z","sourceCount":5,"ledgerOffset":15},{"requestId":"68605682e5c022ece4825ad29c9f1b3b","agentId":"orum","round":4,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Test the new rays by exact saturated intersection, prioritizing a strict support improvement.","costUsd":0.007,"at":"2026-09-03T18:23:48.110Z","sourceCount":5,"ledgerOffset":16},{"requestId":"26f59aefc797aeeb844b3ecaa02e6bc7","agentId":"pip","round":4,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Find an exact 4×5 or 6×4 support improvement using the newly identified high-difference rays.","costUsd":0.007,"at":"2026-09-03T18:23:48.110Z","sourceCount":5,"ledgerOffset":17},{"requestId":"bb70526d864b9c357989a489071a71f1","agentId":"solvi","round":4,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Find an exact 4x5 or 6x4 support pattern by intersecting newly identified divisor rays, without revisiting registry-covered searches.","costUsd":0.007,"at":"2026-09-03T18:23:48.110Z","sourceCount":5,"ledgerOffset":18},{"requestId":"411002b57c0b6e9cb3e64f1d529b153a","agentId":"tess","round":4,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Find an exact 4×5 rectangle by intersecting the new high-difference rays before spending budget on broad scans.","costUsd":0.007,"at":"2026-09-03T18:23:48.110Z","sourceCount":5,"ledgerOffset":19},{"requestId":"d3ddb6002fb52bda6b54852ebde407f2","agentId":"mira","round":5,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run exact, saturated completion/intersection tests on the three strongest untested families, seeking the first strict support improvement.","costUsd":0.007,"at":"2026-09-03T18:33:50.226Z","sourceCount":5,"ledgerOffset":20},{"requestId":"cfe149794d78b94319a18f36cf247537","agentId":"orum","round":5,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Prioritize exact saturation of the 40320, 720000-centered, and t=42 families, seeking the first certified support improvement while recording disjointness obstructions.","costUsd":0.007,"at":"2026-09-03T18:33:50.226Z","sourceCount":5,"ledgerOffset":21},{"requestId":"23d06c1b43b440a9f006552001638d65","agentId":"pip","round":5,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run exact completion and support-ranking tests on the 40320 and 720000-centered rays, seeking a certified 4×5 or 6×4 improvement.","costUsd":0.007,"at":"2026-09-03T18:33:50.226Z","sourceCount":5,"ledgerOffset":22},{"requestId":"a19f65f1922ce9106a560eb303aa81f7","agentId":"solvi","round":5,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Prioritize exact completion-family intersections that can produce a strict support improvement.","costUsd":0.007,"at":"2026-09-03T18:33:50.226Z","sourceCount":5,"ledgerOffset":23},{"requestId":"db44a5b1605b422c65d427bd75a86ce0","agentId":"tess","round":5,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run exact boundary-focused completion and family scans, then rank every returned column by support for strict Pareto improvements.","costUsd":0.007,"at":"2026-09-03T18:33:50.226Z","sourceCount":5,"ledgerOffset":24},{"requestId":"1920a132dead57e035fca9e4f7887c39","agentId":"mira","round":6,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Saturate the fresh divisor rays and four-row seed families, seeking the first exact support improvement.","costUsd":0.007,"at":"2026-09-03T18:43:55.120Z","sourceCount":5,"ledgerOffset":25},{"requestId":"021d0ddf81fca621eb819ad040b5e055","agentId":"orum","round":6,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Obtain the first certified Pareto improvement by saturating the strongest fresh completion rays, then intersecting their exact column sets.","costUsd":0.007,"at":"2026-09-03T18:43:55.120Z","sourceCount":5,"ledgerOffset":26},{"requestId":"841d5c02b6c147d526841428dcdb0a99","agentId":"pip","round":6,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run exact completion and intersection tests on the three fresh outer rays, then expand the strongest surviving 4-row seed without repeating covered scans.","costUsd":0.007,"at":"2026-09-03T18:43:55.120Z","sourceCount":5,"ledgerOffset":27},{"requestId":"d3cb63231652f570a9ea60d8e8c862ae","agentId":"solvi","round":6,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Reduce the next search to exact intersections of the strongest new completion families, targeting a certified 4×5 support pattern.","costUsd":0.007,"at":"2026-09-03T18:43:55.120Z","sourceCount":5,"ledgerOffset":28},{"requestId":"22cee5e5248cfec992c731294ec11ff1","agentId":"tess","round":6,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run exact completion/intersection tests on the strongest uncovered families while targeting a strict support improvement.","costUsd":0.007,"at":"2026-09-03T18:43:55.120Z","sourceCount":5,"ledgerOffset":29},{"requestId":"56bdbfe7a7ab2e519faab6dd1e924e4d","agentId":"mira","round":7,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Find an exact support improvement by saturating the strongest fresh four-row families and testing cross-ray intersections.","costUsd":0.007,"at":"2026-09-03T18:53:57.677Z","sourceCount":5,"ledgerOffset":30},{"requestId":"e8e59757fde7af3aa79fcf65c298a531","agentId":"orum","round":7,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Obtain a machine-checkable saturation certificate for the fresh T_50 five-row ray, or falsify it completely, while preserving exact Pareto accounting.","costUsd":0.007,"at":"2026-09-03T18:53:57.677Z","sourceCount":5,"ledgerOffset":31},{"requestId":"95b2e3e0a86472e6fae2e38e2e9d3cdb","agentId":"pip","round":7,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Seek the first exact support improvement outside covered regions, prioritizing a 4×5 rectangle and then a 5×5 certificate.","costUsd":0.007,"at":"2026-09-03T18:53:57.677Z","sourceCount":5,"ledgerOffset":32},{"requestId":"cef68bed7e5a4f75ae85b43c9c3c2c99","agentId":"solvi","round":7,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Prioritize exact support improvement by intersecting fresh divisor rays, beginning with Orum’s T_50 construction and then Mira’s three seeds.","costUsd":0.007,"at":"2026-09-03T18:53:57.677Z","sourceCount":5,"ledgerOffset":33},{"requestId":"95ab4a5056ab618c410479044a870aba","agentId":"tess","round":7,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run exact, non-overlapping boundary tests prioritized by the T_50 five-row ray, then cross-ray intersections near 720000 and the million-scale block, seeking support profile (4,5,5,5,5) or better.","costUsd":0.007,"at":"2026-09-03T18:53:57.677Z","sourceCount":5,"ledgerOffset":34},{"requestId":"b796a665d235588b5a80e47e8614ea1d","agentId":"mira","round":8,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Test fresh even-shift and 720000-centered cross-ray families for an exact 4×5 or 6×4 improvement.","costUsd":0.007,"at":"2026-09-03T19:03:59.377Z","sourceCount":5,"ledgerOffset":35},{"requestId":"1e6853b10532655acf8cf1a93acdc3e0","agentId":"orum","round":8,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Saturate the fresh T_52 ray before spending compute on broader families, seeking either an exact 4×5 certificate or a cheap complete falsification.","costUsd":0.007,"at":"2026-09-03T19:03:59.377Z","sourceCount":5,"ledgerOffset":36},{"requestId":"65aef6ebb9b2f8fa03f831bf05b93d7f","agentId":"pip","round":8,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run exact saturation and intersection tests on the fresh T_52 ray, while completing the highest-priority 720000-centered searches without repeating covered regions.","costUsd":0.007,"at":"2026-09-03T19:03:59.377Z","sourceCount":5,"ledgerOffset":37},{"requestId":"b6768e0dd2f9fb65d4088d61a53e5147","agentId":"solvi","round":8,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use coordinate changes in the square-difference equations to test fresh even-shift rays and cross-ray intersections for an exact support improvement.","costUsd":0.007,"at":"2026-09-03T19:03:59.377Z","sourceCount":5,"ledgerOffset":38},{"requestId":"9795ab6a13cb13b86b1671de77e1f01e","agentId":"tess","round":8,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Test the fresh high-difference boundary for a strict support improvement while preserving exact completeness certificates.","costUsd":0.007,"at":"2026-09-03T19:03:59.377Z","sourceCount":5,"ledgerOffset":39},{"requestId":"7b3c0b937a20d9001fd4c03d4d092851","agentId":"mira","round":9,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Test fresh primitive four-row families for a strict exact support improvement.","costUsd":0.007,"at":"2026-09-03T19:14:04.674Z","sourceCount":5,"ledgerOffset":40},{"requestId":"977a937e827201878d84ab31a090b55e","agentId":"orum","round":9,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Prioritize cheap exact saturation of the three primitive satellite families and the three missing 720000-centered edges, then test for a certified 4×5 rectangle.","costUsd":0.007,"at":"2026-09-03T19:14:04.674Z","sourceCount":5,"ledgerOffset":41},{"requestId":"f84b0b670eb8ab8b75bd830636d8d09b","agentId":"pip","round":9,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Execute exact saturation of the three missing 720000-centered pair edges, then prioritize complete scans of Mira’s three primitive four-row families.","costUsd":0.007,"at":"2026-09-03T19:14:04.674Z","sourceCount":5,"ledgerOffset":42},{"requestId":"b88b8c1d09b04843b5601408cb411aef","agentId":"solvi","round":9,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Prioritize fresh primitive satellite families above the registry’s covered small-difference regime, while using symmetry reduction to test cross-family column intersections.","costUsd":0.007,"at":"2026-09-03T19:14:04.674Z","sourceCount":5,"ledgerOffset":43},{"requestId":"dd7840395fbadb25685defd5e1a00dac","agentId":"tess","round":9,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Prioritize an exact 4×5 search in the newly exposed primitive high-difference regime, while closing the 720000 block only when completion certificates are available.","costUsd":0.007,"at":"2026-09-03T19:14:04.674Z","sourceCount":5,"ledgerOffset":44},{"requestId":"862b429e36ab5955bc80761820f33955","agentId":"mira","round":10,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Prioritize exact completion of the two 635-bridge four-row families, then test their five-row intersection for a Pareto improvement.","costUsd":0.007,"at":"2026-09-03T19:24:10.394Z","sourceCount":5,"ledgerOffset":45},{"requestId":"cf470570b101b63cf27bdb1890e779a7","agentId":"orum","round":10,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Seek an exact Pareto improvement by cheaply saturating the fresh T_56 and 635-bridge regimes before broader scans.","costUsd":0.007,"at":"2026-09-03T19:24:10.394Z","sourceCount":5,"ledgerOffset":46},{"requestId":"941dce6fb7a7b51232752223e1be14c8","agentId":"pip","round":10,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Turn the 635 bridge into an exact Pareto improvement, using completion intersections to prune impossible rows before broader scans.","costUsd":0.007,"at":"2026-09-03T19:24:10.394Z","sourceCount":5,"ledgerOffset":47},{"requestId":"1984e103af003dea36966100a0c7a672","agentId":"solvi","round":10,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Exploit normalized primitive cross-ray structure while testing the strongest fresh families for an exact Pareto improvement.","costUsd":0.007,"at":"2026-09-03T19:24:10.394Z","sourceCount":5,"ledgerOffset":48},{"requestId":"787444e6c571eed8fcd6ec837cd0768a","agentId":"tess","round":10,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Prioritize exact scans of the fresh 635 bridge and T_56 regime for a strict 4×5 improvement, while recording complete obstructions for failed rays.","costUsd":0.007,"at":"2026-09-03T19:24:10.394Z","sourceCount":5,"ledgerOffset":49},{"requestId":"dc7b34a22666e992c555427e0d085bdc","agentId":"mira","round":11,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Exploit the fresh bridge families to obtain an exact 4×5 or stronger support profile outside covered regions.","costUsd":0.007,"at":"2026-09-03T19:34:12.377Z","sourceCount":5,"ledgerOffset":50},{"requestId":"08d11d9a205271313f838e2e7561d0d5","agentId":"orum","round":11,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Saturate fresh bridge and boundary routes while seeking an exact Pareto improvement.","costUsd":0.007,"at":"2026-09-03T19:34:12.377Z","sourceCount":5,"ledgerOffset":51},{"requestId":"1af06a01206116e186fdc778e2f9c288","agentId":"pip","round":11,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Exploit the two verified 635-bridge families through exact divisor-completion intersections, prioritizing a certified 4×5 or 6×4 improvement.","costUsd":0.007,"at":"2026-09-03T19:34:12.377Z","sourceCount":5,"ledgerOffset":52},{"requestId":"f2f4787eadb0188fa59eeff5e0aa21d1","agentId":"solvi","round":11,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Exploit the fresh 215 bridge to seek an exact Pareto improvement while independently probing the first uncovered smallest-difference boundary.","costUsd":0.007,"at":"2026-09-03T19:34:12.377Z","sourceCount":5,"ledgerOffset":53},{"requestId":"abeff6e64d2319974b5de95be14f85ce","agentId":"tess","round":11,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Exploit the fresh boundary rows above 160, prioritizing exact completion intersections that can certify a 4×5 rectangle or eliminate a bridge family cheaply.","costUsd":0.007,"at":"2026-09-03T19:34:12.377Z","sourceCount":5,"ledgerOffset":54},{"requestId":"ea7c39074939357e10d3d4e738550ec3","agentId":"mira","round":12,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Exploit the exact 215–280–467–635–1640 bridge for a certified support-profile improvement while testing the nearest uncovered difference regime.","costUsd":0.007,"at":"2026-09-03T19:44:15.034Z","sourceCount":5,"ledgerOffset":55},{"requestId":"67f619c3768b35c24d5a58b8079b2078","agentId":"orum","round":12,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Falsify or certify the fresh 467 bridge with deterministic exact scans, then pursue the strongest surviving support profile.","costUsd":0.007,"at":"2026-09-03T19:44:15.034Z","sourceCount":5,"ledgerOffset":56},{"requestId":"d85f044b3e478affbc7ad0b42ff4cf5b","agentId":"pip","round":12,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Search the 467 bridge and the first uncovered boundary for an exact Pareto improvement, prioritizing a certified 4-by-5 or stronger support profile.","costUsd":0.007,"at":"2026-09-03T19:44:15.034Z","sourceCount":5,"ledgerOffset":57},{"requestId":"784d369895d59041f0d3083a0b0ae0dc","agentId":"solvi","round":12,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Exploit the fresh 467 bridge using exact pair-completion intersections, prioritizing a certified 4×5 or 6×4 support profile.","costUsd":0.007,"at":"2026-09-03T19:44:15.034Z","sourceCount":5,"ledgerOffset":58},{"requestId":"0d79d0588e2c6a2801778304e83c6cce","agentId":"tess","round":12,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Search the 467 bridge and the immediately uncovered smallest-difference boundary for an exact Pareto improvement.","costUsd":0.007,"at":"2026-09-03T19:44:15.034Z","sourceCount":5,"ledgerOffset":59},{"requestId":"bc4037ee6f3909fb25d0a5ede1637557","agentId":"mira","round":13,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Prioritize nonduplicate exact searches around the 17640 and 15120 four-row families, then the 401–800 boundary.","costUsd":0.007,"at":"2026-09-03T19:54:20.504Z","sourceCount":5,"ledgerOffset":60},{"requestId":"679cf1e47d65dff17f1e52cb5fc22abb","agentId":"orum","round":13,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Test only genuinely uncovered algebraic regimes with exact saturation certificates, targeting a strict Pareto improvement.","costUsd":0.007,"at":"2026-09-03T19:54:20.504Z","sourceCount":5,"ledgerOffset":61},{"requestId":"5e0460b5e5067ccdecd576af59dcaaa0","agentId":"pip","round":13,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Coordinate only nonredundant exact searches for a Pareto improvement, emphasizing new four-row families and the 401–800 boundary.","costUsd":0.007,"at":"2026-09-03T19:54:20.504Z","sourceCount":5,"ledgerOffset":62},{"requestId":"b3f49bfc86aa17c0580b42da5258d919","agentId":"solvi","round":13,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Search for an exact Pareto improvement through the fresh four-row anchors while avoiding every registry-covered or already completed region.","costUsd":0.007,"at":"2026-09-03T19:54:20.504Z","sourceCount":5,"ledgerOffset":63},{"requestId":"4ffcabc94162d8abf8412fed0f50636c","agentId":"tess","round":13,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Test the two fresh four-row cores and the uncovered 401–800 difference regime for an exact 4×5-or-better support profile.","costUsd":0.007,"at":"2026-09-03T19:54:20.504Z","sourceCount":5,"ledgerOffset":64},{"requestId":"869e9b5d3129ad5e78117f9fff89696d","agentId":"mira","round":14,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Search the new exact seed families for a second full-support column and an exact Pareto improvement.","costUsd":0.007,"at":"2026-09-03T20:04:22.729Z","sourceCount":5,"ledgerOffset":65},{"requestId":"b8a8af8e4294b1e6279459bb735c7c2c","agentId":"orum","round":14,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Find an exact Pareto improvement by saturating the new boundary and seed families without repeating covered regions.","costUsd":0.007,"at":"2026-09-03T20:04:22.729Z","sourceCount":5,"ledgerOffset":66},{"requestId":"da9a7bdccdaa65680a3a423a6292f14a","agentId":"pip","round":14,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Exploit the 24024 seed and the uncovered 401–800 regime while testing whether the 100800 anchor has any exact companion columns.","costUsd":0.007,"at":"2026-09-03T20:04:22.729Z","sourceCount":5,"ledgerOffset":67},{"requestId":"8e7a79eb26d609ec5ff835ee7659a41e","agentId":"solvi","round":14,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Exploit the new 100800 and 24024 anchors through exact divisor-branch enumeration, then compare support profiles for a genuine Pareto improvement.","costUsd":0.007,"at":"2026-09-03T20:04:22.729Z","sourceCount":5,"ledgerOffset":68},{"requestId":"effd3175fddae952f15361c6812de91b","agentId":"tess","round":14,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Seek a nonredundant exact Pareto improvement beyond the reported 401–800 work, while independently testing whether the 100800 six-row anchor has completions.","costUsd":0.007,"at":"2026-09-03T20:04:22.729Z","sourceCount":5,"ledgerOffset":69},{"requestId":"6f715b8fec4dbb5a17668f28b797e499","agentId":"mira","round":15,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Test the new 120-multiple family for an exact Pareto improvement, then use saturated pair atlases for independent confirmation.","costUsd":0.007,"at":"2026-09-03T20:14:25.987Z","sourceCount":5,"ledgerOffset":70},{"requestId":"a0988b5766174fb63c2bdbf537785cda","agentId":"orum","round":15,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Falsify the new 120-multiple family cheaply, then certify any surviving Pareto improvement.","costUsd":0.007,"at":"2026-09-03T20:14:25.987Z","sourceCount":5,"ledgerOffset":71},{"requestId":"91cfdfa9b07558ca2132ac0121b42b61","agentId":"pip","round":15,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Search uncovered families for an exact 4×5 or 6×4 improvement, with deterministic support matrices and no numerical-square heuristics.","costUsd":0.007,"at":"2026-09-03T20:14:25.987Z","sourceCount":5,"ledgerOffset":72},{"requestId":"7b8eb4e90f24702798918fd8fc7c24d7","agentId":"solvi","round":15,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Exploit scale normalization in the new 120-multiple family, then use exact completion intersections to seek a 4×5 or 6×4 improvement.","costUsd":0.007,"at":"2026-09-03T20:14:25.987Z","sourceCount":5,"ledgerOffset":73},{"requestId":"56394983b6679e0bdd98e07b661cc81b","agentId":"tess","round":15,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Prioritize exact searches with the strongest chance of a strict Pareto improvement while avoiding saturated atlases.","costUsd":0.007,"at":"2026-09-03T20:14:25.987Z","sourceCount":5,"ledgerOffset":74},{"requestId":"38cd6022ace67067a332686e002f201f","agentId":"mira","round":16,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Prioritize exact intersection tests that can certify a strict support-profile improvement, while separating exhaustive atlas conclusions from truncated family scans.","costUsd":0.007,"at":"2026-09-03T20:24:29.209Z","sourceCount":5,"ledgerOffset":75},{"requestId":"45e1b22c979020496b61e572c5430e73","agentId":"orum","round":16,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Saturate the new 577024-derived pair intersection before expanding computation.","costUsd":0.007,"at":"2026-09-03T20:24:29.209Z","sourceCount":5,"ledgerOffset":76},{"requestId":"16540d76262607c900d25a61f32c43ab","agentId":"pip","round":16,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Execute the fresh atlas and six-row scans, seeking an exact Pareto improvement.","costUsd":0.007,"at":"2026-09-03T20:24:29.209Z","sourceCount":5,"ledgerOffset":77},{"requestId":"661dcd051dba6bf3fa8a0d026ab6451b","agentId":"solvi","round":16,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Prioritize exact companion searches around the fresh seven-row seed, then independently test the atlas and uncovered boundary regimes for a certified Pareto improvement.","costUsd":0.007,"at":"2026-09-03T20:24:29.209Z","sourceCount":5,"ledgerOffset":78},{"requestId":"8d121c544ae6d0c6745679673505d417","agentId":"tess","round":16,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Prioritize exact Pareto improvement beyond the covered d_min≤160 and 401–480 regimes.","costUsd":0.007,"at":"2026-09-03T20:24:29.209Z","sourceCount":5,"ledgerOffset":79},{"requestId":"f3a486afb5b3f26989b35d9f25c3d433","agentId":"mira","round":17,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Pursue a strict exact support-profile improvement using only unrepeated five-row families and generalized factor-pair completions.","costUsd":0.007,"at":"2026-09-03T20:34:32.351Z","sourceCount":5,"ledgerOffset":80},{"requestId":"2730872d65bb33eef4750f6d5ab79b96","agentId":"orum","round":17,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run only fresh exact intersection tests with the strongest prospect of a 4×5 or 6×4 support improvement.","costUsd":0.007,"at":"2026-09-03T20:34:32.351Z","sourceCount":5,"ledgerOffset":81},{"requestId":"2b1b9febd10474589f74b4cadaa30129","agentId":"pip","round":17,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Prioritize nonduplicative exact intersections from the 124000-, 100276-, and three fresh five-row anchors, seeking a strict support-profile improvement.","costUsd":0.007,"at":"2026-09-03T20:34:32.351Z","sourceCount":5,"ledgerOffset":82},{"requestId":"baee9d760f4f07c8f6e27c8656b6a15e","agentId":"solvi","round":17,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct and test genuinely new difference families by transforming the reported anchor sets, prioritizing disjoint row combinations and exact Pareto gains.","costUsd":0.007,"at":"2026-09-03T20:34:32.351Z","sourceCount":5,"ledgerOffset":83},{"requestId":"bce74b6d85faa62e0ff9366deb0e8ac2","agentId":"tess","round":17,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Find a strict exact support-profile improvement from an unscanned boundary family, prioritizing 4×5 over merely larger one-column anchors.","costUsd":0.007,"at":"2026-09-03T20:34:32.351Z","sourceCount":5,"ledgerOffset":84},{"requestId":"2ef9c7ffb84d8ed67a57b1d94dead71f","agentId":"mira","round":18,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Prioritize the highest-leverage unrepeated family scans and convert any returned 4-support seed into a saturated completion atlas.","costUsd":0.007,"at":"2026-09-03T20:44:37.300Z","sourceCount":5,"ledgerOffset":85},{"requestId":"87e53b57264091abf7fe96acb9f3f2d0","agentId":"orum","round":18,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Attack the exact 4×2 seed first, then test the strongest unrepeated mixed-scale families for a strict support-profile improvement.","costUsd":0.007,"at":"2026-09-03T20:44:37.300Z","sourceCount":5,"ledgerOffset":86},{"requestId":"f0cdbd0675b332e2fcbf0af3e3fc06d5","agentId":"pip","round":18,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Test the freshest exact intersection families for a strict support-profile improvement while avoiding registry-covered regions.","costUsd":0.007,"at":"2026-09-03T20:44:37.300Z","sourceCount":5,"ledgerOffset":87},{"requestId":"dd5953d058cf980eab948fc9e07d45c9","agentId":"solvi","round":18,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Prioritize exact finite atlases that can produce a 4×5 improvement, while extending into mixed-scale high-difference regimes without repeating covered scans.","costUsd":0.007,"at":"2026-09-03T20:44:37.300Z","sourceCount":5,"ledgerOffset":88},{"requestId":"611c2af807f2f821637b720b9a0fe9fd","agentId":"tess","round":18,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Seek the first exact support-profile improvement by saturating the fresh four-row seed and testing genuinely mixed-scale row families.","costUsd":0.007,"at":"2026-09-03T20:44:37.300Z","sourceCount":5,"ledgerOffset":89},{"requestId":"52c2778d73a9f9f600eafe430eeed0fb","agentId":"mira","round":19,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a cross-family divisor-completion atlas targeting a verified (4,5,5,5,5) profile.","costUsd":0.007,"at":"2026-09-03T20:54:41.600Z","sourceCount":5,"ledgerOffset":90},{"requestId":"da4e7390ab860bf9231f02a59cfd2e28","agentId":"orum","round":19,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Saturate the three fresh anchor families, then use finite divisor completions to test whether any near-frontier column extends to five exact rows.","costUsd":0.007,"at":"2026-09-03T20:54:41.600Z","sourceCount":5,"ledgerOffset":91},{"requestId":"225fa80170b6180492fc2380f5b32a2f","agentId":"pip","round":19,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Test the three fresh high-difference families, then build exact completion atlases from any strongest returned columns without entering registry-covered regions.","costUsd":0.007,"at":"2026-09-03T20:54:41.600Z","sourceCount":5,"ledgerOffset":92},{"requestId":"7a3209044be0449ffd211d993ce9c314","agentId":"solvi","round":19,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use coordinate normalization and pair-factorization to couple the fresh high-difference anchors into a new exact 4×5 or 6×4 search.","costUsd":0.007,"at":"2026-09-03T20:54:41.600Z","sourceCount":5,"ledgerOffset":93},{"requestId":"7e9c0665cf48b0e0201977236b3dbd08","agentId":"tess","round":19,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Test the three fresh high-difference families for strict support-profile improvements, then exhaust only genuinely new completion branches.","costUsd":0.007,"at":"2026-09-03T20:54:41.600Z","sourceCount":5,"ledgerOffset":94},{"requestId":"8190ad89db104a970d228aafef086272","agentId":"mira","round":20,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Audit the fresh scan outputs and build an exact support matrix, prioritizing any (4,5,5,5,5) or stronger profile.","costUsd":0.007,"at":"2026-09-03T21:04:41.960Z","sourceCount":5,"ledgerOffset":95},{"requestId":"3a6cc7636d49c9d7b39e3bc9c7743033","agentId":"orum","round":20,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Extract a machine-checkable Pareto improvement from the un-covered exact row atlases, or produce a certified non-improvement.","costUsd":0.007,"at":"2026-09-03T21:04:41.960Z","sourceCount":5,"ledgerOffset":96},{"requestId":"2756453119b59195dc3425c58cf13692","agentId":"pip","round":20,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Test the strongest fresh row families, then exploit any returned columns through exact completion and support intersection.","costUsd":0.007,"at":"2026-09-03T21:04:41.960Z","sourceCount":5,"ledgerOffset":97},{"requestId":"9a2a83962d3acda018d8015f328e320e","agentId":"solvi","round":20,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Test the fresh anchors as an exact row-intersection network, then pursue the strongest Pareto profile without duplicating covered searches.","costUsd":0.007,"at":"2026-09-03T21:04:41.960Z","sourceCount":5,"ledgerOffset":98},{"requestId":"6671a832accb514b8b6b5e83898f995c","agentId":"tess","round":20,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Search the new high-difference and mixed-scale atlases for an exact (4,5,5,5,5) or stronger profile, then complete and saturate the best partial columns.","costUsd":0.007,"at":"2026-09-03T21:04:41.960Z","sourceCount":5,"ledgerOffset":99},{"requestId":"979017daab47f8e32d508623ae83f651","agentId":"mira","round":21,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Join the completed pair atlases into a rigorously verified support matrix, prioritizing a 4×5 or 6×4 profile.","costUsd":0.007,"at":"2026-09-03T21:14:44.315Z","sourceCount":5,"ledgerOffset":100},{"requestId":"abacac9ec91813242af1c2904d3dcb5d","agentId":"orum","round":21,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Obtain deterministic saturation certificates and test the strongest cross-atlas projections for an exact Pareto improvement.","costUsd":0.007,"at":"2026-09-03T21:14:44.315Z","sourceCount":5,"ledgerOffset":101},{"requestId":"dfcc3ffefab1ec5dc9cae9d7a595b416","agentId":"pip","round":21,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Exploit the 15400/105300 cross-atlas using exact support matrices, prioritizing a (4,5,5,5,5) or 6×4 improvement.","costUsd":0.007,"at":"2026-09-03T21:14:44.315Z","sourceCount":5,"ledgerOffset":102},{"requestId":"996742733d36065571cef1ab34a710f0","agentId":"solvi","round":21,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Intersect the newly saturated pair atlases to seek a verified (4,5,5,5,5) or 6×4 support matrix without repeating covered regions.","costUsd":0.007,"at":"2026-09-03T21:14:44.315Z","sourceCount":5,"ledgerOffset":103},{"requestId":"870b2ca591ae2421e27e2be942ccd462","agentId":"tess","round":21,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Exploit the fresh 577024 five-row regime through exact pair-completion intersections, seeking a strict support-profile improvement.","costUsd":0.007,"at":"2026-09-03T21:14:44.315Z","sourceCount":5,"ledgerOffset":104},{"requestId":"9c8d3f8110c043e64a05bb8cfd93c74e","agentId":"mira","round":22,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Exact-test cross-support among the new six-row projections and the 105300 pair atlases, seeking a strict Pareto improvement.","costUsd":0.007,"at":"2026-09-03T21:24:46.788Z","sourceCount":5,"ledgerOffset":105},{"requestId":"0f3278cf2cafbef3a9e6ad146248c1ce","agentId":"orum","round":22,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Prioritize exact atlas intersections capable of certifying a strict support-profile improvement.","costUsd":0.007,"at":"2026-09-03T21:24:46.788Z","sourceCount":5,"ledgerOffset":106},{"requestId":"2485cfb00949c8fb5b97bcec14fa768c","agentId":"pip","round":22,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Join the fresh high-difference projections and pair atlases, targeting either four columns with six common rows or five columns with four common rows.","costUsd":0.007,"at":"2026-09-03T21:24:46.788Z","sourceCount":5,"ledgerOffset":107},{"requestId":"dbe65c26adf02c6f40547e6c35f099b0","agentId":"solvi","round":22,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run exact support-matrix joins across the fresh six-row projections and smooth pair atlases, seeking a strict Pareto improvement.","costUsd":0.007,"at":"2026-09-03T21:24:46.788Z","sourceCount":5,"ledgerOffset":108},{"requestId":"51781ba84c061ce09526dd103fcc4151","agentId":"tess","round":22,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Privately cross-join the fresh atlases and seek a strict exact support improvement, emphasizing columns common to independent pair systems.","costUsd":0.007,"at":"2026-09-03T21:24:46.788Z","sourceCount":5,"ledgerOffset":109},{"requestId":"a95f072df1bb829f9acc3384406f4a56","agentId":"mira","round":23,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct an exact incidence matrix from the fresh pair atlases and prioritize any 4×5 or 5×5 intersection.","costUsd":0.007,"at":"2026-09-03T21:34:52.000Z","sourceCount":5,"ledgerOffset":110},{"requestId":"dfeb9216df56b5277a98aaf431558929","agentId":"orum","round":23,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Prioritize exact completion of the two new pair atlases and cross-score all returned columns against every proposed row set.","costUsd":0.007,"at":"2026-09-03T21:34:52.000Z","sourceCount":5,"ledgerOffset":111},{"requestId":"90393ff945a01544ac474cf3023d07d7","agentId":"pip","round":23,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Turn the fresh 680-based and high-difference projections into an exact Pareto improvement while avoiding every registry-covered region.","costUsd":0.007,"at":"2026-09-03T21:34:52.000Z","sourceCount":5,"ledgerOffset":112},{"requestId":"5a7779327b00469612eba467cdcb0d48","agentId":"solvi","round":23,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Join the fresh exact projections and seek a strict Pareto improvement without entering any registry-covered region.","costUsd":0.007,"at":"2026-09-03T21:34:52.000Z","sourceCount":5,"ledgerOffset":113},{"requestId":"6af4a1305cc871f654d9cec50ab32405","agentId":"tess","round":23,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Exact Pareto search in the first uncovered high-difference bands, prioritizing atlas intersections that can produce five columns with support at least four.","costUsd":0.007,"at":"2026-09-03T21:34:52.000Z","sourceCount":5,"ledgerOffset":114},{"requestId":"0bc95105815351e22e0dc7495d5ee3e4","agentId":"mira","round":24,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Search for exact cross-support columns joining the strongest fresh six-row projections, prioritizing a verified 4×5 or 5×5 rectangle.","costUsd":0.007,"at":"2026-09-03T22:32:33.076Z","sourceCount":5,"ledgerOffset":115},{"requestId":"cf4af68404599c7bbd32e8ec791a8980","agentId":"orum","round":24,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Prioritize exact intersections from the fresh 252/615 and 783/953 atlases, seeking a Pareto improvement rather than more isolated six-row anchors.","costUsd":0.007,"at":"2026-09-03T22:32:33.076Z","sourceCount":5,"ledgerOffset":116},{"requestId":"2a5c22d98879a8a30253bf40dc671cac","agentId":"pip","round":24,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Exploit the fresh high-difference anchors by exact atlas saturation and intersection, seeking five distinct columns with support at least five.","costUsd":0.007,"at":"2026-09-03T22:32:33.076Z","sourceCount":5,"ledgerOffset":117},{"requestId":"fc82d09a9df81c89069e5acf073c0338","agentId":"solvi","round":24,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Exploit the new six-row anchors through exact pair-atlas intersections, seeking a strict Pareto improvement without revisiting registry-covered regions.","costUsd":0.007,"at":"2026-09-03T22:32:33.076Z","sourceCount":5,"ledgerOffset":118},{"requestId":"4f216d6cc29998a740be679abfb9407b","agentId":"tess","round":24,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Exploit the fresh high-difference anchors by exact pair completion and cross-row support maximization.","costUsd":0.007,"at":"2026-09-03T22:32:33.076Z","sourceCount":5,"ledgerOffset":119},{"requestId":"de3ca851bf84887cb2e399a680d6cf1b","agentId":"mira","round":25,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Prioritize exact intersections that can turn the new anchors into at least five columns of support five, while avoiding all completed row sets.","costUsd":0.007,"at":"2026-09-03T22:42:36.770Z","sourceCount":5,"ledgerOffset":120},{"requestId":"2879b2725dc0182e043042e3cafb6526","agentId":"orum","round":25,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Exhaust the fresh atlas intersections for a verified (4,5,5,5,5) or stronger support vector.","costUsd":0.007,"at":"2026-09-03T22:42:36.770Z","sourceCount":5,"ledgerOffset":121},{"requestId":"e0a1a98d6fbf6d1942dda19ba90f3b08","agentId":"pip","round":25,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run exact fresh cross-atlas searches combining the 216/243/405 projection with Orum’s mixed six-row anchor sets, targeting a verified 4×5 or stronger support vector.","costUsd":0.007,"at":"2026-09-03T22:42:36.770Z","sourceCount":5,"ledgerOffset":122},{"requestId":"1f3991e50009a67a0d06ea24d12ba1b8","agentId":"solvi","round":25,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use exact pair-completion intersections to seek a new support vector at least (4,5,5,5,5), prioritizing the fresh high-difference and mixed-family rows.","costUsd":0.007,"at":"2026-09-03T22:42:36.770Z","sourceCount":5,"ledgerOffset":123},{"requestId":"1afe77e92ac10a4a70f4c6810feb0eab","agentId":"tess","round":25,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Intersect the fresh high-difference atlases and extend the common-factor construction into an untouched q=14–22 regime, seeking a verified strict Pareto improvement.","costUsd":0.007,"at":"2026-09-03T22:42:36.770Z","sourceCount":5,"ledgerOffset":124},{"requestId":"e6dbd7b35dd8e6f5ea04f6a2fdd69a94","agentId":"mira","round":26,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a genuinely parameterized, non-Bremner algebraic family for balanced support: study the square-additive variety m_ij^2-d_i^2=4N_j, derive elliptic fibrations or rational families with several independently varying columns, and search positive-rank fibers rather than completing fixed differ","costUsd":0.007,"at":"2026-09-03T23:20:58.625Z","sourceCount":5,"ledgerOffset":125},{"requestId":"d28870d7fd83c39b056a5454ca63d265","agentId":"orum","round":26,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a deduplicated verified incidence graph and search it for balanced K₅,₅ or Pareto improvements using SAT/exact cover, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-03T23:20:58.625Z","sourceCount":5,"ledgerOffset":126},{"requestId":"c85e53df95e749a46699da76cc413e7e","agentId":"pip","round":26,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Round 26 modular/local lane: test the conjectural square-class-clone family d_i≡±d_0 (mod M), using local quadratic-residue incidence and CRT to prune or force balanced compatible rows before exact validation. No divisor-completion job will be submitted.","costUsd":0.007,"at":"2026-09-03T23:20:58.625Z","sourceCount":5,"ledgerOffset":127},{"requestId":"b65527e7af5ebdb53c83f04bd526688d","agentId":"solvi","round":26,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run a stochastic, optimization-guided factor-coordinate search for genuinely new five-row/five-column incidence seeds, followed by independent exact bigint verification; submit no divisor_completion job.","costUsd":0.007,"at":"2026-09-03T23:20:58.625Z","sourceCount":5,"ledgerOffset":128},{"requestId":"3514779d952b85bffe78cabcea98ef03","agentId":"tess","round":26,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Build an adversarial ledger of existing near-misses and select one genuinely adaptive, non-divisor-completion experiment that can either produce balanced support or issue a rigorous obstruction.","costUsd":0.007,"at":"2026-09-03T23:20:58.625Z","sourceCount":5,"ledgerOffset":129},{"requestId":"80ef332fa1cbcf4d60f880837d7aa178","agentId":"mira","round":27,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct an explicit rational parametrization of a three-row square-additive fiber and test whether its elliptic quotient deforms the seed column into additional integral columns, staying in the constructive algebraic-family lane.","costUsd":0.007,"at":"2026-09-03T23:31:04.629Z","sourceCount":5,"ledgerOffset":130},{"requestId":"8e6d04eaa16abaa1a729c910dc2ae3b4","agentId":"orum","round":27,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run an incidence-hypergraph/exact-cover search exclusively on bigint-verified cells from the six-row seed scans and known certificates, seeking a distinct 5-row by 5-column all-one submatrix.","costUsd":0.007,"at":"2026-09-03T23:31:04.629Z","sourceCount":5,"ledgerOffset":131},{"requestId":"3a2f6ff35e15277fef6bf11646d081ef","agentId":"pip","round":27,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Design a two-stage CRT filter that forces compatible square roots modulo prime powers while avoiding the rejected single-modulus clone lifts.","costUsd":0.007,"at":"2026-09-03T23:31:04.629Z","sourceCount":5,"ledgerOffset":132},{"requestId":"d631680e847d2dfd89160db6b1e0f579","agentId":"solvi","round":27,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided seed discovery for new high-difference row sets, then perform exact bigint verification while targeting Pareto improvements such as (4,5,5,5,5).","costUsd":0.007,"at":"2026-09-03T23:31:04.629Z","sourceCount":5,"ledgerOffset":133},{"requestId":"a6d385bad35914177dab0010990fac6f","agentId":"tess","round":27,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run an adversarial, adaptive attack on the q_r=t+r fibers while developing saturation certificates for modular-clone and elliptic-family searches.","costUsd":0.007,"at":"2026-09-03T23:31:04.629Z","sourceCount":5,"ledgerOffset":134},{"requestId":"7d0eeebfc935f485af7748360e2d8ad0","agentId":"mira","round":28,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct an explicit rational parametrization of a three-row genus-one fiber arising from the primitive step-3 family, then seek integral specializations with additional exact square rows.","costUsd":0.007,"at":"2026-09-03T23:41:08.347Z","sourceCount":5,"ledgerOffset":135},{"requestId":"d170ae377973631cdb5030e639d37f37","agentId":"orum","round":28,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct an incidence-hypergraph/SAT exact-cover search using only verified exact cells, targeting five distinct N-columns with five distinct difference rows.","costUsd":0.007,"at":"2026-09-03T23:41:08.347Z","sourceCount":5,"ledgerOffset":136},{"requestId":"7a08b6efdaaa52f469cc8b9263061da0","agentId":"pip","round":28,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use a two-stage local CRT sieve on selected rows of the new step-3 seed columns to force compatible square-root branches before any exact search.","costUsd":0.007,"at":"2026-09-03T23:41:08.347Z","sourceCount":5,"ledgerOffset":137},{"requestId":"a22868124fe5f76c90aa0afa659bfb75","agentId":"solvi","round":28,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run stochastic, optimization-guided discovery over uncovered primitive seed families and translated square-root configurations, then submit only Pareto-promising candidates to exact bigint verification.","costUsd":0.007,"at":"2026-09-03T23:41:08.347Z","sourceCount":5,"ledgerOffset":138},{"requestId":"783b9987d62ad8f3ac750c046b63b022","agentId":"tess","round":28,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run adversarial falsification and adaptive support-first tests on the h=3 family while constructing finite-range saturation certificates; keep modular lifts only as null controls.","costUsd":0.007,"at":"2026-09-03T23:41:08.347Z","sourceCount":5,"ledgerOffset":139},{"requestId":"c91108febb30e623866551e87ca99a83","agentId":"mira","round":29,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct and search a parameterized family of elliptic fibers extending the q_r=t+5r identity, targeting multiple distinct columns with balanced exact support.","costUsd":0.007,"at":"2026-09-03T23:51:10.406Z","sourceCount":5,"ledgerOffset":140},{"requestId":"f17e9fbcebecaa20de4a75896b6fb4bd","agentId":"orum","round":29,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use an incidence-hypergraph search over independently generated, exactly verified cells from the step-5 and inverse-residue methods, targeting a certified 4×5 or 5×5 row-column biclique.","costUsd":0.007,"at":"2026-09-03T23:51:10.406Z","sourceCount":5,"ledgerOffset":141},{"requestId":"59488a24b8dd8bb6b4b03caa8edd2b05","agentId":"pip","round":29,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Build a modular CRT sieve that forces or sharply prunes simultaneous row compatibility for the step-5 and inverse-residue families.","costUsd":0.007,"at":"2026-09-03T23:51:10.406Z","sourceCount":5,"ledgerOffset":142},{"requestId":"8240ba0e0d318b3c3382e1798034697d","agentId":"solvi","round":29,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run stochastic, optimization-guided discovery of new step-h product seeds and inverse-residue CRT wheels, then use family_scan and Orum-only exact bigint verification to seek balanced support.","costUsd":0.007,"at":"2026-09-03T23:51:10.406Z","sourceCount":5,"ledgerOffset":143},{"requestId":"b9f79941e6510c984605a0330b39a156","agentId":"tess","round":29,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run an adversarial, exact-support campaign against the proposed step-5 and cross-anchor constructions, prioritizing strict Pareto improvements over isolated high-support columns.","costUsd":0.007,"at":"2026-09-03T23:51:10.406Z","sourceCount":5,"ledgerOffset":144},{"requestId":"80ab12deb94cc91d6f749aad4bd4573b","agentId":"mira","round":30,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Build a multi-column constructive family by linking the h=7 elliptic fibers with prime-minus-one wheel data, targeting exact support patterns (5,5,5,5,5) or (4,5,5,5,5).","costUsd":0.007,"at":"2026-09-04T00:01:13.838Z","sourceCount":5,"ledgerOffset":145},{"requestId":"1be43100252b99d00195827e7257bead","agentId":"orum","round":30,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct an incidence hypergraph from only bigint-verified cells produced by the new h=7 and modular-wheel regimes, then run SAT/exact-cover searches for 5×5 and 6×4 support.","costUsd":0.007,"at":"2026-09-04T00:01:13.838Z","sourceCount":5,"ledgerOffset":146},{"requestId":"6e9e81215ec0724c912be142869b2c92","agentId":"pip","round":30,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use modular, local, and congruence design to prune or force compatible rows around the new h=7 and prime-minus-one anchors, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-04T00:01:13.838Z","sourceCount":5,"ledgerOffset":147},{"requestId":"19d6c09ede46a2bc3b5f2df92bd2d79d","agentId":"solvi","round":30,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run stochastic, optimization-guided seed discovery over uncovered h=7 elliptic parameters, inverse-residue wheels, and cross-anchor row mutations, then retain only exact bigint Pareto improvements.","costUsd":0.007,"at":"2026-09-04T00:01:13.838Z","sourceCount":5,"ledgerOffset":148},{"requestId":"ca9797675d1aaec81fcd41107963e077","agentId":"tess","round":30,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially falsify the h=7 quartics and inverse-residue wheels, then adaptively build exact incidence saturation certificates targeting strict Pareto improvements.","costUsd":0.007,"at":"2026-09-04T00:01:13.838Z","sourceCount":5,"ledgerOffset":149},{"requestId":"fed2070c2f3e787e1e43479f6e6f42be","agentId":"mira","round":31,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Develop a linked parametrization of the new arithmetic-progression elliptic fibers, seeking shared rational-point structure across h=11 and h=13 rather than isolated seed scans.","costUsd":0.007,"at":"2026-09-04T00:11:17.884Z","sourceCount":5,"ledgerOffset":150},{"requestId":"12c30374be3c4e56f28af0bcb3e6c7fe","agentId":"orum","round":31,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Build an exact incidence hypergraph from newly returned, Tess-verified cells and search it for 5×5, 4×5, or 5×4 bicliques using SAT/exact cover.","costUsd":0.007,"at":"2026-09-04T00:11:17.884Z","sourceCount":5,"ledgerOffset":151},{"requestId":"e8eb8eb2d25a8c4dd84a0bc5a11b1156","agentId":"pip","round":31,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Design a modular and local congruence filter for fresh elliptic and wheel row sets, targeting compatible five-row systems without repeating covered regions.","costUsd":0.007,"at":"2026-09-04T00:11:17.884Z","sourceCount":5,"ledgerOffset":152},{"requestId":"0ab9273607db92700adac5c05bde2aad","agentId":"solvi","round":31,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run stochastic, optimization-guided seed discovery over uncovered progression spacings, wheel rows, and mixed row subsets, targeting Pareto improvement toward sorted supports [4,5,5,5,5] and [5,5,5,5,5], followed by exact bigint verification.","costUsd":0.007,"at":"2026-09-04T00:11:17.884Z","sourceCount":5,"ledgerOffset":153},{"requestId":"f15639d6a490d4c02ecf10f6fc5c9cec","agentId":"tess","round":31,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially falsify the fresh h=11/h=13 and prime-wheel routes, and require saturation certificates before any adaptive expansion.","costUsd":0.007,"at":"2026-09-04T00:11:17.884Z","sourceCount":5,"ledgerOffset":154},{"requestId":"af688d8ef784ca577ac7b6444e160986","agentId":"mira","round":32,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a modular incidence sieve that selects difference rows with compatible local square masks and forces candidate columns into shared CRT classes.","costUsd":0.007,"at":"2026-09-04T00:21:19.092Z","sourceCount":5,"ledgerOffset":155},{"requestId":"e2395321472c9ad9b9121e38ed31d026","agentId":"orum","round":32,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided mutations of the new AP and wheel/CRT seed families to discover balanced high-support candidates, then apply exact bigint verification.","costUsd":0.007,"at":"2026-09-04T00:21:19.092Z","sourceCount":5,"ledgerOffset":156},{"requestId":"358cb34907446b6aa9e1a13630dfc13c","agentId":"pip","round":32,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an incidence-hypergraph/exact-cover search on the newly verified exact cells from the h=17,18,20 fibers and 7560-wheel projections, targeting a 5-row, 5-column biclique and accepting 6×4 as an intermediate.","costUsd":0.007,"at":"2026-09-04T00:21:19.092Z","sourceCount":5,"ledgerOffset":157},{"requestId":"50052312d3e69179b92d9c75f1cec536","agentId":"solvi","round":32,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially test whether the new progression and wheel regimes contain balanced support, while converting negative scans into auditable saturation evidence and adapting only when exact incidence data justify it.","costUsd":0.007,"at":"2026-09-04T00:21:19.092Z","sourceCount":5,"ledgerOffset":158},{"requestId":"4b3ad57319581292f023ff440f89d5cd","agentId":"tess","round":32,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Develop a constructive rational-point parametrization for the arithmetic-progression fibers, prioritizing shared elliptic structure rather than another fixed-seed scan.","costUsd":0.007,"at":"2026-09-04T00:21:19.092Z","sourceCount":5,"ledgerOffset":159},{"requestId":"946eea7173fbcffcdaf999d66360fb94","agentId":"mira","round":33,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design CRT-local row sets whose square conditions share multiple admissible N residue classes, then prune family scans to these compatible classes.","costUsd":0.007,"at":"2026-09-04T00:31:22.072Z","sourceCount":5,"ledgerOffset":160},{"requestId":"afde0bcd314ec51f3b22494b1868ba24","agentId":"orum","round":33,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided mutations of bridge, wheel, and prime-power fiber seeds to discover nonrepeating high-support tuples, followed by exact bigint verification.","costUsd":0.007,"at":"2026-09-04T00:31:22.072Z","sourceCount":5,"ledgerOffset":161},{"requestId":"f50a7cec581c9afef1e002b906e1295e","agentId":"pip","round":33,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an exact-cover search on verified nonanchor incidence cells, targeting a 5×5 certificate or a 6×4 Pareto improvement without repeating covered regions.","costUsd":0.007,"at":"2026-09-04T00:31:22.072Z","sourceCount":5,"ledgerOffset":162},{"requestId":"c437cb792ff149860e8293caa0a20d4b","agentId":"solvi","round":33,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially falsify the proposed fibers and certify saturation before allocating another adaptive search.","costUsd":0.007,"at":"2026-09-04T00:31:22.072Z","sourceCount":5,"ledgerOffset":163},{"requestId":"5a782cca2d2df4f366689422632ff050","agentId":"tess","round":33,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct an elliptic/rational-point family from a fresh six-row P_h seed, then seek five integral columns with support at least five.","costUsd":0.007,"at":"2026-09-04T00:31:22.072Z","sourceCount":5,"ledgerOffset":164},{"requestId":"e321e05e9968302e3c7b928c3ae1b53f","agentId":"mira","round":34,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a modularly pruned row family with several compatible N residue classes, staying in the local/congruence lane and avoiding covered anchor regimes.","costUsd":0.007,"at":"2026-09-04T00:41:28.161Z","sourceCount":5,"ledgerOffset":165},{"requestId":"fa0e17cef3e9f22519270821f7cdda3a","agentId":"orum","round":34,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run stochastic, optimization-guided mutations of verified incidence seeds in uncovered parameter regimes, followed by exact bigint verification of Pareto-leading support patterns.","costUsd":0.007,"at":"2026-09-04T00:41:28.161Z","sourceCount":5,"ledgerOffset":166},{"requestId":"c0153ef5aca96668d2b3357c417ec87e","agentId":"pip","round":34,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an exact-cover search over verified incidence cells from the 10008 fiber and Orum’s cross-cloud mutations, targeting nonanchor 5x5 or 6x4 rectangles.","costUsd":0.007,"at":"2026-09-04T00:41:28.161Z","sourceCount":5,"ledgerOffset":167},{"requestId":"d9df7f9c19e7ef8fbdc03efb1d502b72","agentId":"solvi","round":34,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run adversarial falsification and saturation checks on the freshest row families, with adaptive branching toward exact Pareto improvements.","costUsd":0.007,"at":"2026-09-04T00:41:28.161Z","sourceCount":5,"ledgerOffset":168},{"requestId":"cf62e06eb67d7945bd095588d4f871bf","agentId":"tess","round":34,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a fresh multi-column algebraic family by parameterizing factor pairs q_i=c+Ht_i and imposing simultaneous extra columns through the resulting rational-point equations, including genus-one/elliptic cases.","costUsd":0.007,"at":"2026-09-04T00:41:28.161Z","sourceCount":5,"ledgerOffset":169},{"requestId":"c394c9b75a3a768aa14fb8714c98fd6f","agentId":"mira","round":35,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a fresh composite-CRT row family whose local square classes force at least four compatible difference rows before exact incidence extraction.","costUsd":0.007,"at":"2026-09-04T00:51:30.295Z","sourceCount":5,"ledgerOffset":170},{"requestId":"33ed04c4824c52ce80e7206213980022","agentId":"orum","round":35,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run stochastic, optimization-guided mutations of the fresh divisor fibers and CRT-compatible row sets, then exact-bigint verify the best balanced-support candidates.","costUsd":0.007,"at":"2026-09-04T00:51:30.295Z","sourceCount":5,"ledgerOffset":171},{"requestId":"76ffdf20e21cafc947bb55a69da6a070","agentId":"pip","round":35,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Perform exact-cover/SAT extraction over the fresh verified-cell incidence clouds, targeting a 5×5 biclique while excluding anchor-only support.","costUsd":0.007,"at":"2026-09-04T00:51:30.295Z","sourceCount":5,"ledgerOffset":172},{"requestId":"f203d65dc2c9928b96d9a74cf3b0fd4f","agentId":"solvi","round":35,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially test whether the fresh fibers can exceed the current support frontier, while producing saturation certificates for every exhausted row cloud.","costUsd":0.007,"at":"2026-09-04T00:51:30.295Z","sourceCount":5,"ledgerOffset":173},{"requestId":"2e768e0452b00ee5ca98796c67260ffe","agentId":"tess","round":35,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a new algebraic family by gluing two difference rows through a rational conic, then imposing additional rows through explicit genus-one curves, while retaining only exact integral points outside covered regimes.","costUsd":0.007,"at":"2026-09-04T00:51:30.295Z","sourceCount":5,"ledgerOffset":174},{"requestId":"083219ba2876806ca5489b0aa8cb3a4b","agentId":"mira","round":36,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a modular compatibility atlas that ranks fresh difference sets by simultaneous local square support before exact scanning.","costUsd":0.007,"at":"2026-09-04T01:01:33.781Z","sourceCount":5,"ledgerOffset":175},{"requestId":"a232b098b9336ace541b0a9ef64bc436","agentId":"orum","round":36,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run stochastic, optimization-guided mutations of exact support-3 seed clouds, including the 2052 and 421 rows, then bigint-verify only Pareto-leading candidates.","costUsd":0.007,"at":"2026-09-04T01:01:33.781Z","sourceCount":5,"ledgerOffset":176},{"requestId":"b2cbffe87f01314b94f1deb2d495f67e","agentId":"pip","round":36,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an exact-cover/SAT search on merged verified-cell incidence data, prioritizing a strict 4×5 improvement and then 5×5.","costUsd":0.007,"at":"2026-09-04T01:01:33.781Z","sourceCount":5,"ledgerOffset":177},{"requestId":"397159af32e13207a3319f514bc58bb4","agentId":"solvi","round":36,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially falsify the new CRT and anchor mechanisms, then obtain exact saturation certificates before adapting the search.","costUsd":0.007,"at":"2026-09-04T01:01:33.781Z","sourceCount":5,"ledgerOffset":178},{"requestId":"7877c947a0b3d573450f2d3bb0f290aa","agentId":"tess","round":36,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a new rational-point family for a fifth row from common-divisor equations, using the 2052 and 27720 structures only as seeds.","costUsd":0.007,"at":"2026-09-04T01:01:33.781Z","sourceCount":5,"ledgerOffset":179},{"requestId":"8a1865c92a19d75469a2e5937b144c71","agentId":"mira","round":37,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an exact-cover/SAT search on the merged verified-cell incidence hypergraph, prioritizing a strict 4×5 improvement before testing 5×5.","costUsd":0.007,"at":"2026-09-06T10:30:23.834Z","sourceCount":5,"ledgerOffset":180},{"requestId":"c5163dd6b26c9061bca5b39706cbbe10","agentId":"orum","round":37,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially test the claimed fresh geometries while building exact saturation certificates.","costUsd":0.007,"at":"2026-09-06T10:30:23.834Z","sourceCount":5,"ledgerOffset":181},{"requestId":"6b29b6024c34cffa0898145fc8f40518","agentId":"pip","round":37,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run stochastic, optimization-guided seed discovery from the fresh balanced incidence seeds, then perform exact bigint verification of Pareto-improving bicliques.","costUsd":0.007,"at":"2026-09-06T10:30:23.834Z","sourceCount":5,"ledgerOffset":182},{"requestId":"f058ce7794cfce197f9ef236a5734a9c","agentId":"solvi","round":37,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Develop a constructive rational-point family from the fresh support-5 seed 30400, using coordinate normalization t_i=2x_i+d_i so that t_i^2-t_j^2=d_i^2-d_j^2 and common N is automatic.","costUsd":0.007,"at":"2026-09-06T10:30:23.834Z","sourceCount":5,"ledgerOffset":183},{"requestId":"27fd76709193157e0d2bd976612b63fb","agentId":"tess","round":37,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a modular compatibility sieve for the fresh Pip bridge rows and Mira’s H=27720 pools, without numerical approximation or divisor completion.","costUsd":0.007,"at":"2026-09-06T10:30:23.834Z","sourceCount":5,"ledgerOffset":184},{"requestId":"5c3a417289fbfdd52f6065f5e6e85c80","agentId":"mira","round":38,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an exact-cover/SAT search on the merged verified-cell incidence hypergraph, prioritizing a 4×5 biclique and then testing 5×5 extensions.","costUsd":0.007,"at":"2026-09-06T10:40:30.280Z","sourceCount":5,"ledgerOffset":185},{"requestId":"96ccdf8128f8fc5eac801a25b056be68","agentId":"orum","round":38,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially saturate the union of the reported exact row clouds, then adaptively select only genuinely new rows or families that can improve the weakest-first support profile.","costUsd":0.007,"at":"2026-09-06T10:40:30.280Z","sourceCount":5,"ledgerOffset":186},{"requestId":"ee4b8e4c1e3f3b5f3572ab8a601da696","agentId":"pip","round":38,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run stochastic, optimization-guided seed discovery around Mira’s four-row anchor, Pip’s mixed bridge supports, and Orum’s finite union cloud; then exact-verify only Pareto-improving bicliques.","costUsd":0.007,"at":"2026-09-06T10:40:30.280Z","sourceCount":5,"ledgerOffset":187},{"requestId":"fe664f98945a97f47be3a223ac02d287","agentId":"solvi","round":38,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct an exact genus-one family from Mira’s four-row seed, seeking five integral N-columns and then a common fifth difference.","costUsd":0.007,"at":"2026-09-06T10:40:30.280Z","sourceCount":5,"ledgerOffset":188},{"requestId":"85d98831eb63fc73d2679b79587108c2","agentId":"tess","round":38,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use the h=27720 multiplier geometry for a modular/CRT search that prunes or forces compatible added rows without divisor completion.","costUsd":0.007,"at":"2026-09-06T10:40:30.280Z","sourceCount":5,"ledgerOffset":189},{"requestId":"7cfe4caa51917a7e2796d8159a92dfd9","agentId":"mira","round":39,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an exact-cover/SAT search on the merged, bigint-verified incidence hypergraph, using the 30400 support-5 anchor and Mira’s four-row columns without requesting divisor completion.","costUsd":0.007,"at":"2026-09-06T10:50:32.152Z","sourceCount":5,"ledgerOffset":190},{"requestId":"bd7f1a7276bd6fa46911634540017cbf","agentId":"orum","round":39,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially reconstruct the claimed incidence frontier, eliminate duplicate scans, and adapt the next experiment to the sparsest missing edges.","costUsd":0.007,"at":"2026-09-06T10:50:32.152Z","sourceCount":5,"ledgerOffset":191},{"requestId":"d8026b412f8cbb70eeefd032be2b20fe","agentId":"pip","round":39,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run stochastic, optimization-guided family-seed discovery around the Mira multiplier layers and mixed 411 bridge geometry, then exact bigint-verify only Pareto-improving incidence structures.","costUsd":0.007,"at":"2026-09-06T10:50:32.152Z","sourceCount":5,"ledgerOffset":192},{"requestId":"b5f12d89528720fc4f9cc951fed925c7","agentId":"solvi","round":39,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct a genuinely new rational-point family by treating the square-additive matrix symmetrically. For distinct rational row parameters r1,r2,r3, use the genus-one curve y_i^2=4n+r_i^2 and generate columns from its Mordell–Weil points, then impose two further rows exactly.","costUsd":0.007,"at":"2026-09-06T10:50:32.152Z","sourceCount":5,"ledgerOffset":193},{"requestId":"3922a8028e54d74871d4718d83c364f8","agentId":"tess","round":39,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Design a modular local sieve that forces compatible additional rows around the verified support-4 anchors without repeating covered regions.","costUsd":0.007,"at":"2026-09-06T10:50:32.152Z","sourceCount":5,"ledgerOffset":194},{"requestId":"feeb1ca08b1c94ada13e9da9f55ad7be","agentId":"mira","round":40,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct an exact incidence hypergraph from all verified seed columns and run support-sorted exact-cover/SAT branching for 5×5, then 6×4 and 4×5 configurations.","costUsd":0.007,"at":"2026-09-06T11:00:36.214Z","sourceCount":5,"ledgerOffset":195},{"requestId":"2d635fdf8af3a0ef5c712df73731c7c3","agentId":"orum","round":40,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially test the fresh mutation and multiplier masks by demanding complete incidence streams and exact saturation certificates before expanding the search.","costUsd":0.007,"at":"2026-09-06T11:00:36.214Z","sourceCount":5,"ledgerOffset":196},{"requestId":"a3e9e74aa8b9b54522388e52e13f78d7","agentId":"pip","round":40,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use stochastic, optimization-guided mutations of the fresh Mira-layer and mutation-bridge seeds, followed by exact bigint verification.","costUsd":0.007,"at":"2026-09-06T11:00:36.214Z","sourceCount":5,"ledgerOffset":197},{"requestId":"8d2b0a02dd1dae82abddbdf7d7565743","agentId":"solvi","round":40,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct an exact rational-point family from the strongest support-4 seeds, then seek five integral columns and a fifth common difference.","costUsd":0.007,"at":"2026-09-06T11:00:36.214Z","sourceCount":5,"ledgerOffset":198},{"requestId":"d436fc4b9e326649c24cab104d3ac822","agentId":"tess","round":40,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use modular, local congruence design around the exact support-4 seeds to prune candidate rows and force compatible extensions, without divisor completion.","costUsd":0.007,"at":"2026-09-06T11:00:36.214Z","sourceCount":5,"ledgerOffset":199},{"requestId":"1591b30d42f9bbc4d542a1044ff17bb0","agentId":"mira","round":41,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run exact-cover/SAT extraction over the newly verified cross-layer masks, targeting a strict 4×5 or 6×4 certificate before 5×5.","costUsd":0.007,"at":"2026-09-06T11:10:38.478Z","sourceCount":5,"ledgerOffset":200},{"requestId":"97ab674cbb23abac6b919c8e60ff7f60","agentId":"orum","round":41,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially test whether the reported seed structures can saturate to a strict 4×5, 6×4, or 5×5 certificate without repeating registry-covered computation.","costUsd":0.007,"at":"2026-09-06T11:10:38.478Z","sourceCount":5,"ledgerOffset":201},{"requestId":"71241bc2edcf82660f225927b5f68aae","agentId":"pip","round":41,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run stochastic, optimization-guided mutations around the fresh balanced seed palettes, targeting Pareto improvements in weakest column and row support, then exact bigint verification.","costUsd":0.007,"at":"2026-09-06T11:10:38.478Z","sourceCount":5,"ledgerOffset":202},{"requestId":"a1e794c444b5ff802d1d39903df060a8","agentId":"solvi","round":41,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct normalized rational families for simultaneous cells, using conic parametrizations and elliptic-curve fibers, while avoiding fixed Bremner specializations.","costUsd":0.007,"at":"2026-09-06T11:10:38.478Z","sourceCount":5,"ledgerOffset":203},{"requestId":"f7220fd7ba52236152a2ba3f0aaea5c4","agentId":"tess","round":41,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use modular local congruence filters to force or prune compatible rows around the strongest exact seed palettes, without repeating covered global scans.","costUsd":0.007,"at":"2026-09-06T11:10:38.478Z","sourceCount":5,"ledgerOffset":204},{"requestId":"7f972b2388431056eccb403616b116e4","agentId":"mira","round":42,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run stochastic, optimization-guided mutation of fresh difference palettes, then apply exact bigint verification to retain only Pareto-improving support structures.","costUsd":0.007,"at":"2026-09-06T11:20:41.477Z","sourceCount":5,"ledgerOffset":205},{"requestId":"04a4e056821a2f15c31dea8b7a3a6a76","agentId":"orum","round":42,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a new rational-point family from the equations N=t(t+d), centered on the exact 4×3 core, rather than performing another stochastic palette scan.","costUsd":0.007,"at":"2026-09-06T11:20:41.477Z","sourceCount":5,"ledgerOffset":206},{"requestId":"646d9e06d956e28f2590bd31776ba317","agentId":"pip","round":42,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially decide whether the fresh 411/438 branches saturate or yield a strict Pareto improvement, using exact support masks only.","costUsd":0.007,"at":"2026-09-06T11:20:41.477Z","sourceCount":5,"ledgerOffset":207},{"requestId":"e902c057eb2de037db7fdfb23e91cc1d","agentId":"solvi","round":42,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Design a modular local sieve that prunes row palettes before exact validation and seeks candidates with minimum row and column support at least 3.","costUsd":0.007,"at":"2026-09-06T11:20:41.477Z","sourceCount":5,"ledgerOffset":208},{"requestId":"2aba5036afac2d8906ed590783fcc455","agentId":"tess","round":42,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run an exact-cover/SAT search on recomputed verified-cell incidence, targeting a strict 4×5 or 6×4 Pareto improvement.","costUsd":0.007,"at":"2026-09-06T11:20:41.477Z","sourceCount":5,"ledgerOffset":209},{"requestId":"009740ca9b1d087c713745d557acfb7b","agentId":"mira","round":43,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run stochastic, optimization-guided seed discovery in fresh multiplier and gcd-diversified layers, then exact-bigint verify only support-balanced candidates.","costUsd":0.007,"at":"2026-09-06T11:30:42.365Z","sourceCount":5,"ledgerOffset":210},{"requestId":"99d88f041ac9125f029e07038501993a","agentId":"orum","round":43,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Develop a genuinely new constructive rational/elliptic family for simultaneous factor differences, avoiding the covered fixed specializations and broad denominator search.","costUsd":0.007,"at":"2026-09-06T11:30:42.365Z","sourceCount":5,"ledgerOffset":211},{"requestId":"e547281d2c5024d5ef54104ca733b0c8","agentId":"pip","round":43,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially falsify the fresh 372/690 pivot branches, certify scoped saturation exactly, and adapt only from genuine Pareto improvements.","costUsd":0.007,"at":"2026-09-06T11:30:42.365Z","sourceCount":5,"ledgerOffset":212},{"requestId":"ba51d3434ad25dce88395ef1fc68ba7d","agentId":"solvi","round":43,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use modular and local congruence sieves to construct fresh compatible row palettes, avoiding broad repeats and all divisor_completion requests.","costUsd":0.007,"at":"2026-09-06T11:30:42.365Z","sourceCount":5,"ledgerOffset":213},{"requestId":"0b346a423e5c371b7cabf293112c26a7","agentId":"tess","round":43,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Build an incidence-hypergraph exact-cover search from all verified cells in the 372/690 pivot branches and prior certified seeds, targeting five difference vertices and five N vertices with complete incidence.","costUsd":0.007,"at":"2026-09-06T11:30:42.365Z","sourceCount":5,"ledgerOffset":214},{"requestId":"363b2666b8ca77539f501e236a8bdc9a","agentId":"mira","round":44,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run stochastic, optimization-guided family_scan seed discovery on fresh high-layer projective mutations and large-difference bridge mutations, then perform exact bigint verification through orum only.","costUsd":0.007,"at":"2026-09-08T03:02:11.172Z","sourceCount":5,"ledgerOffset":215},{"requestId":"b3233be23b434e394bc2b187a92581b3","agentId":"orum","round":44,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Develop a genuinely new constructive rational/elliptic family for simultaneous factor differences, avoiding the covered fixed specializations and broad denominator search.","costUsd":0.007,"at":"2026-09-08T03:02:11.172Z","sourceCount":5,"ledgerOffset":216},{"requestId":"3bc72869370d9744060a481523182a24","agentId":"pip","round":44,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially test the fresh large-difference and high-layer regimes while seeking an exact Pareto improvement, using only saturation certificates and adaptive scans.","costUsd":0.007,"at":"2026-09-08T03:02:11.172Z","sourceCount":5,"ledgerOffset":217},{"requestId":"a7f5919f53aaf85f8067e2e8f95f5655","agentId":"solvi","round":44,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use modular and local congruence sieves to construct fresh compatible row palettes, avoiding broad repeats and all divisor_completion requests.","costUsd":0.007,"at":"2026-09-08T03:02:11.172Z","sourceCount":5,"ledgerOffset":218},{"requestId":"d7e6e5789a032e888f71b58506923e27","agentId":"tess","round":44,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Build an incidence-hypergraph exact-cover search from all verified cells in the 372/690 pivot branches and prior certified seeds, targeting five difference vertices and five N vertices with complete incidence.","costUsd":0.007,"at":"2026-09-08T03:02:11.172Z","sourceCount":5,"ledgerOffset":219},{"requestId":"c89c90938226285f075eac9b1a3b653e","agentId":"mira","round":45,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run stochastic, optimization-guided family_scan seed discovery in uncovered high-difference and prime-layer regimes, then submit only Pareto-promising seeds for exact bigint verification.","costUsd":0.007,"at":"2026-09-08T03:12:13.028Z","sourceCount":5,"ledgerOffset":220},{"requestId":"f5c9215e5432a9a0b3007425eae2ea1b","agentId":"orum","round":45,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct rational parametrizations of multi-difference columns and seek a genus-one family whose rational points produce a fifth shared difference.","costUsd":0.007,"at":"2026-09-08T03:12:13.028Z","sourceCount":5,"ledgerOffset":221},{"requestId":"231151df3c13d8c211509a1c4178bad8","agentId":"pip","round":45,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially test whether the 51099/61599 boundary promotions or fresh >149 layers yield a genuine Pareto improvement, while producing scoped saturation certificates.","costUsd":0.007,"at":"2026-09-08T03:12:13.028Z","sourceCount":5,"ledgerOffset":222},{"requestId":"16804d91c62d382c9b1186972c8718e3","agentId":"solvi","round":45,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Build a modular local sieve that forces compatible non-boundary rows against exact four-row seeds, while pruning covered and structurally degenerate regimes.","costUsd":0.007,"at":"2026-09-08T03:12:13.028Z","sourceCount":5,"ledgerOffset":223},{"requestId":"58537916607ac9eb4aa860e5d64303f9","agentId":"tess","round":45,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct an incidence hypergraph from every exact verified cell in the current frontier and search for a strict 4×5 or 5×4 improvement via exact-cover/SAT constraints.","costUsd":0.007,"at":"2026-09-08T03:12:13.028Z","sourceCount":5,"ledgerOffset":224},{"requestId":"fd17f6090d7e5fc64949ed72425f3e0c","agentId":"mira","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run stochastic, optimization-guided family-seed discovery only in uncovered cross-core and high-layer regimes, then pass Pareto-promising seeds to Tess for exact bigint verification.","costUsd":0.007,"at":"2026-09-08T03:22:16.168Z","sourceCount":5,"ledgerOffset":225},{"requestId":"d08f3fa9482ad19f6f9e73ee940a0f67","agentId":"mira","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run stochastic, optimization-guided family-seed discovery only in uncovered cross-core and high-layer regimes, then pass Pareto-promising seeds to Tess for exact bigint verification.","costUsd":0.007,"at":"2026-09-08T04:07:40.674Z","sourceCount":5,"ledgerOffset":226},{"requestId":"5d1aad14b690f8bf17db19f865993589","agentId":"orum","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a genuinely new rational/elliptic deformation of the square-additive condition X_i+Y_j=Z_{ij}^2, beginning from the exact 3-row, 4-column scaffold and targeting an additional shared row.","costUsd":0.007,"at":"2026-09-08T03:22:16.168Z","sourceCount":5,"ledgerOffset":227},{"requestId":"8f7e077a94d02f1d49a4e4a4ac08cae2","agentId":"orum","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a genuinely new rational/elliptic deformation of the square-additive condition X_i+Y_j=Z_{ij}^2, beginning from the exact 3-row, 4-column scaffold and targeting an additional shared row.","costUsd":0.007,"at":"2026-09-08T04:07:40.674Z","sourceCount":5,"ledgerOffset":228},{"requestId":"9694ee029609589b919047222d037055","agentId":"pip","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use adversarial falsification, scoped saturation certificates, and adaptive family experiments to seek a Pareto improvement without repeating covered regions.","costUsd":0.007,"at":"2026-09-08T03:22:16.168Z","sourceCount":5,"ledgerOffset":229},{"requestId":"7db6af5dccb5873a1f04ff5c3e01a7db","agentId":"pip","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use adversarial falsification, scoped saturation certificates, and adaptive family experiments to seek a Pareto improvement without repeating covered regions.","costUsd":0.007,"at":"2026-09-08T04:07:40.674Z","sourceCount":5,"ledgerOffset":230},{"requestId":"69ed488f16f42c30c3db997946c34327","agentId":"solvi","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use modular, local congruence design to prune or force compatible difference rows before any broad search.","costUsd":0.007,"at":"2026-09-08T03:22:16.168Z","sourceCount":5,"ledgerOffset":231},{"requestId":"0adc9f6acba5457bb878eef272d95802","agentId":"solvi","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use modular, local congruence design to prune or force compatible difference rows before any broad search.","costUsd":0.007,"at":"2026-09-08T04:07:40.674Z","sourceCount":5,"ledgerOffset":232},{"requestId":"d4b5a80761538d1158befbda5541ded9","agentId":"tess","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct an exact-cover/SAT incidence search using only verified cells from the 4×3 scaffold and Pip’s exact boundary columns, seeking a Pareto improvement in the weakest support vector.","costUsd":0.007,"at":"2026-09-08T03:22:16.168Z","sourceCount":5,"ledgerOffset":233},{"requestId":"71afb3c4e6e69b6a5947a59c446333f7","agentId":"tess","round":46,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct an exact-cover/SAT incidence search using only verified cells from the 4×3 scaffold and Pip’s exact boundary columns, seeking a Pareto improvement in the weakest support vector.","costUsd":0.007,"at":"2026-09-08T04:07:40.674Z","sourceCount":5,"ledgerOffset":234},{"requestId":"8d78495bfb50c9486aec6c39ce3278bf","agentId":"mira","round":47,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially test the lifted palettes, certify pairwise saturation, and adaptively generate only rows that improve the weakest exact support.","costUsd":0.007,"at":"2026-09-08T04:17:45.099Z","sourceCount":5,"ledgerOffset":235},{"requestId":"d6eefde5e91a1085737cf89988002a24","agentId":"orum","round":47,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Design modular and local congruence filters for Pip-style lifted rows, forcing compatible difference intersections before exact family scanning.","costUsd":0.007,"at":"2026-09-08T04:17:45.099Z","sourceCount":5,"ledgerOffset":236},{"requestId":"ef45f715143f4a5e4537997c3d8b3ed9","agentId":"pip","round":47,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct an iterated rational-point family from the two-row factor parametrization, targeting a genuinely new balanced support palette rather than another truncated boundary scan.","costUsd":0.007,"at":"2026-09-08T04:17:45.099Z","sourceCount":5,"ledgerOffset":237},{"requestId":"e1ac90134fd24b6cc13b76de677787d3","agentId":"solvi","round":47,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Build an incidence-hypergraph/SAT exact-cover search using only verified cells from the current reservoirs, targeting five difference vertices and five number vertices with every required edge present.","costUsd":0.007,"at":"2026-09-08T04:17:45.099Z","sourceCount":5,"ledgerOffset":238},{"requestId":"f7c8e0f5202a9227033527a973d9a0a1","agentId":"tess","round":47,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run stochastic, optimization-guided seed discovery in uncovered high-layer and cross-core family palettes, then send only Pareto-promising seeds for exact bigint verification.","costUsd":0.007,"at":"2026-09-08T04:17:45.099Z","sourceCount":5,"ledgerOffset":239},{"requestId":"f12159661fd47e8390304efdcb6b2c89","agentId":"mira","round":48,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially test collision-generated support while building saturation certificates and adaptively selecting only exact Pareto-improving rows.","costUsd":0.007,"at":"2026-09-08T04:27:48.568Z","sourceCount":5,"ledgerOffset":240},{"requestId":"030adb6523e286bc23e65206634334c6","agentId":"orum","round":48,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use modular, local congruence filters to prune collision-derived candidates and force compatible additional rows before exact support testing.","costUsd":0.007,"at":"2026-09-08T04:27:48.568Z","sourceCount":5,"ledgerOffset":241},{"requestId":"a6178cd6b757a82d0f8f743db2aa7c14","agentId":"pip","round":48,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct a new rational-point family by converting three simultaneous factor-difference conditions into an elliptic quartic, then seek five rational points whose scaled integer columns share additional rows.","costUsd":0.007,"at":"2026-09-08T04:27:48.568Z","sourceCount":5,"ledgerOffset":242},{"requestId":"986ad2b90245b543201c297421c2a807","agentId":"solvi","round":48,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run an exact SAT/exact-cover search on the incidence hypergraph formed solely from pip-certified cells, prioritizing the new collision rows and the saturated-pair rows.","costUsd":0.007,"at":"2026-09-08T04:27:48.568Z","sourceCount":5,"ledgerOffset":243},{"requestId":"980c4db190a3e18a23da6256de831cdb","agentId":"tess","round":48,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Perform stochastic, optimization-guided mutation of collision parameters and divisor-pair seeds, targeting Pareto improvements in weakest-column exact support before bigint verification.","costUsd":0.007,"at":"2026-09-08T04:27:48.568Z","sourceCount":5,"ledgerOffset":244},{"requestId":"4af6282c424a946bb83955617090db15","agentId":"mira","round":49,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially test the new CRT and collision palettes while retaining only exact saturation certificates and adaptive Pareto improvements toward (4,5,5,5,5) or (5,5,5,5,5).","costUsd":0.007,"at":"2026-09-08T04:37:50.770Z","sourceCount":5,"ledgerOffset":245},{"requestId":"74c3d3bd37a732e6638f72410d488eb2","agentId":"orum","round":49,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use a fresh c=2 odd-z shell and modular intersections to force compatible rows, then exact-test only the surviving CRT classes without repeating registry-covered searches.","costUsd":0.007,"at":"2026-09-08T04:37:50.770Z","sourceCount":5,"ledgerOffset":246},{"requestId":"2badb171172c079a71ec069aced89950","agentId":"pip","round":49,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct a new elliptic or rational-point family of three-row common completions, then seek a fifth compatible row without repeating covered searches.","costUsd":0.007,"at":"2026-09-08T04:37:50.770Z","sourceCount":5,"ledgerOffset":247},{"requestId":"1d0a95da74829c5799c953e09b149eec","agentId":"solvi","round":49,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Build an incidence-hypergraph and exact-cover search exclusively from Orum-validated cells in the new z=63–73 palettes, Mira’s collision palettes, and the complete CRT-root palette.","costUsd":0.007,"at":"2026-09-08T04:37:50.770Z","sourceCount":5,"ledgerOffset":248},{"requestId":"670fdd18a19113bfa439d6f47bf3cfe3","agentId":"tess","round":49,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use stochastic, optimization-guided sampling of unscanned CRT-root subsets and collision-layer seeds, then submit only Pareto-promising supports for exact bigint verification.","costUsd":0.007,"at":"2026-09-08T04:37:50.770Z","sourceCount":5,"ledgerOffset":249},{"requestId":"150c7f9866b7116b4a4c6473ce1103c5","agentId":"mira","round":50,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially falsify or certify the fresh elliptic, divisor-lattice, and z-shell regimes using exact support and saturation criteria, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T04:47:57.982Z","sourceCount":5,"ledgerOffset":250},{"requestId":"e76b337fa61eddd4da1e9336dc3f4740","agentId":"orum","round":50,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use modular, local congruence design to prune compatible rows before any broad search.","costUsd":0.007,"at":"2026-09-08T04:47:57.982Z","sourceCount":5,"ledgerOffset":251},{"requestId":"b066dd82b071709d938157274285f0c2","agentId":"pip","round":50,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct and test an exact rational-point family from the (204,225,411) genus-one intersection, targeting a balanced 4-by-5 incidence without divisor_completion requests.","costUsd":0.007,"at":"2026-09-08T04:47:57.982Z","sourceCount":5,"ledgerOffset":252},{"requestId":"3a1c85618165dfb1689b21a7810d26d1","agentId":"solvi","round":50,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run an exact-cover/SAT search on a bigint-verified incidence hypergraph, combining Pip’s genus-one palette with Orum’s audited shell while excluding Mira’s disputed cells.","costUsd":0.007,"at":"2026-09-08T04:47:57.982Z","sourceCount":5,"ledgerOffset":253},{"requestId":"d72ded448d84906f5404041da12570f1","agentId":"tess","round":50,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use stochastic, optimization-guided seed discovery in uncovered elliptic, divisor-lattice, and shell boundary regimes, then submit only exact bigint survivors for validation.","costUsd":0.007,"at":"2026-09-08T04:47:57.982Z","sourceCount":5,"ledgerOffset":254},{"requestId":"90939453fb40714f2e97f88a1cdb7792","agentId":"mira","round":51,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially test the new elliptic and mixed-lattice palettes for a strict exact-support Pareto improvement, while seeking saturation certificates rather than isolated anchors.","costUsd":0.007,"at":"2026-09-08T05:02:58.285Z","sourceCount":5,"ledgerOffset":255},{"requestId":"84e8ec4247088d7d33b201e0b6b60184","agentId":"orum","round":51,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Design modular and local congruence filters around the new anchors before any further exact family scan, seeking residue classes that force compatible multi-row columns.","costUsd":0.007,"at":"2026-09-08T05:02:58.285Z","sourceCount":5,"ledgerOffset":256},{"requestId":"b81574b995f79806dabfabee6302de58","agentId":"pip","round":51,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct rational-point families from the 260820 and 647500 factor-pair seeds, targeting multiple integral columns on one fixed five-row palette.","costUsd":0.007,"at":"2026-09-08T05:02:58.285Z","sourceCount":5,"ledgerOffset":257},{"requestId":"b78a66d6a1a698077c5c3cd41932a589","agentId":"solvi","round":51,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run an exact-cover search on the union of verified cells, targeting a strict 4x5 improvement before 5x5.","costUsd":0.007,"at":"2026-09-08T05:02:58.285Z","sourceCount":5,"ledgerOffset":258},{"requestId":"7c2db9dd950c09afb232a6b6c40785c6","agentId":"tess","round":51,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use stochastic, optimization-guided mutations of exact factor-pair seeds from the new anchors, targeting a strict (4,5,5,5,5) improvement before attempting (5,5,5,5,5), while excluding registry-covered regions.","costUsd":0.007,"at":"2026-09-08T05:02:58.285Z","sourceCount":5,"ledgerOffset":259},{"requestId":"5a19efd6ca61dfbe4094b3a1ad40464f","agentId":"mira","round":52,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct elliptic and rational-point families from the exact support-four anchors, seeking multiple integral columns on one palette rather than isolated collisions.","costUsd":0.007,"at":"2026-09-08T05:13:07.219Z","sourceCount":5,"ledgerOffset":260},{"requestId":"f7776714dfd9da97358eb83a80fb0ad5","agentId":"orum","round":52,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run an exact incidence-hypergraph search over newly verified cells, targeting a certified 5x5 or 6x4 subgraph.","costUsd":0.007,"at":"2026-09-08T05:13:07.219Z","sourceCount":5,"ledgerOffset":261},{"requestId":"a58f87bcdd43d9a413d090d6237641db","agentId":"pip","round":52,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Design modular and local congruence sieves for replacement palettes around the 260820, 647500, and 70536 structures.","costUsd":0.007,"at":"2026-09-08T05:13:07.219Z","sourceCount":5,"ledgerOffset":262},{"requestId":"0dceaa5b02806764d84507aefd7c904f","agentId":"solvi","round":52,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided mutation of exact collision palettes to discover new balanced seeds, then submit survivors to deterministic bigint verification.","costUsd":0.007,"at":"2026-09-08T05:13:07.219Z","sourceCount":5,"ledgerOffset":263},{"requestId":"40fb168376e1c90eefb0fbb34e215dae","agentId":"tess","round":52,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially falsify the new bridge and replacement palettes while designing exact, saturation-certified experiments toward a strict support improvement.","costUsd":0.007,"at":"2026-09-08T05:13:07.219Z","sourceCount":5,"ledgerOffset":264},{"requestId":"7111bddd3f72d94962d324590405f8b3","agentId":"mira","round":53,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a rational-point/elliptic family from the 260820 six-difference factorization, then seek parameter values that retain four or more rows and extend to a balanced 5-by-5 configuration.","costUsd":0.007,"at":"2026-09-08T05:23:07.083Z","sourceCount":5,"ledgerOffset":265},{"requestId":"1acd2170b8b6967a04b0174d7250759c","agentId":"orum","round":53,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run an exact incidence-hypergraph/SAT search over verified cells from the new palettes, targeting a 5x5 rectangle or the Pareto improvement (4,5,5,5,5).","costUsd":0.007,"at":"2026-09-08T05:23:07.083Z","sourceCount":5,"ledgerOffset":266},{"requestId":"6e802d3b8e1c5ce35dce637a16756f8b","agentId":"pip","round":53,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Build a modular local sieve for the new mixed palettes, forcing compatible four-row residue classes before any exact search.","costUsd":0.007,"at":"2026-09-08T05:23:07.083Z","sourceCount":5,"ledgerOffset":267},{"requestId":"2636f697667d6116f578bd6a527ce982","agentId":"solvi","round":53,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided mutations of the exact seed palettes to discover novel balanced support patterns, then submit only Pareto-promising survivors for exact bigint verification.","costUsd":0.007,"at":"2026-09-08T05:23:07.083Z","sourceCount":5,"ledgerOffset":268},{"requestId":"a95fa2d966b602d5bf6da2e1c8591a23","agentId":"tess","round":53,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially test the mixed palettes while producing saturation-grade certificates and adapting toward the sparsest viable residue branches.","costUsd":0.007,"at":"2026-09-08T05:23:07.083Z","sourceCount":5,"ledgerOffset":269},{"requestId":"237e5b04d8485f609ac154c33b729af3","agentId":"mira","round":54,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a rational-point parametrization linking the 260820 and 647500 anchor systems through their shared difference 411, then specialize to integral columns with support at least four.","costUsd":0.007,"at":"2026-09-08T05:33:10.460Z","sourceCount":5,"ledgerOffset":270},{"requestId":"d8de003b1cce02d0d9c498fee7e87c45","agentId":"orum","round":54,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run a SAT/incidence-hypergraph search using only explicitly verified cells from the new Mira/Pip palettes, targeting a saturated 5-by-5 rectangle and then the Pareto fallback (4,5,5,5,5).","costUsd":0.007,"at":"2026-09-08T05:33:10.460Z","sourceCount":5,"ledgerOffset":271},{"requestId":"da7cecb621eb321ce4a8283e48bf111b","agentId":"pip","round":54,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Design a new modular sieve for the 647500-derived and 260820 bridge palettes, targeting columns with exact support at least four while avoiding all registry-covered regions.","costUsd":0.007,"at":"2026-09-08T05:33:10.460Z","sourceCount":5,"ledgerOffset":272},{"requestId":"ea70e937b6d2c240e1b2a17ac82c273e","agentId":"solvi","round":54,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided mutations and crossovers of the new exact anchor palettes to discover distinct high-support columns, then submit only exact candidates for bigint validation.","costUsd":0.007,"at":"2026-09-08T05:33:10.460Z","sourceCount":5,"ledgerOffset":273},{"requestId":"458c9e35f6e43f3fdb8998d1a26a231e","agentId":"tess","round":54,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially test the mixed palettes while producing saturation-grade certificates and adapting toward the sparsest viable residue branches.","costUsd":0.007,"at":"2026-09-08T05:33:10.460Z","sourceCount":5,"ledgerOffset":274},{"requestId":"77fe36afcb4b4c37e381b8be38481421","agentId":"mira","round":55,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct and specialize the q=411 elliptic fiber product linking the 260820 and 647500 conic systems, seeking a genuinely multi-column rational-point family rather than isolated anchors.","costUsd":0.007,"at":"2026-09-08T05:43:14.137Z","sourceCount":5,"ledgerOffset":275},{"requestId":"6431ec2791f571a676cf17067751a320","agentId":"orum","round":55,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct an incidence hypergraph solely from exact verified cells and search it by SAT/exact cover for balanced K5,5 or strict Pareto improvements.","costUsd":0.007,"at":"2026-09-08T05:43:14.137Z","sourceCount":5,"ledgerOffset":276},{"requestId":"8c5b63aa7a35f1c9f18965b2d18bf2b4","agentId":"pip","round":55,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Design a modular, local congruence sieve for a new d≡3 (mod 16) palette, forcing compatible four-row or five-row support before exact scanning.","costUsd":0.007,"at":"2026-09-08T05:43:14.137Z","sourceCount":5,"ledgerOffset":277},{"requestId":"3226871ef844a46fd6ef02395b57e6eb","agentId":"solvi","round":55,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided seed discovery in new difference regimes, then perform deterministic exact-bigint verification of the highest-support candidates.","costUsd":0.007,"at":"2026-09-08T05:43:14.137Z","sourceCount":5,"ledgerOffset":278},{"requestId":"67839067aa6fcb72169dadb6ee8574a4","agentId":"tess","round":55,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially test the q=411 and CRT proposals while constructing bounded saturation certificates and adapting the next palette to observed exact support.","costUsd":0.007,"at":"2026-09-08T05:43:14.137Z","sourceCount":5,"ledgerOffset":279},{"requestId":"4f6e5f15ace2517757dd7f689cfb6acf","agentId":"mira","round":56,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct an explicit parameterized q=411 rational-point family from the shared-t equations, targeting multiple integral columns with common rows.","costUsd":0.007,"at":"2026-09-08T07:59:11.685Z","sourceCount":5,"ledgerOffset":280},{"requestId":"dc6b380901495ec5668cdd77960a45e3","agentId":"orum","round":56,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run an exact-cover/SAT search on the verified-cell incidence hypergraph, using the q=411 and 1459 seed cells only as certified input.","costUsd":0.007,"at":"2026-09-08T07:59:11.685Z","sourceCount":5,"ledgerOffset":281},{"requestId":"0832cd7fca8ddc974e15c31961d255ca","agentId":"pip","round":56,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Strengthen the 1459 modular lane by intersecting local support constraints before any exact family scan.","costUsd":0.007,"at":"2026-09-08T07:59:11.685Z","sourceCount":5,"ledgerOffset":282},{"requestId":"ca05e9bc8607d4ee4947240af51d8e74","agentId":"solvi","round":56,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided seed discovery on unscanned difference palettes, then submit only exact bigint-verified support improvements.","costUsd":0.007,"at":"2026-09-08T07:59:11.685Z","sourceCount":5,"ledgerOffset":283},{"requestId":"be360daf7076b126a2e234d973353943","agentId":"tess","round":56,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially falsify the proposed q=411 and 1459 lanes, then design one adaptive exact scan only if a saturation certificate leaves an uncovered support regime.","costUsd":0.007,"at":"2026-09-08T07:59:11.685Z","sourceCount":5,"ledgerOffset":284},{"requestId":"ba85304c42410dfe2eec37ac57d86276","agentId":"mira","round":57,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a new row palette by intersecting local quadratic-residue constraints modulo several small primes, centered on exact seed incidences but disjoint from covered palettes and difference regions.","costUsd":0.007,"at":"2026-09-08T08:09:18.660Z","sourceCount":5,"ledgerOffset":285},{"requestId":"73ee999c32b04b5cf80ce3720e3b1189","agentId":"orum","round":57,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided seed discovery over new difference palettes and progressions, followed only by exact family-scan bigint verification.","costUsd":0.007,"at":"2026-09-08T08:09:18.660Z","sourceCount":5,"ledgerOffset":286},{"requestId":"da73eb9e3f203fd6a5304280fded10cf","agentId":"pip","round":57,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an exact-cover search over the union of verified cells from the 1459 palette and CRT-flat family, targeting a 4x5 or 5x5 biclique without requesting divisor completion.","costUsd":0.007,"at":"2026-09-08T08:09:18.660Z","sourceCount":5,"ledgerOffset":287},{"requestId":"b2e583c5554e63ff5484956612848c26","agentId":"solvi","round":57,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially falsify the two new palettes, then use exact modular saturation to choose the next bounded experiment without dispatching divisor_completion.","costUsd":0.007,"at":"2026-09-08T08:09:18.660Z","sourceCount":5,"ledgerOffset":288},{"requestId":"877052715e400d622eeaa1ef2300caa7","agentId":"tess","round":57,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a rational-point family by extending the four-row square-difference equations on an intersection of quadrics, targeting a symbolic fifth row rather than an unbounded incidence scan.","costUsd":0.007,"at":"2026-09-08T08:09:18.660Z","sourceCount":5,"ledgerOffset":289},{"requestId":"29df531fc7aff58a66f26aedf66ebeae","agentId":"mira","round":58,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a new modular row palette that prunes N classes while forcing shared local compatibility, without requesting divisor completion.","costUsd":0.007,"at":"2026-09-08T08:19:20.044Z","sourceCount":5,"ledgerOffset":290},{"requestId":"e9d3eb1adef73795fd42b594761c10ab","agentId":"orum","round":58,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided seed discovery to select new five-row CRT/sign palettes and seed N-columns, maximizing the Pareto score of weakest-column exact support; then perform deterministic bigint verification only on the best seeds.","costUsd":0.007,"at":"2026-09-08T08:19:20.044Z","sourceCount":5,"ledgerOffset":291},{"requestId":"68285ae0e00c2e999a754624ecbfaf2d","agentId":"pip","round":58,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an exact-cover incidence search on verified cells from the union of the 312 and 1459 palettes, requiring selected row sets to contain both differences and targeting 5×5 before 6×4 or 4×5.","costUsd":0.007,"at":"2026-09-08T08:19:20.044Z","sourceCount":5,"ledgerOffset":292},{"requestId":"094d2f04f6d4dd4142d8d37a147acb40","agentId":"solvi","round":58,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially falsify both proposed slices, certify their support saturation, and choose any follow-up experiment only from an uncovered high-support frontier.","costUsd":0.007,"at":"2026-09-08T08:19:20.044Z","sourceCount":5,"ledgerOffset":293},{"requestId":"07f0ca7b5906b1a01f3a8b35c29a3f54","agentId":"tess","round":58,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a rational algebraic family by fixing three differences, parametrizing the two-row conic, and converting the third-row square condition into an explicit nonsingular quartic or elliptic curve; seek rank-positive specializations that support five common differences and multiple distinct N co","costUsd":0.007,"at":"2026-09-08T08:19:20.044Z","sourceCount":5,"ledgerOffset":294},{"requestId":"f231e91efefee2d7716e1cfbacbaf846","agentId":"mira","round":59,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a new modular row palette using local square-residue branches and CRT intersections, with no divisor_completion request; route any candidate-cell validation solely through Orum.","costUsd":0.007,"at":"2026-09-08T08:29:21.775Z","sourceCount":5,"ledgerOffset":295},{"requestId":"759bedaf29576a03530c501fbdae198f","agentId":"orum","round":59,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided seed discovery to generate new factor-pair palettes outside the ledger, then apply deterministic bigint verification only to the highest-scoring candidates.","costUsd":0.007,"at":"2026-09-08T08:29:21.775Z","sourceCount":5,"ledgerOffset":296},{"requestId":"7708b5cfaa3c163baa18c8ee974cf32e","agentId":"pip","round":59,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Build an exact-cover incidence hypergraph from independently verified cells in the new Mira and Pip palettes, prioritizing 5x5, then common 4x5 and 6x4 structures.","costUsd":0.007,"at":"2026-09-08T08:29:21.775Z","sourceCount":5,"ledgerOffset":297},{"requestId":"9e33b95b529ca721789f2f684ecfaac9","agentId":"solvi","round":59,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially falsify both new families, then extract saturation certificates and adapt the next experiment to whichever row intersections survive.","costUsd":0.007,"at":"2026-09-08T08:29:21.775Z","sourceCount":5,"ledgerOffset":298},{"requestId":"0f19c20827b50b989362137721d83d2a","agentId":"tess","round":59,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a genuine rational-point family by extending the Bremner four-row rectangle: eliminate each column N from the square equations, derive the resulting genus-one curves for a fifth difference, and search exact rational points whose denominator clearing yields multiple distinct positive column","costUsd":0.007,"at":"2026-09-08T08:29:21.775Z","sourceCount":5,"ledgerOffset":299},{"requestId":"97b0e6fdd6e0ced5a44385ca5ae65450","agentId":"mira","round":60,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a new modular-local family_scan using CRT-compatible row palettes, selecting congruence branches that maximize common admissible N residue density while avoiding all recorded search regions.","costUsd":0.007,"at":"2026-09-08T08:39:26.090Z","sourceCount":5,"ledgerOffset":300},{"requestId":"f9020a53524fc34556c96bfb4e705695","agentId":"orum","round":60,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Generate stochastic, optimization-guided row palettes outside covered regimes, then exact-bigint verify only Pareto-promising support structures; request no divisor_completion.","costUsd":0.007,"at":"2026-09-08T08:39:26.090Z","sourceCount":5,"ledgerOffset":301},{"requestId":"01e5435ea5676e21c0af10c83971ceb9","agentId":"pip","round":60,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an incidence-hypergraph/exact-cover search on the mixed scale-5 palette, using only independently verified exact cells and prioritizing 5×5, then common 4×5 support.","costUsd":0.007,"at":"2026-09-08T08:39:26.090Z","sourceCount":5,"ledgerOffset":302},{"requestId":"d8ab3af6f404029cf2bbf1bca2e29548","agentId":"solvi","round":60,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially falsify both palettes, then choose the next experiment from exact support saturation rather than isolated dense seeds.","costUsd":0.007,"at":"2026-09-08T08:39:26.090Z","sourceCount":5,"ledgerOffset":303},{"requestId":"d42d2d12090b6af41a46bf8e881df936","agentId":"tess","round":60,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a genuinely parametrized square-additive family by treating the cell equations d_i^2+4N_j=m_{ij}^2 as an intersection of quadrics, beginning with rational parametrization of a two-row/two-column seed and identifying the resulting elliptic curve for an added row or column.","costUsd":0.007,"at":"2026-09-08T08:39:26.090Z","sourceCount":5,"ledgerOffset":304},{"requestId":"6ca9c2295fbd973dd7b877c914d35339","agentId":"mira","round":61,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a new modular-local row palette using CRT branches distinct from 23·29 and 31·37, maximizing forced common admissible N residues before exact scanning.","costUsd":0.007,"at":"2026-09-08T08:49:32.086Z","sourceCount":5,"ledgerOffset":305},{"requestId":"9d11f6c72c5e28c2895e8470438df3cd","agentId":"orum","round":61,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided seed discovery for new CRT-compatible, nonhomothetic difference palettes, then submit only the highest-scoring seeds to Tess for exact bigint incidence verification.","costUsd":0.007,"at":"2026-09-08T08:49:32.086Z","sourceCount":5,"ledgerOffset":306},{"requestId":"f6281c9f1fdf8220fd8a502896024ce8","agentId":"pip","round":61,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an incidence-hypergraph/exact-cover search using only Tess-verified cells from the new prime-31 and 31·37 palettes, prioritizing a true 5×5 block and then exact 4×5 or 5×4 support.","costUsd":0.007,"at":"2026-09-08T08:49:32.086Z","sourceCount":5,"ledgerOffset":307},{"requestId":"b32deaffd22337fd5bcddc75a114a651","agentId":"solvi","round":61,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Apply adversarial falsification, saturation certificates, and adaptive branch selection to the two new palettes without requesting divisor completion.","costUsd":0.007,"at":"2026-09-08T08:49:32.086Z","sourceCount":5,"ledgerOffset":308},{"requestId":"0d0e60af37ef52cf3e914e08b41f737a","agentId":"tess","round":61,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a non-homothetic rational-point family from common square conditions, using the new CRT row palette only as a seed and seeking elliptic-curve or rational parametrizations rather than another raw scan.","costUsd":0.007,"at":"2026-09-08T08:49:32.086Z","sourceCount":5,"ledgerOffset":309},{"requestId":"7024c3d429e9a3fb9b495b8516702349","agentId":"mira","round":62,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an incidence-hypergraph/SAT exact-cover search on independently verified cells from the new CRT and mutation families, prioritizing balanced 5-by-5 support and then strict 4-by-6 or 5-by-4 improvements.","costUsd":0.007,"at":"2026-09-08T08:59:34.935Z","sourceCount":5,"ledgerOffset":310},{"requestId":"112ea5c2cb92153298460685df1b3b19","agentId":"orum","round":62,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially audit the 41·43 construction and obtain a genuine saturation certificate with one exact family scan; request no divisor_completion.","costUsd":0.007,"at":"2026-09-08T08:59:34.935Z","sourceCount":5,"ledgerOffset":311},{"requestId":"72f5116960b03dd6a6e57569c6aae001","agentId":"pip","round":62,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use stochastic and optimization-guided CRT/linear-mutation seed discovery to maximize balanced support floors, followed by exact bigint verification.","costUsd":0.007,"at":"2026-09-08T08:59:34.935Z","sourceCount":5,"ledgerOffset":312},{"requestId":"b00506dea7b29f964a6144bb66cae516","agentId":"solvi","round":62,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Develop a constructive rational-point family from the square-additive matrix model X_{ij}^2=A_i+B_j, with A_i=d_i^2 and B_j=4N_j, starting from the known 4-row rectangle and treating a fifth row as a genus-one lifting problem.","costUsd":0.007,"at":"2026-09-08T08:59:34.935Z","sourceCount":5,"ledgerOffset":313},{"requestId":"0ced93140886dfd339ea347b35122408","agentId":"tess","round":62,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Design a modular-local row palette whose congruence conditions force compatible pair-completion classes before any broad incidence scan.","costUsd":0.007,"at":"2026-09-08T08:59:34.935Z","sourceCount":5,"ledgerOffset":314},{"requestId":"16bb0704350a28cb7c8c66be2d3736c9","agentId":"mira","round":63,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Use only reconciled, bigint-verified cells from the CRT and corrected endpoint palettes to build an incidence hypergraph, then search exact-cover/SAT encodings for a 5×5 block and strict 4×6 or 5×4 frontiers.","costUsd":0.007,"at":"2026-09-08T09:09:35.756Z","sourceCount":5,"ledgerOffset":315},{"requestId":"27b65e951fda6c1223549636404448f9","agentId":"orum","round":63,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct an adaptive, adversarially falsifiable family scan whose finite domain and saturation profile are fully reproducible, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T09:09:35.756Z","sourceCount":5,"ledgerOffset":316},{"requestId":"dcc12192e760925a95b6e55c33099883","agentId":"pip","round":63,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use stochastic, optimization-guided endpoint/CRT seed discovery to maximize balanced exact support, then submit only internally consistent palettes to deterministic bigint verification.","costUsd":0.007,"at":"2026-09-08T09:09:35.756Z","sourceCount":5,"ledgerOffset":317},{"requestId":"2a987b20a51f3f3e1bf68d2851e37fde","agentId":"solvi","round":63,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Develop a low-dimensional constructive family by rewriting simultaneous square completions as rational-point conditions, using CRT lifts only to select promising fibers.","costUsd":0.007,"at":"2026-09-08T09:09:35.756Z","sourceCount":5,"ledgerOffset":318},{"requestId":"34ffe8605afd01083b5780aa45f08848","agentId":"tess","round":63,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Design a new local CRT palette that forces compatible factor-pair endpoint residues and targets balanced four- or five-row support, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T09:09:35.756Z","sourceCount":5,"ledgerOffset":319},{"requestId":"2dfa7e01812723cf80aa924e4681308e","agentId":"mira","round":64,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct an exact-cover/SAT search over only independently bigint-verified incidence cells from the 59·61 and smooth-factor palettes, maximizing balanced biclique support.","costUsd":0.007,"at":"2026-09-08T09:19:39.213Z","sourceCount":5,"ledgerOffset":320},{"requestId":"a2fb4f7695c4a665862f7562450d36ad","agentId":"orum","round":64,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially filter the proposed palettes and require a saturated, replayable incidence certificate before authorizing substantial computation.","costUsd":0.007,"at":"2026-09-08T09:19:39.213Z","sourceCount":5,"ledgerOffset":321},{"requestId":"81c5894e8f2b413e8398fb1a5cc952a0","agentId":"pip","round":64,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run the rotating stochastic, optimization-guided seed-discovery lane on reproducible smooth/CRT hyperbola seeds, then submit only the best balanced palettes for exact bigint verification.","costUsd":0.007,"at":"2026-09-08T09:19:39.213Z","sourceCount":5,"ledgerOffset":322},{"requestId":"a6f8b78c0ddf1c5e3570e33cc77d5bff","agentId":"solvi","round":64,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct a rational-point family from the Bremner rectangle: for selected differences define C by y_i^2=4N+d_i^2, convert the resulting genus-one intersection to Weierstrass form, and use its group law to generate new common columns before imposing additional rows.","costUsd":0.007,"at":"2026-09-08T09:19:39.213Z","sourceCount":5,"ledgerOffset":323},{"requestId":"817546a12e1b4256253d37143c62626c","agentId":"tess","round":64,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Design a modular, local-congruence sieve that forces or prunes compatible rows for the 59·61 palette and a disjoint auxiliary-prime variant, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T09:19:39.213Z","sourceCount":5,"ledgerOffset":324},{"requestId":"9525adb23d7701b5636dad96da0e6f08","agentId":"mira","round":65,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an incidence-hypergraph/SAT exact-cover search using only independently verified cells from the admissible new palette, targeting 5x5 first and strict 6x4 or 4x5 support second.","costUsd":0.007,"at":"2026-09-08T09:29:41.390Z","sourceCount":5,"ledgerOffset":325},{"requestId":"8ed099b3a319fd17a5d01f103ede8870","agentId":"orum","round":65,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially prefilter both palettes, then adaptively select at most one exact incidence experiment with a reportable saturation profile.","costUsd":0.007,"at":"2026-09-08T09:29:41.390Z","sourceCount":5,"ledgerOffset":326},{"requestId":"41535c9263cd782274f610ac4599a6a5","agentId":"pip","round":65,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use stochastic, optimization-guided seed discovery to generate one genuinely new balanced-incidence family, then perform exact bigint verification through a family scan only.","costUsd":0.007,"at":"2026-09-08T09:29:41.390Z","sourceCount":5,"ledgerOffset":327},{"requestId":"64a451975fbc934edf9bd9c921f89143","agentId":"solvi","round":65,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct a reproducible low-dimensional algebraic family by reparameterizing shared factor-difference cells as rational points on conics, then derive and inspect the resulting elliptic fibers for five-row incidence.","costUsd":0.007,"at":"2026-09-08T09:29:41.390Z","sourceCount":5,"ledgerOffset":328},{"requestId":"0f2b9424c67e2fcb85e6a89998b23bb6","agentId":"tess","round":65,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Develop a modular local-congruence palette that prunes incompatible rows and preferentially forces shared square-completion residues, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T09:29:41.390Z","sourceCount":5,"ledgerOffset":329},{"requestId":"8acbf9c40045d12e3f8efd6ae8bab5ae","agentId":"mira","round":66,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an incidence-hypergraph/SAT exact-cover search on Pip’s Markov palette using only independently verified exact cells, targeting 5×5 first and then 6×4 or 4×5.","costUsd":0.007,"at":"2026-09-08T09:39:43.224Z","sourceCount":5,"ledgerOffset":330},{"requestId":"33a35b0a0846e0686546f8bde4a23113","agentId":"orum","round":66,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially replay both palettes, then adapt computation only after obtaining a parser-clean input and a saturation certificate.","costUsd":0.007,"at":"2026-09-08T09:39:43.224Z","sourceCount":5,"ledgerOffset":331},{"requestId":"03b441784132d03334a25eec0847c0fb","agentId":"pip","round":66,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Generate stochastic, optimization-guided seeds from a hybrid Markov/CRT incidence model, then submit one family_scan for exact bigint verification only.","costUsd":0.007,"at":"2026-09-08T09:39:43.224Z","sourceCount":5,"ledgerOffset":332},{"requestId":"936c2c74a58894102c95e0347e3af5ee","agentId":"solvi","round":66,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct a reproducible algebraic-family lift by fixing the Markov seed M_0=3524578, M_1=9227465 and deriving exact rational parametrizations of shared-cell intersections, including any genus-one subfamilies.","costUsd":0.007,"at":"2026-09-08T09:39:43.224Z","sourceCount":5,"ledgerOffset":333},{"requestId":"a5be52a65e661562a11968b52967bfac","agentId":"tess","round":66,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Design a fresh modular palette that locally forces compatible row completions before any exact search, staying outside covered regions.","costUsd":0.007,"at":"2026-09-08T09:39:43.224Z","sourceCount":5,"ledgerOffset":334},{"requestId":"84cc4127a6e2ee0c97900d6be4eabfab","agentId":"mira","round":67,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Use stochastic, optimization-guided seed discovery in a new hybrid incidence space, then perform deterministic bigint verification of retained seeds.","costUsd":0.007,"at":"2026-09-08T09:49:45.711Z","sourceCount":5,"ledgerOffset":335},{"requestId":"e22bf56af284ebfb637b58e7ab2c5e75","agentId":"orum","round":67,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a verified rational/elliptic family of simultaneous factor-difference cells, using Markov mutations only as algebraic seeds rather than as unverified numeric palettes.","costUsd":0.007,"at":"2026-09-08T09:49:45.711Z","sourceCount":5,"ledgerOffset":336},{"requestId":"ca8f025170543b8eb7ebba1a488eb11e","agentId":"pip","round":67,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially falsify the CRT and off-axis Markov proposals while extracting exact saturation certificates, then adapt the next experiment only toward branches showing shared support at least three.","costUsd":0.007,"at":"2026-09-08T09:49:45.711Z","sourceCount":5,"ledgerOffset":337},{"requestId":"a1cd9fb4c913ed61077fb6df805c4c9c","agentId":"solvi","round":67,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use modular and local congruence signatures to force compatible four- or five-row subsets before any exact incidence search, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T09:49:45.711Z","sourceCount":5,"ledgerOffset":338},{"requestId":"1888ee844d826a0b2d6896f0793c5233","agentId":"tess","round":67,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Build an incidence-hypergraph exact-cover/SAT search using only independently verified cells, targeting 5×5 first and strict Pareto shapes second.","costUsd":0.007,"at":"2026-09-08T09:49:45.711Z","sourceCount":5,"ledgerOffset":339},{"requestId":"94e8d77af2c992892d068896b6cf8ad2","agentId":"mira","round":68,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run stochastic, optimization-guided seed discovery on a fresh modular/CRT family, maximizing balanced incidence support before exact bigint verification.","costUsd":0.007,"at":"2026-09-08T09:59:51.655Z","sourceCount":5,"ledgerOffset":340},{"requestId":"d328f3f68b954d8ae1668d733f6092ab","agentId":"orum","round":68,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a genuine fifth-row algebraic family from a known 4×6 square-additive rectangle: use three columns to derive the rational-point/elliptic structure, then test the remaining columns exactly.","costUsd":0.007,"at":"2026-09-08T09:59:51.655Z","sourceCount":5,"ledgerOffset":341},{"requestId":"a6b82dde3c8ecd9007aebd3b9af20943","agentId":"pip","round":68,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially falsify both proposed families, extract exact saturation certificates, and adapt only toward incidence patterns with support at least three.","costUsd":0.007,"at":"2026-09-08T09:59:51.655Z","sourceCount":5,"ledgerOffset":342},{"requestId":"acb22a416711247953a62d300e885874","agentId":"solvi","round":68,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Design a deterministic local-congruence palette that forces compatible pair shells, using moduli and residue classes distinct from the registry, the 83·89 and 97·101 lifts, and the powers-of-two lane; retain only patterns capable of support at least three before extension toward 4×5 or 5×5.","costUsd":0.007,"at":"2026-09-08T09:59:51.655Z","sourceCount":5,"ledgerOffset":343},{"requestId":"44330c1972d291ef8278d569ccfdf078","agentId":"tess","round":68,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use only exact verified cells to build an incidence hypergraph and run an exact-cover/SAT search for a strict Pareto improvement over the 4×4 frontier.","costUsd":0.007,"at":"2026-09-08T09:59:51.655Z","sourceCount":5,"ledgerOffset":344},{"requestId":"3c923837e2c34a5178703b4cd6901698","agentId":"mira","round":69,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Use stochastic, optimization-guided pair-shell seed discovery on a fresh high-index Pell–Lucas block, then perform exact bigint incidence verification and submit only retained records to orum.","costUsd":0.007,"at":"2026-09-08T10:09:50.491Z","sourceCount":5,"ledgerOffset":345},{"requestId":"561ccde7e3d12328757e45810e6aa931","agentId":"orum","round":69,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a new exact algebraic family by eliminating N from simultaneous equations m_i^2-d_i^2=4N, prioritizing rational-point and elliptic-curve parametrizations rather than stochastic scans.","costUsd":0.007,"at":"2026-09-08T10:09:50.491Z","sourceCount":5,"ledgerOffset":346},{"requestId":"63613afd5eb9081aed3fdda192874642","agentId":"pip","round":69,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially test whether the CRT and Pell families escape support-2 saturation, using exact pair-completion certificates and adaptive family scans only; no divisor_completion request.","costUsd":0.007,"at":"2026-09-08T10:09:50.491Z","sourceCount":5,"ledgerOffset":347},{"requestId":"03fe17b7927aebe44c790dbb0b272daa","agentId":"solvi","round":69,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Design a row-selective modular family in which local congruence classes force at least five differences to remain simultaneously admissible for the same N residue, then run only a deterministic family_scan.","costUsd":0.007,"at":"2026-09-08T10:09:50.491Z","sourceCount":5,"ledgerOffset":348},{"requestId":"4c2b57207c6fe5c30f11967e466a886a","agentId":"tess","round":69,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use only deterministically verified cells to construct an incidence hypergraph and run exact-cover/SAT searches for 5x5, then strict Pareto improvements such as 4x5 or 6x4.","costUsd":0.007,"at":"2026-09-08T10:09:50.491Z","sourceCount":5,"ledgerOffset":349},{"requestId":"c72130d609a599bbda1a3e2921af61e4","agentId":"mira","round":70,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run stochastic, optimization-guided seed discovery on the certified six-root CRT family, followed only by exact bigint verification.","costUsd":0.007,"at":"2026-09-08T10:19:54.811Z","sourceCount":5,"ledgerOffset":350},{"requestId":"ffef1a3d33599c6b610c9b5b00527631","agentId":"orum","round":70,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a non-specialized rational square-additive rectangle by parametrizing the conic fibers q^2-d^2=4N and studying the resulting elliptic-curve fiber product for a fifth row or column.","costUsd":0.007,"at":"2026-09-08T10:19:54.811Z","sourceCount":5,"ledgerOffset":351},{"requestId":"3b8c6e411e68eb646d8f10906227ec23","agentId":"pip","round":70,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use adversarial falsification, exact saturation certificates, and adaptive experiment design to compare the CRT and odd-Lucas palettes without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T10:19:54.811Z","sourceCount":5,"ledgerOffset":352},{"requestId":"450df0b4edaf40134de067ca983b0e70","agentId":"solvi","round":70,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Build a fresh CRT palette in which all selected differences share prescribed square classes modulo several small primes, then use these local conditions to prune pair-shell candidates and seek balanced exact support.","costUsd":0.007,"at":"2026-09-08T10:19:54.811Z","sourceCount":5,"ledgerOffset":353},{"requestId":"51772ce4128bcd5170db16660b6e00d2","agentId":"tess","round":70,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct an incidence-hypergraph/SAT exact-cover search using only bigint-verified cells, targeting a 5x5 biclique and strict 4x5 or 5x4 improvements.","costUsd":0.007,"at":"2026-09-08T10:19:54.811Z","sourceCount":5,"ledgerOffset":354},{"requestId":"f4b2d37af00a756598fd6d3864645619","agentId":"mira","round":71,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run stochastic, optimization-guided seed discovery on a fresh CRT-lift family, then perform exact bigint support verification; request no divisor_completion.","costUsd":0.007,"at":"2026-09-08T10:29:56.763Z","sourceCount":5,"ledgerOffset":355},{"requestId":"1e9f76145c0fb4ac2f8b6cda47389069","agentId":"orum","round":71,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a non-CRT algebraic family by parametrizing simultaneous difference-of-squares equations and imposing a third or fourth incidence condition as an explicit elliptic-curve or rational-point problem.","costUsd":0.007,"at":"2026-09-08T10:29:56.763Z","sourceCount":5,"ledgerOffset":356},{"requestId":"b0a42183d8711c96b37de7b7e62f6d1a","agentId":"pip","round":71,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially falsify both proposed palettes, then adapt toward the family with the strongest exact Pareto support while producing saturation certificates rather than heuristic density claims.","costUsd":0.007,"at":"2026-09-08T10:29:56.763Z","sourceCount":5,"ledgerOffset":357},{"requestId":"279072d46c8ad8c941cdd015fc212fa2","agentId":"solvi","round":71,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct a new modular/local CRT palette that forces compatible row pairs and prunes impossible five-row incidence before any global scan.","costUsd":0.007,"at":"2026-09-08T10:29:56.763Z","sourceCount":5,"ledgerOffset":358},{"requestId":"f931e00d9309ed8a3f7094aab7de0e37","agentId":"tess","round":71,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run an incidence-hypergraph/exact-cover search over independently verified cells from the strongest nonzero family output, targeting a strict Pareto improvement before K5.","costUsd":0.007,"at":"2026-09-08T10:29:56.763Z","sourceCount":5,"ledgerOffset":359},{"requestId":"0ab6c55aaecb61067df8f9b3a37ca4e0","agentId":"mira","round":72,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially falsify both proposed families, certify exact support saturation where applicable, and adapt the next experiment only from balanced-support evidence.","costUsd":0.007,"at":"2026-09-08T10:40:00.047Z","sourceCount":5,"ledgerOffset":360},{"requestId":"b565515cd5e3f9e4b2e953a48a0f1ad9","agentId":"orum","round":72,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Develop a modular local sieve that prunes or forces compatible rows before any expensive search, using the CRT-root and quadratic-unit identities only as sources of residue constraints; do not request divisor_completion.","costUsd":0.007,"at":"2026-09-08T10:40:00.047Z","sourceCount":5,"ledgerOffset":361},{"requestId":"cd802ef7387463fbcca4d872aafddabc","agentId":"pip","round":72,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct a distinct simultaneous-square family from a tunable norm identity or rational-point/elliptic parametrization, avoiding the CRT-root and quadratic-unit palettes.","costUsd":0.007,"at":"2026-09-08T10:40:00.047Z","sourceCount":5,"ledgerOffset":362},{"requestId":"1b16bc9bcd2e4e264e74973057a4d9a9","agentId":"solvi","round":72,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use only verified incidence cells to build an N–difference hypergraph, then solve exact-cover/SAT instances for balanced 5-by-5 and strict Pareto-improving supports.","costUsd":0.007,"at":"2026-09-08T10:40:00.047Z","sourceCount":5,"ledgerOffset":363},{"requestId":"080d0bf1bb1f2ca4e4bc14af66322673","agentId":"tess","round":72,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run stochastic, optimization-guided discovery of fresh CRT-lift and quadratic-unit seed palettes, then apply exact bigint verification only to Pareto-leading candidates; request no divisor_completion.","costUsd":0.007,"at":"2026-09-08T10:40:00.047Z","sourceCount":5,"ledgerOffset":364},{"requestId":"026573e0792e50e4b90f198433a86de7","agentId":"mira","round":73,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially falsify both proposed families, then build exact finite saturation certificates and adapt only from balanced-support evidence.","costUsd":0.007,"at":"2026-09-08T10:50:03.418Z","sourceCount":5,"ledgerOffset":365},{"requestId":"7c182b1ce699cd6562b753f0e1d364cb","agentId":"orum","round":73,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Design a modular sign-code family: choose a modulus M and distinct CRT sign patterns so every selected difference satisfies d_i^2 congruent to d_0^2 modulo M, forcing identical local square-residue conditions for N.","costUsd":0.007,"at":"2026-09-08T10:50:03.418Z","sourceCount":5,"ledgerOffset":366},{"requestId":"4d822aed503a043788a3852a835728b7","agentId":"pip","round":73,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Develop a constructive rational-point or elliptic-curve family extending a square-additive four-row seed, varying parameters rather than repeating fixed Bremner specializations.","costUsd":0.007,"at":"2026-09-08T10:50:03.418Z","sourceCount":5,"ledgerOffset":367},{"requestId":"62a0349c40ecbb5996b76d69d2edee09","agentId":"solvi","round":73,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct the incidence hypergraph from already verified exact cells and solve for K5,5 first, then Pareto-improving 4x5 or 5x4 bicliques.","costUsd":0.007,"at":"2026-09-08T10:50:03.418Z","sourceCount":5,"ledgerOffset":368},{"requestId":"06be506f1c9c26c18d800f2f0cdaaf3b","agentId":"tess","round":73,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use stochastic, optimization-guided sampling of new algebraic seed families, then exact bigint verification of only the highest-scoring balanced-support seeds; request no divisor_completion job.","costUsd":0.007,"at":"2026-09-08T10:50:03.418Z","sourceCount":5,"ledgerOffset":369},{"requestId":"c51e555d8ef50936333f3af1d13f60d9","agentId":"mira","round":74,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially test Pip’s quartic family first, certify only the exact finite traversal actually completed, and adapt toward a cofactor permutation only if balanced-support data—not raw candidate counts—justifies it. Use Orum only for validation; request no divisor_completion job.","costUsd":0.007,"at":"2026-09-08T11:00:06.056Z","sourceCount":5,"ledgerOffset":370},{"requestId":"c708316271b432fa719f7f8e6d8426db","agentId":"orum","round":74,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Apply a complete local congruence sieve to quartic and reversed-primorial row subsets before exact lifting.","costUsd":0.007,"at":"2026-09-08T11:00:06.056Z","sourceCount":5,"ledgerOffset":371},{"requestId":"75b9fcf9c0b134e155ed8dc1b0dae64b","agentId":"pip","round":74,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct a rational-point or elliptic-curve family extending a verified four-row square-additive rectangle, with the fifth row generated algebraically rather than by an isolated quartic scan.","costUsd":0.007,"at":"2026-09-08T11:00:06.056Z","sourceCount":5,"ledgerOffset":372},{"requestId":"894553890d9b96e3756ce5e38f0469ef","agentId":"solvi","round":74,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run an exact-cover/SAT search on the union of all already verified incidence cells, seeking a balanced 5-by-5 biclique and Pareto improvements without generating unverified numerical candidates.","costUsd":0.007,"at":"2026-09-08T11:00:06.056Z","sourceCount":5,"ledgerOffset":373},{"requestId":"f3e13eea77fd0a8d781271f8a5573355","agentId":"tess","round":74,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use stochastic, optimization-guided sampling of uncovered quartic and primorial-cofactor seeds, rank them by balanced-support proxies, then exact-bigint verify only the best seed through a family scan; request no divisor_completion.","costUsd":0.007,"at":"2026-09-08T11:00:06.056Z","sourceCount":5,"ledgerOffset":374},{"requestId":"e6c7701274bd20d7bf70ea5a91dd5ab4","agentId":"mira","round":75,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially falsify the cubic slice, obtain a deterministic saturation certificate, and adapt only from its balanced-support profile.","costUsd":0.007,"at":"2026-09-08T11:10:08.656Z","sourceCount":5,"ledgerOffset":375},{"requestId":"8446b33b9e8014e8431a8cb3a9a7dee8","agentId":"orum","round":75,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Build a modular, local congruence sieve for the cubic family before any large scan.","costUsd":0.007,"at":"2026-09-08T11:10:08.656Z","sourceCount":5,"ledgerOffset":376},{"requestId":"55225b081382e0ffef2fdb536a424b2d","agentId":"pip","round":75,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct a genuinely parametrized algebraic family by deforming the common-N equations for three or four difference rows into genus-one curves, then use rational points to generate new columns and test whether two additional rows can be imposed without repeating fixed Bremner specializations.","costUsd":0.007,"at":"2026-09-08T11:10:08.656Z","sourceCount":5,"ledgerOffset":377},{"requestId":"3b8f97da2b7a16d57050acc1e024cd76","agentId":"solvi","round":75,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Build an incidence-hypergraph/SAT exact-cover search from all currently verified cells, targeting a balanced 5x5 biclique and strict 4x5 improvements.","costUsd":0.007,"at":"2026-09-08T11:10:08.656Z","sourceCount":5,"ledgerOffset":378},{"requestId":"dbfe23e0f37d8eddf8f4c3a002318a13","agentId":"tess","round":75,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run stochastic, optimization-guided discovery over registry-separated cubic and primorial-shell seeds, then perform exact bigint verification of the highest-balanced-support candidates.","costUsd":0.007,"at":"2026-09-08T11:10:08.656Z","sourceCount":5,"ledgerOffset":379},{"requestId":"466597e2d79020a83f98f15cfab02af0","agentId":"mira","round":76,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially falsify the elliptic scaling claim, obtain exact saturation certificates for the active fixed families, and adapt the next bounded experiment only from its balanced-support profile.","costUsd":0.007,"at":"2026-09-08T11:20:13.709Z","sourceCount":5,"ledgerOffset":380},{"requestId":"1a23485907a1f165408624ba7ed4a273","agentId":"orum","round":76,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Design a modular local sieve that forces retained candidate columns into residue classes simultaneously admissible for at least five selected differences, then test exact support only inside those classes.","costUsd":0.007,"at":"2026-09-08T11:20:13.709Z","sourceCount":5,"ledgerOffset":381},{"requestId":"266d16ce1ea202fbcc169fe9a2357ee1","agentId":"pip","round":76,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Develop a genuinely varying elliptic or genus-one parametrization for simultaneous factor differences, rather than another fixed scaled five-row specialization.","costUsd":0.007,"at":"2026-09-08T11:20:13.709Z","sourceCount":5,"ledgerOffset":382},{"requestId":"e8f5fc61c9d23d828f47755c5b56263b","agentId":"solvi","round":76,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Using only Tess-verified cells from the accepted family scans, construct the incidence hypergraph and search for exact 5×5, 4×5, or 5×4 bicliques via SAT/exact-cover methods.","costUsd":0.007,"at":"2026-09-08T11:20:13.709Z","sourceCount":5,"ledgerOffset":383},{"requestId":"0737196b89b5679146c24ce539424dec","agentId":"tess","round":76,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run the rotating stochastic lane: generate algebraically specified five-row seeds by annealed or cross-entropy mutation of square-cofactor and elliptic parameters, optimize exact modular-support surrogates, then bigint-verify the strongest survivors.","costUsd":0.007,"at":"2026-09-08T11:20:13.709Z","sourceCount":5,"ledgerOffset":384},{"requestId":"4afc721753425a672def12688dcd8fe6","agentId":"mira","round":77,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a genuinely varying rational-point family of square-additive rectangles: derive an elliptic/genus-one fiber for adding a fifth difference, then vary the base parameters rather than repeating fixed Bremner specializations.","costUsd":0.007,"at":"2026-09-08T11:30:17.443Z","sourceCount":5,"ledgerOffset":385},{"requestId":"4cb55e96926cacc8aea470e08bd3ff6e","agentId":"orum","round":77,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Build an incidence hypergraph from independently verified exact cells returned by the new family scans, then search it by SAT or exact cover for 5x5, 6x4, and 4x5 bicliques.","costUsd":0.007,"at":"2026-09-08T11:30:17.443Z","sourceCount":5,"ledgerOffset":386},{"requestId":"b055b9f3cb32eea5b639199aeb83f32d","agentId":"pip","round":77,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Design a genuinely varying modular sieve that forces compatible local square roots across at least five difference rows, while excluding scaled Bremner and the tested CRT ±1 slice.","costUsd":0.007,"at":"2026-09-08T11:30:17.443Z","sourceCount":5,"ledgerOffset":387},{"requestId":"697bebc700e1dca0ddb6af68a47bd4f2","agentId":"solvi","round":77,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided seed discovery to generate new balanced-support difference families, then submit only high-scoring seeds for exact bigint verification; request no divisor_completion job.","costUsd":0.007,"at":"2026-09-08T11:30:17.443Z","sourceCount":5,"ledgerOffset":388},{"requestId":"38b212b57cbd32ccb8c5eff9cca4b011","agentId":"tess","round":77,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially falsify both proposed mechanisms, obtain exact saturation certificates for their finite incidence structures, and adapt the next experiment only from weakest-first biclique support.","costUsd":0.007,"at":"2026-09-08T11:30:17.443Z","sourceCount":5,"ledgerOffset":389},{"requestId":"f31705e72aedf5d1db558405732ffd05","agentId":"mira","round":78,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a new rational-point/elliptic family of square-additive rectangles, with parameters varying independently rather than another fixed row list.","costUsd":0.007,"at":"2026-09-08T11:40:17.925Z","sourceCount":5,"ledgerOffset":390},{"requestId":"584468686c09efddba4de91c0cd58db5","agentId":"orum","round":78,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct an incidence hypergraph from independently verified exact cells in Pip’s eight-branch CRT family, then search it for exact 5×5, 6×4, and 4×5 bicliques.","costUsd":0.007,"at":"2026-09-08T11:40:17.925Z","sourceCount":5,"ledgerOffset":391},{"requestId":"e0119879c69bccf05dc424f666d47bfb","agentId":"pip","round":78,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Design a modular local sieve for shifted-Pell and CRT-derived rows: enumerate prime-power residue classes of N and retain only classes supporting at least five selected differences simultaneously.","costUsd":0.007,"at":"2026-09-08T11:40:17.925Z","sourceCount":5,"ledgerOffset":392},{"requestId":"e900b0e592905bb1e68cbd3a59e6845b","agentId":"solvi","round":78,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided seed discovery for new affine-recursive and modular-hybrid difference families, then send only the best non-overlapping family_scan candidates for exact bigint verification by Pip.","costUsd":0.007,"at":"2026-09-08T11:40:17.925Z","sourceCount":5,"ledgerOffset":393},{"requestId":"19932b730021e504694e12e1ba8305a7","agentId":"tess","round":78,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially falsify the CRT and affine-Pell proposals, then adapt experiment design from verified saturation data toward an uncovered high-support boundary.","costUsd":0.007,"at":"2026-09-08T11:40:17.925Z","sourceCount":5,"ledgerOffset":394},{"requestId":"1c2862b50c637f0fe29069e53d94a3a6","agentId":"mira","round":79,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Develop a constructive rational-point family of square-additive rectangles, using the c=10054 CRT branch as a congruence sieve rather than as a standalone search; prioritize a parameterized extension from a verified lower-dimensional rectangle toward five common differences.","costUsd":0.007,"at":"2026-09-08T11:50:21.920Z","sourceCount":5,"ledgerOffset":395},{"requestId":"24df33db0860385ec28dffdccb62c603","agentId":"orum","round":79,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct an incidence hypergraph from independently verified cells in the fresh c=10054 CRT family and search it by exact-cover/SAT constraints for 5×5, 6×4, or 4×5 support.","costUsd":0.007,"at":"2026-09-08T11:50:21.920Z","sourceCount":5,"ledgerOffset":396},{"requestId":"f6bb7a39e354882667e62ee44c244591","agentId":"pip","round":79,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct a modular local sieve for CRT-derived difference rows, retaining residue classes of N that can support at least five selected differences simultaneously.","costUsd":0.007,"at":"2026-09-08T11:50:21.920Z","sourceCount":5,"ledgerOffset":397},{"requestId":"6c300318e8721c8f7fbf69f7ba18cc5b","agentId":"solvi","round":79,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided seed discovery on fresh CRT/local-residue and affine-parameter mutations, then submit only the highest exact-support seeds for bigint verification.","costUsd":0.007,"at":"2026-09-08T11:50:21.920Z","sourceCount":5,"ledgerOffset":398},{"requestId":"e39b291704ae293f3107bface79b9ba4","agentId":"tess","round":79,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially falsify the new CRT family, seek exact saturation certificates, and adapt the next family experiment from observed support patterns without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T11:50:21.920Z","sourceCount":5,"ledgerOffset":399},{"requestId":"0c4e4bbd257792770718f45cc95c83d6","agentId":"mira","round":80,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct new parametrized square-additive families for a fifth row, emphasizing elliptic curves and rational points outside the registry’s covered regimes.","costUsd":0.007,"at":"2026-09-08T12:00:28.128Z","sourceCount":5,"ledgerOffset":400},{"requestId":"903deb1d9a68f1e3ebfcbdc30be568f4","agentId":"orum","round":80,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Run an incidence-hypergraph/exact-cover search using only independently verified square-additive cells, seeking a 5-by-5 all-support certificate outside the covered registry regions.","costUsd":0.007,"at":"2026-09-08T12:00:28.128Z","sourceCount":5,"ledgerOffset":401},{"requestId":"c8c87fcef37b83301ab644425884d362","agentId":"pip","round":80,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Develop modular, local congruence filters that prune or force compatible fifth rows without repeating registry-covered regions.","costUsd":0.007,"at":"2026-09-08T12:00:28.128Z","sourceCount":5,"ledgerOffset":402},{"requestId":"49497229ca78223db4c87a6f46b89870","agentId":"solvi","round":80,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided seed discovery in an uncovered parameter regime, then perform exact bigint verification of support.","costUsd":0.007,"at":"2026-09-08T12:00:28.128Z","sourceCount":5,"ledgerOffset":403},{"requestId":"8f3f1906dded920f9e571f99bd04a7fd","agentId":"tess","round":80,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run adversarial falsification and saturation checks on uncovered algebraic or boundary regimes, without divisor_completion.","costUsd":0.007,"at":"2026-09-08T12:00:28.128Z","sourceCount":5,"ledgerOffset":404},{"requestId":"9f326cb289bd4c697ca37ed45642b946","agentId":"mira","round":81,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a genuine rational-point family by parametrizing the three-column intersection y_i²=x²+4A_i, then use elliptic group-law multiples and denominator clearing to generate integral rows and columns outside covered bounds.","costUsd":0.007,"at":"2026-09-08T12:10:36.410Z","sourceCount":5,"ledgerOffset":405},{"requestId":"3fee65b8cebf79afb7e5208de20d4536","agentId":"orum","round":81,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Build an incidence-hypergraph and exact-cover search over only exact verified cells returned by the new scans, targeting Pareto-optimal 5-row and 5-column supports.","costUsd":0.007,"at":"2026-09-08T12:10:36.410Z","sourceCount":5,"ledgerOffset":406},{"requestId":"15662a6fe3d3fff645ad7d66b2215228","agentId":"pip","round":81,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Design a genuinely new paired-CRT row family whose local square classes force a common N residue while avoiding all fixed Bremner specializations.","costUsd":0.007,"at":"2026-09-08T12:10:36.410Z","sourceCount":5,"ledgerOffset":407},{"requestId":"cfb1031b66e48b9bcaa598c72f507106","agentId":"solvi","round":81,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use stochastic, optimization-guided seed discovery across perturbed height and CRT shells outside covered regions, then perform exact bigint verification of the highest-support seeds.","costUsd":0.007,"at":"2026-09-08T12:10:36.410Z","sourceCount":5,"ledgerOffset":408},{"requestId":"3d0a06d906c8ee8507ae0e789c97c73b","agentId":"tess","round":81,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Adversarially falsify the claimed family mechanisms, then adapt the next exact experiment to the weakest uncovered support while seeking saturation certificates.","costUsd":0.007,"at":"2026-09-08T12:10:36.410Z","sourceCount":5,"ledgerOffset":409},{"requestId":"2b9475ec83df03a8dc9db8e92c8f62e7","agentId":"mira","round":82,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Develop a modular sign-allocation filter that forces compatible witness residues across several rows before any global square search.","costUsd":0.007,"at":"2026-09-08T12:20:40.439Z","sourceCount":5,"ledgerOffset":410},{"requestId":"14aba5bac87abb77fc442a4172959e8d","agentId":"orum","round":82,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided seed discovery to search fresh paired-CRT and algebraic parameter seeds, then perform exact bigint verification for Pareto-improving support patterns.","costUsd":0.007,"at":"2026-09-08T12:20:40.439Z","sourceCount":5,"ledgerOffset":411},{"requestId":"3daf8d2b27077d01a9fc9766e947eccf","agentId":"pip","round":82,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use only verified cells to build an incidence hypergraph and search exact-cover/biclique structures in the new CRT shell.","costUsd":0.007,"at":"2026-09-08T12:20:40.439Z","sourceCount":5,"ledgerOffset":412},{"requestId":"f2e6210340864ab461e8bda7a239f506","agentId":"solvi","round":82,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use adversarial falsification, saturation certificates, and adaptive experiment design on the two proposed CRT shells without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T12:20:40.439Z","sourceCount":5,"ledgerOffset":413},{"requestId":"f4c23659e8ffde6ee1df17fbc48d239b","agentId":"tess","round":82,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Develop a genuine rational-point family: parametrize common completions for three differences by an elliptic quartic, then seek group-law-generated columns and an algebraic fourth- or fifth-row extension outside covered regimes.","costUsd":0.007,"at":"2026-09-08T12:20:40.439Z","sourceCount":5,"ledgerOffset":414},{"requestId":"20dbe7e70608c6b37506bca7fdcfb51d","agentId":"mira","round":83,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a new modular shell whose local root/sign assignments are simultaneously compatible across at least five rows, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T12:30:43.637Z","sourceCount":5,"ledgerOffset":415},{"requestId":"457086d00fe5670663f60a523cbc10a1","agentId":"orum","round":83,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Generate stochastic, optimization-ranked seeds from fresh affine/AP–CRT hybrid families outside the registry, then route the highest-support seeds to Pip for deterministic exact bigint verification; request no divisor_completion job.","costUsd":0.007,"at":"2026-09-08T12:30:43.637Z","sourceCount":5,"ledgerOffset":416},{"requestId":"d48db27d0855821602f953bc06fbbc18","agentId":"pip","round":83,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an exact incidence-hypergraph/SAT search over independently verified cells, prioritizing Mira’s p-adic shell and recording local sign-allocation masks before biclique search.","costUsd":0.007,"at":"2026-09-08T12:30:43.637Z","sourceCount":5,"ledgerOffset":417},{"requestId":"70779b510ee92685b8a6038c4940edfe","agentId":"solvi","round":83,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Run an adversarial, saturation-focused round that tests whether either proposed family has genuine high-support structure rather than merely useful necessary conditions.","costUsd":0.007,"at":"2026-09-08T12:30:43.637Z","sourceCount":5,"ledgerOffset":418},{"requestId":"bf34c2018186f16ac10fcbffde06f33d","agentId":"tess","round":83,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a genuinely new rational-point family of square-additive rectangles, using the projective equations m_{ij}^2=d_i^2+4N_j and an elliptic-curve model for extending a variable 4-row or 4-column configuration. Target a verified 5×4 improvement first, then seek a fifth column.","costUsd":0.007,"at":"2026-09-08T12:30:43.637Z","sourceCount":5,"ledgerOffset":419},{"requestId":"5df0730b85f660aea41d14f871c62aa0","agentId":"mira","round":84,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a rotating-lane modular sieve that couples several proposed rows through simultaneous local root and sign constraints, seeking compatible five-row residue patterns before exact validation by Orum.","costUsd":0.007,"at":"2026-09-08T12:40:47.051Z","sourceCount":5,"ledgerOffset":420},{"requestId":"ef482c0d9127b5738c07d9e4b1e80759","agentId":"orum","round":84,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided sampling of untested p-adic/CRT sign states and parameter seeds to discover high-support candidates, then certify every surviving cell with exact bigint arithmetic.","costUsd":0.007,"at":"2026-09-08T12:40:47.051Z","sourceCount":5,"ledgerOffset":421},{"requestId":"5afd9eca300da571645ed61d66cd8a65","agentId":"pip","round":84,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run a deterministic incidence-hypergraph/SAT or exact-cover search over independently verified cells from the accepted shells, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T12:40:47.051Z","sourceCount":5,"ledgerOffset":422},{"requestId":"773122a79a71994f3128d5b8e6bc1a9b","agentId":"solvi","round":84,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially falsify both shells, derive saturation certificates for their attainable row and column supports, and adapt the next experiment only if an exact support frontier survives.","costUsd":0.007,"at":"2026-09-08T12:40:47.051Z","sourceCount":5,"ledgerOffset":423},{"requestId":"e57d594d72c2b5bed1ec1f57c7eed2c9","agentId":"tess","round":84,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct an exact algebraic bridge from Mira’s p-adic shell to a rational-point or elliptic-curve family capable of extending a four-row rectangle by a fifth row, while avoiding the covered denominator regime.","costUsd":0.007,"at":"2026-09-08T12:40:47.051Z","sourceCount":5,"ledgerOffset":424},{"requestId":"d21d89a4a495e090b7427183fc9f49de","agentId":"mira","round":85,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Construct a modular local sieve that preselects five-row difference tuples with simultaneous witness residues before any large exact scan.","costUsd":0.007,"at":"2026-09-08T12:50:50.925Z","sourceCount":5,"ledgerOffset":425},{"requestId":"f9068ab71831fdbc93f940e56e438bcb","agentId":"orum","round":85,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided seed discovery across the Q=729 shell and smooth-step progression, then perform exact bigint verification for maximal support.","costUsd":0.007,"at":"2026-09-08T12:50:50.925Z","sourceCount":5,"ledgerOffset":426},{"requestId":"b60b1704250ea910cdfffebdd1d8e072","agentId":"pip","round":85,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use an incidence-hypergraph/SAT exact-cover search over independently bigint-verified cells from the accepted modular and progression families, targeting 5×5 first and strict 6×4 or 4×5 alternatives.","costUsd":0.007,"at":"2026-09-08T12:50:50.925Z","sourceCount":5,"ledgerOffset":427},{"requestId":"2e88829605bb7dc035b4ec3d9cb16b59","agentId":"solvi","round":85,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially compare the 3^6 shell and the 27720-step progression using exact support saturation and adaptive allocation, while reserving divisor completion only for a candidate independently produced by a primary search.","costUsd":0.007,"at":"2026-09-08T12:50:50.925Z","sourceCount":5,"ledgerOffset":428},{"requestId":"ca8d84803fe877c7abec00bd299b097b","agentId":"tess","round":85,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a new algebraic-family route by parametrizing simultaneous square conditions through rational points on conics and their elliptic-curve intersections, targeting five-row shared columns outside the registry.","costUsd":0.007,"at":"2026-09-08T12:50:50.925Z","sourceCount":5,"ledgerOffset":429},{"requestId":"00f5b3d67b2f60bfed15312d2d627376","agentId":"mira","round":86,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Design a modular CRT sieve that forces compatible five-row residue classes while explicitly retaining the divisibility and parity branches needed for genuine completions.","costUsd":0.007,"at":"2026-09-08T13:00:55.575Z","sourceCount":5,"ledgerOffset":430},{"requestId":"563bc5e41313b52f16a508a70509e601","agentId":"orum","round":86,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Use stochastic, optimization-guided seed discovery over new parameterized difference families, ranking seeds by predicted multi-row and multi-column support, then perform exact bigint verification only on the highest-scoring seeds.","costUsd":0.007,"at":"2026-09-08T13:00:55.575Z","sourceCount":5,"ledgerOffset":431},{"requestId":"062c11cf92cea4e163e7f3ee15547bc8","agentId":"pip","round":86,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct an incidence hypergraph from independently verified cells in the two new families and solve exact-cover/SAT instances for 5×5, then strict 6×4 and 4×5 support.","costUsd":0.007,"at":"2026-09-08T13:00:55.575Z","sourceCount":5,"ledgerOffset":432},{"requestId":"968f0f7541d4278eebdb56792b72c102","agentId":"solvi","round":86,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Adversarially falsify both proposed families, then build saturation certificates and choose the next experiment from exact support density rather than residue plausibility.","costUsd":0.007,"at":"2026-09-08T13:00:55.575Z","sourceCount":5,"ledgerOffset":433},{"requestId":"ba2a28518f1139373aca341f95c50840","agentId":"tess","round":86,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a new rational/elliptic parametrization of simultaneous cells, seeking a reusable family with five prescribed differences rather than scanning residue shells.","costUsd":0.007,"at":"2026-09-08T13:00:55.575Z","sourceCount":5,"ledgerOffset":434},{"requestId":"d3eed94a85fd7790e2d39969da6e3f7a","agentId":"mira","round":87,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an incidence-hypergraph/exact-cover search using only deterministically verified cells, targeting five differences shared by five distinct N values.","costUsd":0.007,"at":"2026-09-08T13:11:03.239Z","sourceCount":5,"ledgerOffset":435},{"requestId":"3a190b8adbe74b04555b6569d227c6b9","agentId":"orum","round":87,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Recover with an adversarial, non-validation experiment targeting an uncovered algebraic family or parameter regime, with an exact saturation certificate.","costUsd":0.007,"at":"2026-09-08T13:11:03.239Z","sourceCount":5,"ledgerOffset":436},{"requestId":"d971f581ed828d7ef0f241889d8a9f01","agentId":"pip","round":87,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run optimization-guided stochastic seed discovery in an uncovered family or parameter regime, followed by deterministic bigint support verification.","costUsd":0.007,"at":"2026-09-08T13:11:03.239Z","sourceCount":5,"ledgerOffset":437},{"requestId":"535f5746a9b8ad2cc0313c75918087f0","agentId":"solvi","round":87,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct a new rational-point parametrization for simultaneous cells N=x(x+d), seeking a five-row common-support family outside the covered searches.","costUsd":0.007,"at":"2026-09-08T13:11:03.239Z","sourceCount":5,"ledgerOffset":438},{"requestId":"b16a6e5a9bd2361508afd644840e262e","agentId":"tess","round":87,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Design a modular, local congruence filter for compatible rows without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T13:11:03.239Z","sourceCount":5,"ledgerOffset":439},{"requestId":"f88dbd59faa6d2be2626d64895ca6692","agentId":"mira","round":88,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Use an exact-cover search over only verified cells returned by the family scans, prioritizing balanced 5-row/5-column incidence and strict Pareto improvements.","costUsd":0.007,"at":"2026-09-08T13:21:07.122Z","sourceCount":5,"ledgerOffset":440},{"requestId":"f4d408cf2eca9bb6de6f6408242e25a9","agentId":"orum","round":88,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially eliminate unsupported shells before any new scan, then select only a regime with a certifiable path to balanced support.","costUsd":0.007,"at":"2026-09-08T13:21:07.122Z","sourceCount":5,"ledgerOffset":441},{"requestId":"0f196785ae739cfab0e9a623a109d57f","agentId":"pip","round":88,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use stochastic, optimization-guided seed discovery in uncovered high-difference algebraic families, then perform deterministic exact-bigint incidence verification; use no divisor_completion job unless another primary method first produces a concrete candidate.","costUsd":0.007,"at":"2026-09-08T13:21:07.122Z","sourceCount":5,"ledgerOffset":442},{"requestId":"b8c656eab02f9d7e3ffd1a6eb9078c6d","agentId":"solvi","round":88,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct a new rational genus-one family by normalizing a square-additive rectangle, eliminating the column parameters, and deriving an elliptic or quartic curve whose rational points generate an additional row or column.","costUsd":0.007,"at":"2026-09-08T13:21:07.122Z","sourceCount":5,"ledgerOffset":443},{"requestId":"7bd5c0459c0f3f17919478ec4c779283","agentId":"tess","round":88,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Design a new family_scan whose rows share deliberately engineered local square conditions modulo a composite modulus, while using nonlinear offsets to avoid covered arithmetic-progressions and shells.","costUsd":0.007,"at":"2026-09-08T13:21:07.122Z","sourceCount":5,"ledgerOffset":444},{"requestId":"78ff7e910b55a4032d5da8bd6aefbc99","agentId":"mira","round":89,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Build an incidence-hypergraph/exact-cover search using only cells independently verified by Orum from the new slates, targeting balanced 5×5 and strict 6×4 or 4×5 bicliques.","costUsd":0.007,"at":"2026-09-08T13:31:12.968Z","sourceCount":5,"ledgerOffset":445},{"requestId":"c24718b3a43c8109a35d698a3ef5be73","agentId":"orum","round":89,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially select the better primary family by cheap compatibility tests, then demand a deterministic saturation certificate for its finite incidence domain.","costUsd":0.007,"at":"2026-09-08T13:31:12.968Z","sourceCount":5,"ledgerOffset":446},{"requestId":"4c44150b4da6faab32d098667da60a05","agentId":"pip","round":89,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use stochastic, optimization-guided seed discovery across uncovered high-difference offset and square-anchor parameter tuples, scoring only exact incidence support, then submit the strongest masks for deterministic bigint verification.","costUsd":0.007,"at":"2026-09-08T13:31:12.968Z","sourceCount":5,"ledgerOffset":447},{"requestId":"7f7513bc55dda6fc08640e455f25dc05","agentId":"solvi","round":89,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Develop a constructive rational/elliptic family from the intersection y_i^2=x+d_i^2, with x=4N, then glue exact rational points from overlapping difference triples toward balanced support.","costUsd":0.007,"at":"2026-09-08T13:31:12.968Z","sourceCount":5,"ledgerOffset":448},{"requestId":"b2eb017f3ba4facd8472741a13ed6958","agentId":"tess","round":89,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a modular local-congruence sieve for equal-offset shells before requesting any exact scan, targeting compatible five-row signatures and excluding the previously tested parameter window.","costUsd":0.007,"at":"2026-09-08T13:31:12.968Z","sourceCount":5,"ledgerOffset":449},{"requestId":"573d02a8901bef0eb70456e02b312018","agentId":"mira","round":90,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an incidence-hypergraph/SAT exact-cover search using only independently verified cells, seeking K_{5,5}, K_{6,4}, and K_{4,5} without requesting divisor completion.","costUsd":0.007,"at":"2026-09-08T13:41:17.754Z","sourceCount":5,"ledgerOffset":450},{"requestId":"e8933ea33423dbd4258cdff2f69617ce","agentId":"orum","round":90,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially test the two new family scans for false support and seek a saturation certificate through adaptive exact incidence analysis.","costUsd":0.007,"at":"2026-09-08T13:41:17.754Z","sourceCount":5,"ledgerOffset":451},{"requestId":"7572299f65caad2af828add1245ef373","agentId":"pip","round":90,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run stochastic, optimization-guided seed discovery over uncovered high-difference offset and quartic parameter slates, optimizing weakest row/column exact-support floors, then submit only the strongest slates for deterministic bigint verification.","costUsd":0.007,"at":"2026-09-08T13:41:17.754Z","sourceCount":5,"ledgerOffset":452},{"requestId":"f77c7bbdbfdcff169ae6fda4b905ef38","agentId":"solvi","round":90,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct an algebraic completion family by fixing three common differences and reducing the equations m_i^2-4N=d_i^2 to an explicit genus-one curve, then use rational-point generation to seek five distinct integral N-columns and two additional common differences.","costUsd":0.007,"at":"2026-09-08T13:41:17.754Z","sourceCount":5,"ledgerOffset":453},{"requestId":"3b637bf20ed6e86ba7b951246f05938c","agentId":"tess","round":90,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Construct a modular and local compatibility sieve for shared rows, prioritizing five-row support without numerical near-square scoring or divisor_completion.","costUsd":0.007,"at":"2026-09-08T13:41:17.754Z","sourceCount":5,"ledgerOffset":454},{"requestId":"8146625fcb312d2ec477591c4106be9c","agentId":"mira","round":91,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an incidence-hypergraph/SAT search using only independently verified cells from the cross-block support seeds, targeting balanced bicliques.","costUsd":0.007,"at":"2026-09-08T13:51:17.601Z","sourceCount":5,"ledgerOffset":455},{"requestId":"a1ef033f2a9c8cbe34a4477f604e4afc","agentId":"orum","round":91,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Adversarially falsify both proposed scans before accepting any support improvement, then design the smallest adaptive exact experiment with a saturation certificate.","costUsd":0.007,"at":"2026-09-08T13:51:17.601Z","sourceCount":5,"ledgerOffset":456},{"requestId":"cb61969318ae5eb8b1efcac4ac6fce0a","agentId":"pip","round":91,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use stochastic, optimization-guided generation of uncovered high-difference shell slates, optimizing weakest row and column exact-support floors, followed by deterministic exact-bigint verification.","costUsd":0.007,"at":"2026-09-08T13:51:17.601Z","sourceCount":5,"ledgerOffset":457},{"requestId":"7711a5ef1d59b4154ffd984b8fd211ba","agentId":"solvi","round":91,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Construct a new difference-of-squares family by changing to U=m+d and V=m-d coordinates, quotienting the scaling symmetry, and reducing the remaining simultaneous-incidence conditions to rational conics and genus-one fibers; submit only a family_scan, never divisor_completion.","costUsd":0.007,"at":"2026-09-08T13:51:17.601Z","sourceCount":5,"ledgerOffset":458},{"requestId":"ae70e00aa6636c264ae58485bc80e0a0","agentId":"tess","round":91,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Design a modular local filter that forces at least five differences to remain simultaneously compatible with candidate N columns before exact bigint verification.","costUsd":0.007,"at":"2026-09-08T13:51:17.601Z","sourceCount":5,"ledgerOffset":459},{"requestId":"fd204048c7da39be12b4a2504ea60937","agentId":"mira","round":92,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Use stochastic, optimization-guided seed discovery in uncovered algebraic families, maximizing lexicographic minimum row and column exact support, then perform deterministic bigint verification.","costUsd":0.007,"at":"2026-09-08T14:01:22.568Z","sourceCount":5,"ledgerOffset":460},{"requestId":"aa61d60de55b9113e2038f5a7cf3a4f7","agentId":"orum","round":92,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Derive an explicit rational-point family for square-additive rectangles, preferably by extending a known 4-row configuration through an elliptic-curve or conic-intersection parametrization, before requesting any scan.","costUsd":0.007,"at":"2026-09-08T14:01:22.568Z","sourceCount":5,"ledgerOffset":461},{"requestId":"5e0ca603a34d09473afb5d8f295d1b61","agentId":"pip","round":92,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially falsify both proposed saturation mechanisms, certify maximal support profiles, and adapt the next experiment only from independently verified residual neighborhoods.","costUsd":0.007,"at":"2026-09-08T14:01:22.568Z","sourceCount":5,"ledgerOffset":462},{"requestId":"d11a04063dc46a6f12f12ffcbb6b3928","agentId":"solvi","round":92,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Design a new local congruence filter for shell parameters that prunes incompatible rows before exact biclique extraction, without requesting divisor completion.","costUsd":0.007,"at":"2026-09-08T14:01:22.568Z","sourceCount":5,"ledgerOffset":463},{"requestId":"2bacdc7a1762bd7b2aade22c28760e2b","agentId":"tess","round":92,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run an incidence-hypergraph/SAT search over the union of independently verified shell cells, targeting K_{5,5}, then strict K_{4,5} and K_{6,4} support profiles.","costUsd":0.007,"at":"2026-09-08T14:01:22.568Z","sourceCount":5,"ledgerOffset":464},{"requestId":"6ff23a445534d971154ac25f85c6fd8b","agentId":"mira","round":93,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Use stochastic, optimization-guided seed discovery across new affine-shell parameter combinations, then exact bigint verification of the strongest support neighborhoods.","costUsd":0.007,"at":"2026-09-08T14:11:22.893Z","sourceCount":5,"ledgerOffset":465},{"requestId":"8874f66800016c4198e8fc21fd563ecb","agentId":"orum","round":93,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a genuinely shared-N algebraic family by intersecting the affine square-root shell with rational-point or elliptic-curve constraints, rather than scanning unrelated difference slates.","costUsd":0.007,"at":"2026-09-08T14:11:22.893Z","sourceCount":5,"ledgerOffset":466},{"requestId":"fdb02a86f2df33c7731ebef831fc9960","agentId":"pip","round":93,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Adversarially compare the T=1009 affine shell with the offset-Fibonacci shell through deterministic saturation certificates and adaptive support-neighborhood experiments.","costUsd":0.007,"at":"2026-09-08T14:11:22.893Z","sourceCount":5,"ledgerOffset":467},{"requestId":"5a492f2b4b9934af788752d2e73433a8","agentId":"solvi","round":93,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Develop a modular, local congruence filter for affine-shell rows that forces shared column neighborhoods before exact validation; issue no divisor_completion request.","costUsd":0.007,"at":"2026-09-08T14:11:22.893Z","sourceCount":5,"ledgerOffset":468},{"requestId":"c962aae95a902d55f43b2ecf0d336d47","agentId":"tess","round":93,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use only independently verified cells from the T=1009 shell and any validated residuals to build an incidence hypergraph, then run exact-cover/biclique search for strict support improvements.","costUsd":0.007,"at":"2026-09-08T14:11:22.893Z","sourceCount":5,"ledgerOffset":469},{"requestId":"c632f81e8830834d5bec142bcd4c883d","agentId":"mira","round":94,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Use stochastic, optimization-guided seed discovery over mixed affine-shell parameters, then perform exact bigint verification of the highest-support seeds.","costUsd":0.007,"at":"2026-09-08T14:21:29.323Z","sourceCount":5,"ledgerOffset":470},{"requestId":"cc2a00560a1fe27ad536dcd7091d93ac","agentId":"orum","round":94,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a lower-dimensional rational or elliptic parametrization of simultaneous square conditions, prioritizing shared five-row neighborhoods rather than broad shell scans.","costUsd":0.007,"at":"2026-09-08T14:21:29.323Z","sourceCount":5,"ledgerOffset":471},{"requestId":"c90ff9c3c950f7f905cdb80b4d8acb02","agentId":"pip","round":94,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Use adversarial falsification, saturation certificates, and adaptive family design to test whether the new affine-shell incidence geometry genuinely improves the support frontier.","costUsd":0.007,"at":"2026-09-08T14:21:29.323Z","sourceCount":5,"ledgerOffset":472},{"requestId":"de8e7c5b62efd311102e2d58c284e360","agentId":"solvi","round":94,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Develop a deterministic modular sieve for the new affine and quadratic-unit shells, pruning or forcing compatible row neighborhoods before any family scan; send only surviving witnesses to Orum for validation.","costUsd":0.007,"at":"2026-09-08T14:21:29.323Z","sourceCount":5,"ledgerOffset":473},{"requestId":"21e18aaa34db40f9636bc6ec824557ab","agentId":"tess","round":94,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use only Orum-verified cells from the two proposed regimes to construct a deterministic incidence hypergraph, then solve exact-cover/SAT instances for K_{5,5}, K_{6,4}, and K_{4,5}.","costUsd":0.007,"at":"2026-09-08T14:21:29.323Z","sourceCount":5,"ledgerOffset":474},{"requestId":"92ab5ba6f4190a462eb22090ac2d3aed","agentId":"mira","round":95,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Use stochastic, optimization-guided seed discovery over previously untested mixed-gap affine-shell parameters, then exact bigint verification of the highest-support seeds; request only family_scan jobs.","costUsd":0.007,"at":"2026-09-08T14:31:33.019Z","sourceCount":5,"ledgerOffset":475},{"requestId":"2795aa6f8599a5d4fb8df0b327845eb3","agentId":"orum","round":95,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a genuinely coupled rational-point family by expressing five shared differences and five N-columns as an intersection of exact quadrics, then identify and exploit any genus-one or elliptic quotient rather than scanning an arbitrary difference slate.","costUsd":0.007,"at":"2026-09-08T14:31:33.019Z","sourceCount":5,"ledgerOffset":476},{"requestId":"414a274f91971766acc65af88055bf7a","agentId":"pip","round":95,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run an adaptive family-scan-only round centered on adversarial falsification and exact saturation certificates, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T14:31:33.019Z","sourceCount":5,"ledgerOffset":477},{"requestId":"b8636e3904fad9926f72d9a7e6741d1b","agentId":"solvi","round":95,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Develop a modular and local CRT sieve that forces compatible difference rows before any large search, using a primary algebraic construction rather than another blind family scan.","costUsd":0.007,"at":"2026-09-08T14:31:33.019Z","sourceCount":5,"ledgerOffset":478},{"requestId":"abdabd293cd79bf091daf60d28742d0a","agentId":"tess","round":95,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run an incidence-hypergraph exact-cover search over only independently verified cells from the new family scans, targeting K_{5,5} first and strict K_{4,5} or K_{6,4} second.","costUsd":0.007,"at":"2026-09-08T14:31:33.019Z","sourceCount":5,"ledgerOffset":479},{"requestId":"20461d3aba54c32fecf012edd53dc707","agentId":"mira","round":96,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Use stochastic, optimization-guided seed discovery in an untested affine-shell/Pell-phase family, then apply deterministic bigint verification; request no divisor_completion and keep Tess as the sole validation-only verifier.","costUsd":0.007,"at":"2026-09-08T14:41:35.019Z","sourceCount":5,"ledgerOffset":480},{"requestId":"02d410cee712b2bc83ea66cc6215760d","agentId":"orum","round":96,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Develop a genuinely shared-N parametrization by intersecting the affine-shell equations and reducing the resulting rational-point conditions to elliptic-curve or higher-genus algebraic families.","costUsd":0.007,"at":"2026-09-08T14:41:35.019Z","sourceCount":5,"ledgerOffset":481},{"requestId":"62401037cd33d2be83ef50598115a99f","agentId":"pip","round":96,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Run one adaptive family_scan-only round centered on adversarial falsification and exact saturation certificates, using the strongest residual support pattern rather than another blind parameter slate.","costUsd":0.007,"at":"2026-09-08T14:41:35.019Z","sourceCount":5,"ledgerOffset":482},{"requestId":"89fcadfeb8427119039a6a64a3523631","agentId":"solvi","round":96,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Design a modular and local congruence prefilter that identifies difference sets capable of sharing candidate N residues, then use those signatures to prioritize a deterministic family scan; request no divisor_completion.","costUsd":0.007,"at":"2026-09-08T14:41:35.019Z","sourceCount":5,"ledgerOffset":483},{"requestId":"6bde470f968fb9e6e79eae35142b7aca","agentId":"tess","round":96,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use the rotating incidence-hypergraph/SAT/exact-cover lane on deterministically verified cells only, seeking a strict support-profile improvement.","costUsd":0.007,"at":"2026-09-08T14:41:35.019Z","sourceCount":5,"ledgerOffset":484},{"requestId":"5439d17076266d2d002fc444c18aef6f","agentId":"mira","round":97,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run an adversarial, saturation-first comparison of new shell families, using adaptive experiment design rather than raw candidate volume.","costUsd":0.007,"at":"2026-09-08T14:51:39.641Z","sourceCount":5,"ledgerOffset":485},{"requestId":"ad98f791ded91c5f70b16407278a7113","agentId":"orum","round":97,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Develop a modular, local congruence sieve that forces compatible shared-N rows before exact support search.","costUsd":0.007,"at":"2026-09-08T14:51:39.641Z","sourceCount":5,"ledgerOffset":486},{"requestId":"c6559ffdae03407a2e45648ca51b344a","agentId":"pip","round":97,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Develop a corrected constructive affine-shell parametrization and reduce shared five-row support to explicit rational-point conditions on intersections of quadrics, preferably an elliptic fibration.","costUsd":0.007,"at":"2026-09-08T14:51:39.641Z","sourceCount":5,"ledgerOffset":487},{"requestId":"de0214151943970b4b2f23185de7a21f","agentId":"solvi","round":97,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Build an exact incidence-hypergraph/SAT search over independently verified cells from corrected affine-shell and CRT-shell families, targeting K_{5,5}, then K_{6,4} or K_{4,5}.","costUsd":0.007,"at":"2026-09-08T14:51:39.641Z","sourceCount":5,"ledgerOffset":488},{"requestId":"3d72661b7f2fee5f115048a6d678eb70","agentId":"tess","round":97,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use stochastic, optimization-guided sampling of new affine-shell parameter seeds, scoring predicted row/column support floors, then submit only the strongest non-registry seeds to exact bigint family scans.","costUsd":0.007,"at":"2026-09-08T14:51:39.641Z","sourceCount":5,"ledgerOffset":489},{"requestId":"28b53aa218d1f5a8a1bdef230f58856c","agentId":"mira","round":98,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run the next round in the adversarial-falsification lane: demand saturation certificates before interpreting any negative shell result, then adaptively narrow toward families with demonstrably high support.","costUsd":0.007,"at":"2026-09-08T15:01:42.385Z","sourceCount":5,"ledgerOffset":490},{"requestId":"419071712956824fb54795bb3944054a","agentId":"orum","round":98,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Construct a modular row-compatibility filter that prunes or forces candidate five-row intersections before exact bigint square testing, without requesting divisor_completion.","costUsd":0.007,"at":"2026-09-08T15:01:42.385Z","sourceCount":5,"ledgerOffset":491},{"requestId":"73305ac7e6ee5f1135a211407fc19759","agentId":"pip","round":98,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct a rational elliptic-family lift from a four-row square-additive configuration to a fifth common difference and five distinct positive columns.","costUsd":0.007,"at":"2026-09-08T15:01:42.385Z","sourceCount":5,"ledgerOffset":492},{"requestId":"999515a4ff638d1461b0de0ef53936d7","agentId":"solvi","round":98,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use only independently verified incidence cells to build an incidence-hypergraph and run exact-cover/SAT searches for a (5,5), (6,4), or (4,5) biclique; do not request divisor_completion.","costUsd":0.007,"at":"2026-09-08T15:01:42.385Z","sourceCount":5,"ledgerOffset":493},{"requestId":"d99bcd6e26cc7c20b3e4ea99de877097","agentId":"tess","round":98,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Run stochastic, optimization-guided discovery of undercovered affine/Pell shell seeds, then submit only the highest Pareto-support seeds to exact bigint verification.","costUsd":0.007,"at":"2026-09-08T15:01:42.385Z","sourceCount":5,"ledgerOffset":494},{"requestId":"4e88c4f35ae604620337a42e80e2edb2","agentId":"mira","round":99,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Run the next round in the adversarial-falsification lane: stress-test both proposed shell families, seek saturation certificates, and adaptively retain only slices showing strong exact support.","costUsd":0.007,"at":"2026-09-08T15:11:44.470Z","sourceCount":5,"ledgerOffset":495},{"requestId":"d2c531efced0c4021a603e0c6f996976","agentId":"orum","round":99,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Develop a modular and local congruence sieve for the two shell families before any broad computation; request no divisor-completion job unless an independently generated exact candidate appears.","costUsd":0.007,"at":"2026-09-08T15:11:44.470Z","sourceCount":5,"ledgerOffset":496},{"requestId":"3aa6a1b348bbac0080a6c7e59fcf90af","agentId":"pip","round":99,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct a generic rational elliptic lift from a four-row square-additive rectangle to a fifth common difference, avoiding fixed Bremner specializations and requesting no divisor-completion job.","costUsd":0.007,"at":"2026-09-08T15:11:44.470Z","sourceCount":5,"ledgerOffset":497},{"requestId":"c625159f251cda43ac3a349eb0d5c873","agentId":"solvi","round":99,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use only independently verified cells to construct an incidence hypergraph and run exact-cover/SAT searches for balanced 5×5 and Pareto-improving 6×4 or 4×5 supports; request no divisor completion.","costUsd":0.007,"at":"2026-09-08T15:11:44.470Z","sourceCount":5,"ledgerOffset":498},{"requestId":"7f77f092e733608a1555af1ad4a1f04d","agentId":"tess","round":99,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use stochastic mutation and crossover over nonlinear-shell and Pell-shell parameters, ranking seeds by exact support potential, then submit only the highest-scoring family_scan seeds for deterministic bigint verification.","costUsd":0.007,"at":"2026-09-08T15:11:44.470Z","sourceCount":5,"ledgerOffset":499},{"requestId":"ec02e9b3dd3e1833c03e2be2b82a374a","agentId":"mira","round":100,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Cartographer: Adversarially test shell support while building saturation certificates and adaptively retaining only exact self-incidence survivors.","costUsd":0.007,"at":"2026-09-08T15:21:48.619Z","sourceCount":5,"ledgerOffset":500},{"requestId":"a29e8ce24401c292c076204d3342ed6e","agentId":"orum","round":100,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Skeptic: Design a modular local sieve that retains only row pairs and completions compatible with five-way support, without requesting divisor completion.","costUsd":0.007,"at":"2026-09-08T15:21:48.619Z","sourceCount":5,"ledgerOffset":501},{"requestId":"e966e4f98aba2c860f2d62d920d55d4a","agentId":"pip","round":100,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Factor Gardener: Construct a rational-point/algebraic-family extension of the known 4x4 Bremner rectangle, then derive a fifth row or fifth column rather than broad-scanning shell differences.","costUsd":0.007,"at":"2026-09-08T15:21:48.619Z","sourceCount":5,"ledgerOffset":502},{"requestId":"a129d18b5c0df5224c4c6362dfef899f","agentId":"solvi","round":100,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Symmetry Thief: Use only independently verified incidence cells to build a row-column hypergraph, then run an exact-cover/SAT search for K_{5,5}, K_{6,4}, and K_{4,5} support across the cubic and corrected 4-adic slices.","costUsd":0.007,"at":"2026-09-08T15:21:48.619Z","sourceCount":5,"ledgerOffset":503},{"requestId":"05c4ab213c806a2ac4e1d9fba5e01cbb","agentId":"tess","round":100,"query":"Erdos problem 885 Diophantine rectangles simultaneous square differences The Boundary Runner: Use stochastic, optimization-guided mutations of the cubic-shell and 4-adic-shell parameters to discover seeds with improved exact weakest-first support, followed by deterministic bigint verification; do not request divisor_completion.","costUsd":0.007,"at":"2026-09-08T15:21:48.619Z","sourceCount":5,"ledgerOffset":504}],"codeJobs":[{"id":"competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r1-mira-0","agentId":"mira","round":1,"jobType":"divisor_completion","status":"complete","error":null,"createdAt":"2026-09-03T17:58:32.268Z","completedAt":"2026-09-03T17:59:06.158Z"},{"id":"competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r1-mira-1","agentId":"mira","round":1,"jobType":"divisor_completion","status":"complete","error":null,"createdAt":"2026-09-03T17:58:32.717Z","completedAt":"2026-09-03T17:59:02.516Z"},{"id":"competition-f6aadf00-aeb3-465b-9ba2-8679d70f28b1-r1-mira-2","agentId":"mira","round":1,"jobType":"boundary_scan","status":"failed","error":"endDifference must be a safe integer from 161 to 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