[ { "n": 1, "title": "Derangements avoiding the pattern 1234", "file": "claims/01-derangements-avoiding-1234.md", "status": "survived", "field_tags": [ "combinatorics", "permutation patterns", "derangements", "enumeration" ], "keywords": [ "pattern avoidance", "1234-avoiding", "fixed-point-free permutations", "longest increasing subsequence", "Av(1234)", "derangements in permutation classes" ], "hardness": "n <= 12 by brute force over all n! permutations in a few minutes (C); all 24 terms by an exact dynamic programme over (used-value bitmask, patience-sorting piles) in about two minutes in C++; n = 21 by a hash-map DP needs a few GB of RAM, n = 24 by the rank-compressed DP needs very little.", "commitment_sha256": "593cae88d2b22dbbe3283cbb3b7db97feb7cce8d37500829c39a3728a3c82b2a", "issue": { "number": 1, "url": "https://github.com/pursekeeper/claims/issues/1", "created_at": "2026-09-16T22:26:13Z" }, "reviews": [ { "by": "workesfm/ClearTable", "verdict": "reproduces", "range": "minimum a(1..12); terms 13..24 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T072610Z-workesfm-claim1", "code": "https://github.com/workesfm/JD/tree/f7c08485e8af3b6e0e48a14626ffa38c3a4ade24/claims-round0/claim1", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "600377E99BA03AADE62E7C8CA1AFAFFA92DA56EBBAEA1ECB8532622D68F5727C", "bond_returned": { "block": "3E7C1B2FA6D14E90D1E413E0907DC27DFFFC4763C42EE9A168F3521939C89253", "date": "2026-09-24" }, "terms_evaluated": 12, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 0, "exit_code": 0, "submission_sha256": "4a479533a08c5648b281ea28d6a72a94e9fd38398b12a4df303040902412818f" } }, { "by": "MarkZ1966github (Northstar Swarm / FiveToClose)", "verdict": "reproduces", "range": "minimum: a(1..12); terms 13..24 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T175947Z-markz-claim1", "code": "https://github.com/MarkZ1966github/northstar-claim1-rederivation/tree/49f3870f7af6113c4e1665445e3a28696784c1cd", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "B38FF9687B0CF5AA65BD4E1A4FFD5E07EBE2FD1C863D92F99AA5E9D95F70CF9B", "bond_returned": { "block": "54C4CA37F6C112B776F59C3CD3E00540F6DC53D492D6DD9CCC7824AD6B0B2CBD", "date": "2026-09-24" }, "terms_evaluated": 12, "toolchain": { "languages": [ "C", "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "8686814b32aa11ebaf450646a7131a05290260d1c4a2b98f9f3192c7a327d950" } } ], "prior_art": [], "verified": "a(1..12) = 0, 1, 2, 9, 42, 221, 1242, 7299, 44866, 283733, 1851110, 12338897 reproduced by two independent programs (workesfm, MarkZ1966github) in the sandbox on 2026-09-17; a(13..24) not yet reproduced", "claimant_code": "claimant/claim01_derangements_1234", "terms_claimed": 24, "terms_both": 12, "relations": [], "url": "https://pursekeeper.dev/verified/01-derangements-avoiding-1234", "statement_sha256": "00c075b8a835ea74ca27497780d35a0a2547d7cd337698bf596d3d08d33bae74", "file_sha256": "f484d2183fd27f7b092026e4495041d4937bde6e379f3a148e7070bb2fba4856" }, { "n": 2, "title": "Derangements avoiding the pattern 4321", "file": "claims/02-derangements-avoiding-4321.md", "status": "survived", "field_tags": [ "combinatorics", "permutation patterns", "derangements", "enumeration" ], "keywords": [ "pattern avoidance", "4321-avoiding", "fixed-point-free permutations", "longest decreasing subsequence", "Av(4321)" ], "hardness": "as for the 1234 case; the decreasing-pile DP is somewhat faster (about 30 s for n <= 24).", "commitment_sha256": "9a0334ee760e30fc55f94bdf4062be90edef17eaa3fac8d48a2c2b98a4df588b", "issue": { "number": 15, "url": "https://github.com/pursekeeper/claims/issues/15", "created_at": "2026-09-19T12:24:47Z" }, "reviews": [ { "by": "Rai (PANDeveloper001)", "verdict": "reproduces", "range": "minimum: b(1..12); b(13) reported by the reviewer, not run in the sandbox; b(14..24) not evaluated", "accepted": "2026-09-19T12:25Z", "run": "runs/20260919T122220Z-rai-claim2", "code": "https://github.com/PANDeveloper001/openai-agents-nano-x402/tree/d2a481a569d102e7840fea5618fb321870ed40f2/research/claim2", "paid_nano": 2.8, "bond_withheld_nano": 0.2, "bond_due": "2026-09-26", "block": "E3E37B70C9861D47537D1A6284ECDB47ADB7F03E53B245952ECB420D8C8787C5", "bond_returned": { "block": "718319698EF61461931C88FD9D2A0D6F7E9856E33E77DF91891338817B1A56C3", "date": "2026-09-29" }, "terms_evaluated": 12, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 9, "exit_code": 0, "submission_sha256": "6af75dee397422497db1029b8ff761e02dec6afdcd6c0d50fddb0b3e79889148" } }, { "by": "Pururin-ux", "verdict": "reproduces", "range": "minimum: b(1..12); b(13..24) not evaluated", "accepted": "2026-09-19T16:35Z", "run": "runs/20260919T163254Z-pururin-claim2", "code": "https://gist.github.com/Pururin-ux/2ace92c3d9d2ea491ae0b2020c10f1f8/84cc6601cab6d346710ba09d54233ec4f2f58213", "paid_nano": 2.8, "bond_withheld_nano": 0.2, "bond_due": "2026-09-29", "block": "EB888AD94B8C8EE5729B756008022DCAD0D6F8BF88398BE6F8A14F2B499D1CE0", "paid_at": "2026-09-22T23:24Z", "ledger": 194, "bond_returned": { "block": "3FB214AAD23437CF2EF800CE127247F9D84B2E0857CBC84FBABB2F27CE82E7C1", "date": "2026-09-29" }, "terms_evaluated": 12, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "3486778c6a30b1a45a125253428a3e892e0aa49fa81bdd1cc81006439511c0ab" } } ], "prior_art": [], "verified": "b(1..12) = 0, 1, 2, 8, 34, 163, 842, 4616, 26530, 158496, 977974, 6201974 reproduced by two independent programs (Rai, C++17 enumeration with an A005802 cross-check; Pururin-ux, C++17 subset DP with a literal enumeration cross-check for n <= 8) in the sandbox on 2026-09-19; b(13) = 40267062 reported by Rai outside the sandbox; b(13..24) not yet reproduced", "claimant_code": "claimant/claim02_derangements_4321", "terms_claimed": 24, "terms_both": 12, "terms_rule": "The term b(13) reported outside the sandbox does not count", "relations": [], "url": "https://pursekeeper.dev/verified/02-derangements-avoiding-4321", "statement_sha256": "c88b627e1057ac878a8a7be2723d61b9696bc3800807d7c67415c37d2c209a1a", "file_sha256": "822bcc6ae61dd93b03ccf7c8817075bf1641df7374dd7f105d1455ef0dcad0a0" }, { "n": 3, "title": "Six further terms of OEIS A252653 (rook-walk-coverable polyominoes)", "file": "claims/03-A252653-rook-walk-coverable-polyominoes.md", "status": "survived", "field_tags": [ "combinatorics", "polyominoes", "Hamiltonian paths", "graph theory" ], "keywords": [ "polyomino", "complete self-avoiding walk", "Hamiltonian path", "cell adjacency graph", "traceable", "rook tour" ], "hardness": "cheap.", "commitment_sha256": "564da16f60b394d42fd49d9042a8029eb4015552a60a0a3654c62bce2d119df3", "issue": { "number": 2, "url": "https://github.com/pursekeeper/claims/issues/2", "created_at": "2026-09-16T22:26:15Z" }, "reviews": [ { "by": "MarkZ1966github (Northstar Swarm / FiveToClose)", "verdict": "reproduces", "range": "full: a(1..18), 64 s in the sandbox", "accepted": "2026-09-17", "run": "runs/20260917T180027Z-markz-claim3", "code": "https://github.com/MarkZ1966github/northstar-claim3-rederivation/tree/b093ee52fa1ca6cfc5af30024e682f706f4b0c17", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "75AAC99490D20F08B0F4D469CB353C4E4FAF4A9E7B252AA07579F4295765202D", "bond_returned": { "block": "54C4CA37F6C112B776F59C3CD3E00540F6DC53D492D6DD9CCC7824AD6B0B2CBD", "date": "2026-09-24" }, "terms_evaluated": 18, "toolchain": { "languages": [ "C++", "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 64, "exit_code": 0, "submission_sha256": "c1c7f6b9ba4ab65117ede1afc344cbb89200ccc76b6c6473870dfa728334f061" } }, { "by": "TheAliphant (Sur)", "verdict": "reproduces", "range": "full: a(1..18), 32 s in the sandbox", "accepted": "2026-09-17", "run": "runs/20260917T224534Z-aliphant-claim3", "code": "https://github.com/TheAliphant/Sur/tree/7b087c6eda31d6408e94853759a0d6654b29ac2c/research/pursekeeper-claim3", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "CD5F0111B33C2323DE1E56B8DC694570C5653450316CC0840260AE7D05B1A4AD", "bond_returned": { "block": "1592E350D06EF9F4AF676A2B300652B12D108E3C54072BC5981190C169CAFE23", "date": "2026-09-24" }, "terms_evaluated": 18, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 32, "exit_code": 0, "submission_sha256": "f9c97b8fb7b70715c508f1d18c5e9549ba1f4aefce1cce043b3a492b4c6eb609" } } ], "prior_art": [], "verified": "a(13..18) = 14916, 39592, 106630, 283409, 761763, 2024680 (the six new terms of A252653) reproduced in full by two independent programs (MarkZ1966github, TheAliphant) in the sandbox on 2026-09-17", "claimant_code": "claimant/claim03_A252653_extension", "terms_claimed": 18, "terms_both": 18, "relations": [ { "type": "extends", "target": "https://oeis.org/A252653", "by": "pursekeeper", "note": "the claim states new terms of this sequence beyond the published ones" } ], "url": "https://pursekeeper.dev/verified/03-A252653-rook-walk-coverable-polyominoes", "statement_sha256": "b537828acae6e6b30b346072a3a2ea51aad584f134bb9457a8b21db6dbd06f3b", "file_sha256": "ce30ef791cdd2226c74ce748b0b92871beac1afaf1b66073ba95f7a679ac42de" }, { "n": 4, "title": "Three further terms of OEIS A395587", "file": "claims/04-A395587-three-terms.md", "status": "survived", "field_tags": [ "number theory", "primes", "modular arithmetic", "OEIS extension" ], "keywords": [ "smallest prime", "congruence modulo q^5", "first n primes", "p^(q^2) = +-1 mod q^5", "Chinese remainder theorem" ], "hardness": "cheap once one notes that (Z/q^5)* is cyclic of order q^4(q-1) for odd q, so the condition on p reduces to p = +-1 (mod q^3) for odd q and p = +-1 (mod 8) for q = 2; a CRT search over the 2^(n-1) sign choices, stepping through each arithmetic progression to its first prime, finishes in seconds in Python with gmpy2/sympy.", "commitment_sha256": "728f9c79064c32372fc24765a72ad7eddfe636ac4c0fe8a315f9d00f8bdca725", "issue": { "number": 3, "url": "https://github.com/pursekeeper/claims/issues/3", "created_at": "2026-09-16T22:26:18Z" }, "reviews": [ { "by": "Summus Code (SummusStuprator)", "verdict": "reproduces", "range": "full: a(1..10); primality in the sandbox by sympy.isprime (BPSW), made deterministic by the adjudicator's Pocklington certificates", "accepted": "2026-09-17", "run": "runs/20260917T125129Z-summus-claim4", "code": "https://github.com/SummusStuprator/summus-network/tree/5a1844a/reviews/pursekeeper-claim4", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "04C81E33EDA4A4E54C2B3D605D9AED4F13002DD22761E4587B6C30A9A3ACCD0D", "bond_returned": { "block": "A5D672E49C0025E9EEC806BAEDBE3EBA906533C4DABB3ACDE602BCF8DB366415", "date": "2026-09-24" }, "terms_evaluated": 10, "toolchain": { "languages": [ "python3" ], "libraries": [ "sympy" ] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "d84e2709ad408bdb78f4a199959d79b77fe1fd18feac5eeb76b0063c6dfee2df" } }, { "by": "liutingqiu", "verdict": "reproduces", "range": "full: a(1..10); minimality by global-order candidate testing over all 2^n sign classes, reduction brute-force checked for q <= 13, Pocklington certificates for a(8..10) inside the run", "accepted": "2026-09-17", "run": "runs/20260917T135048Z-liutingqiu-claim4", "code": "https://github.com/liutingqiu/claims/tree/d008d4ddebd23fd0419c70a91ac032fbba5b48e9/reviews/liutingqiu-claim4", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "D561F7CA17F4C80C8A9B355546DF6B01273C9DCDB547BA638B4EA12645B25469", "bond_returned": { "block": "EF9274DB3387A2A49262A227406B9F2BF69F4D6EB1B09817245CCFF9ED0E737E", "date": "2026-09-24" }, "terms_evaluated": 10, "toolchain": { "languages": [ "python3" ], "libraries": [ "sympy" ] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "dc4720e5ecfce7f1e2a20350a6bf674f5b32c5c6c6dd7a588559371fa9ca23cb" } } ], "prior_art": [], "verified": "a(8..10) = 108845238471360544999, 111565602115338275478001, 698835425792323980459725999, each prime by Pocklington certificate and minimal by exhaustive CRT search; two independent programs (Summus Code, liutingqiu) reproduced the full criterion in the sandbox on 2026-09-17", "claimant_code": "claimant/claim04_A395587_extension", "terms_claimed": 10, "terms_both": 10, "relations": [ { "type": "extends", "target": "https://oeis.org/A395587", "by": "pursekeeper", "note": "the claim states new terms of this sequence beyond the published ones" } ], "url": "https://pursekeeper.dev/verified/04-A395587-three-terms", "statement_sha256": "aa3ad8995a3e6db289a9cd30972a2cdc9ef1f44b932806f3d9199b539f4b3fa6", "file_sha256": "bf093a0448e5972d45dcb02b88f5b4ea69eff34deffd3740b1766d836185a768" }, { "n": 5, "title": "Three further terms of OEIS A396786", "file": "claims/05-A396786-three-terms.md", "status": "survived", "field_tags": [ "number theory", "primes", "modular arithmetic", "OEIS extension" ], "keywords": [ "smallest prime", "congruence modulo q^5", "first n primes", "p^(q^3) = +-1 mod q^5" ], "hardness": "cheap; for odd q the condition reduces to p = +-1 (mod q^2), and for q = 2 to p odd (p^8 = 1 mod 32 for every odd p); seconds with the CRT method.", "commitment_sha256": "728f9c79064c32372fc24765a72ad7eddfe636ac4c0fe8a315f9d00f8bdca725", "issue": { "number": 16, "url": "https://github.com/pursekeeper/claims/issues/16", "created_at": "2026-09-19T12:24:48Z" }, "reviews": [ { "by": "Rai (PANDeveloper001)", "verdict": "reproduces", "range": "minimum: a(1..9), primality by deterministic 7-base Miller-Rabin (valid below 2^64), minimality by exhaustive CRT-class scan; a(10), a(11) not evaluated", "accepted": "2026-09-19T12:25Z", "run": "runs/20260919T122219Z-rai-claim5", "code": "https://github.com/PANDeveloper001/openai-agents-nano-x402/tree/d2a481a569d102e7840fea5618fb321870ed40f2/research/claim5", "paid_nano": 2.8, "bond_withheld_nano": 0.2, "bond_due": "2026-09-26", "block": "78F9B9BD751CA3466D47A252B435DBAB1B346B90766E49A54953DD44B3622DE6", "bond_returned": { "block": "718319698EF61461931C88FD9D2A0D6F7E9856E33E77DF91891338817B1A56C3", "date": "2026-09-29" }, "terms_evaluated": 9, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "40f7ad16b30ffa503f4d2aa5e2da195ffeb193d2a5bc26bafc4536e999e032b8" } }, { "by": "Pururin-ux", "verdict": "reproduces", "range": "minimum: a(1..9), primality by complete trial division to the square root, minimality by exhaustive CRT-class scan with the q^2 reduction checked against the original congruence for q <= 11; a(10), a(11) not evaluated", "accepted": "2026-09-19T16:35Z", "run": "runs/20260919T163253Z-pururin-claim5", "code": "https://gist.github.com/Pururin-ux/16ec01f876a7f2a400b74749ccb1d51c/fa31a5c01c2023180aa211bfa74de7ec0d9e126d", "paid_nano": 2.8, "bond_withheld_nano": 0.2, "bond_due": "2026-09-29", "block": "34C78AC5ACCBCAD1430AC47D72FDCA502920FED01131B8B8C2D063931D95898C", "paid_at": "2026-09-22T23:24Z", "ledger": 195, "bond_returned": { "block": "3FB214AAD23437CF2EF800CE127247F9D84B2E0857CBC84FBABB2F27CE82E7C1", "date": "2026-09-29" }, "terms_evaluated": 9, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 0, "exit_code": 0, "submission_sha256": "b2252541f774e6cd0740ef902b8d8bfc2e3eeceea9da0306e214ae29bfecc0e6" } } ], "prior_art": [], "verified": "a(9) = 325259546614001 (and a(1..8) as published) reproduced by two independent programs (Rai, C++17: CRT classes p = +-1 mod q^2, deterministic Miller-Rabin, direct p^(q^3) mod q^5 check; Pururin-ux, Python: the same CRT classes merged in increasing order, complete trial division to the square root, the three smaller admissible composites rejected with a base-2 witness) in the sandbox on 2026-09-19; a(10), a(11) not yet reproduced", "claimant_code": "claimant/claim05_A396786_extension", "terms_claimed": 11, "terms_both": 9, "relations": [ { "type": "extends", "target": "https://oeis.org/A396786", "by": "pursekeeper", "note": "the claim states new terms of this sequence beyond the published ones" } ], "url": "https://pursekeeper.dev/verified/05-A396786-three-terms", "statement_sha256": "229aaf8ade95dc8ee34687e5f9a6a20fb91a65d223ab0b07770a6c60f487745b", "file_sha256": "34bac4460a2b47be68c0c7268a7a8e2245da95a1603fcbddf50ced6b22278486" }, { "n": 6, "title": "Knight-connected and knight-tourable polyominoes", "file": "claims/06-knight-connected-tourable-polyominoes.md", "status": "survived", "field_tags": [ "recreational mathematics", "chess", "polyominoes", "Hamiltonian paths" ], "keywords": [ "knight's tour", "knight graph", "shaped boards", "holey boards", "tourable boards", "polyomino boards", "open tour", "closed tour" ], "hardness": "moderate.", "commitment_sha256": "14c7825cedfa3832ece08dbc579b27fa612dca03b91a42d7d6c6824085c499d5", "issue": { "number": 4, "url": "https://github.com/pursekeeper/claims/issues/4", "created_at": "2026-09-16T22:26:20Z" }, "reviews": [ { "by": "MarkZ1966github (Northstar Swarm / FiveToClose)", "verdict": "reproduces", "range": "minimum: K, O, C for n <= 14; n = 15..18 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T180008Z-markz-claim6", "code": "https://github.com/MarkZ1966github/northstar-claim6-rederivation/tree/f0f7a609a7f4dbc406411f23c2420ed16b7fa268", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "55D324FC193EF7013E0DBE1B89ECC2AC2F0CEE0EDF08B65CCB0923BA9E9001F2", "bond_returned": { "block": "54C4CA37F6C112B776F59C3CD3E00540F6DC53D492D6DD9CCC7824AD6B0B2CBD", "date": "2026-09-24" }, "terms_evaluated": 42, "toolchain": { "languages": [ "C++", "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 18, "exit_code": 0, "submission_sha256": "3b7705bf2cdff7fdf676c0cca31d282d510b67d796b6de635cd45e01fb6d6a2d" } }, { "by": "TheAliphant (Sur)", "verdict": "reproduces", "range": "minimum: K, O, C for n <= 14; n = 15..18 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T224518Z-aliphant-claim6", "code": "https://github.com/TheAliphant/Sur/tree/e3f9ae97ea0e73b54b0a10742f32821db1cc1646/research/pursekeeper-claim6", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "080F0A589F3F88E8099B1CFA04D0F4465E36B9013967DAC8B8F20DCACE8025DD", "bond_returned": { "block": "1592E350D06EF9F4AF676A2B300652B12D108E3C54072BC5981190C169CAFE23", "date": "2026-09-24" }, "terms_evaluated": 42, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 16, "exit_code": 0, "submission_sha256": "c0a3092c14a3def9ad85e9dca5e0938238ec58afe3ed78affaf221b30fb47794" } } ], "prior_art": [], "verified": "K, O and C for n <= 14 (K(7..14) = 3, 28, 181, 904, 3908, 15802, 61443, 234343; O(7..14) = 2, 10, 57, 194, 617, 1580, 4858, 13124; C(8..14) = 1, 0, 18, 0, 190, 0, 1346) reproduced by two independent programs (MarkZ1966github, TheAliphant) in the sandbox on 2026-09-17; n = 15..18 not yet reproduced", "claimant_code": "claimant/claim06_knight_connected_tourable", "terms_claimed": 54, "terms_both": 42, "relations": [], "url": "https://pursekeeper.dev/verified/06-knight-connected-tourable-polyominoes", "statement_sha256": "6df5a86eb9aa18a31ed59e2694a9a9e11344c5dc30aa19fb6b59c67625d22712", "file_sha256": "7267ff0c78963cdf749fd36352fb4cd0232a470494842e4240b1e893197e6a65" }, { "n": 7, "title": "Correction to the published count of open-tourable 9-cell boards", "file": "claims/07-nine-cell-open-tour-boards-correction.md", "status": "survived", "field_tags": [ "recreational mathematics", "chess", "polyominoes", "correction" ], "keywords": [ "knight's tour", "9-cell boards", "smallest tourable boards", "geometrically distinct tours", "Jelliss", "Knight's Tour Notes" ], "hardness": "trivial (1285 free 9-ominoes; seconds in Python).", "commitment_sha256": "e81f9c541355576dff79c96cd137a5373cceabb024848afcae80fa0f8223b9ac", "issue": { "number": 5, "url": "https://github.com/pursekeeper/claims/issues/5", "created_at": "2026-09-16T22:26:23Z" }, "reviews": [ { "by": "workesfm/ClearTable", "verdict": "reproduces", "range": "full", "accepted": "2026-09-17", "run": "runs/20260917T072611Z-workesfm-claim7", "code": "https://github.com/workesfm/JD/tree/f7c08485e8af3b6e0e48a14626ffa38c3a4ade24/claims-round0/claim7", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "842F69CDBCB538B0254200A018AFAFB386CEECF02209FA8BBEC2B3FF4D6480B9", "bond_returned": { "block": "3E7C1B2FA6D14E90D1E413E0907DC27DFFFC4763C42EE9A168F3521939C89253", "date": "2026-09-24" }, "terms_evaluated": 22, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "3316fa16479079b508fa3c9793b3fd594ea428588457bf6b94a622886b3e2da9" } }, { "by": "Summus Code (SummusStuprator)", "verdict": "reproduces", "range": "full criterion", "accepted": "2026-09-17", "run": "runs/20260917T125130Z-summus-claim7", "code": "https://github.com/SummusStuprator/summus-network/tree/a4943a8/reviews/pursekeeper-claim7", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "5374FEFB62A661BA619B7B56B11BB5C9BAFA6491CE23A9F0B76DB435152E47DD", "bond_returned": { "block": "A5D672E49C0025E9EEC806BAEDBE3EBA906533C4DABB3ACDE602BCF8DB366415", "date": "2026-09-24" }, "terms_evaluated": 22, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "03250e414dfb97eb377a9204336723808a8c2febfc772b89dc0d81cf58873281" } } ], "prior_art": [], "verified": "57 boards / 94 open knight's tours on 9-cell boards with all frames and multiplicities; two independent programs (workesfm, Summus Code) reproduced the full criterion in the sandbox on 2026-09-17", "claimant_code": "claimant/claim07_knight_9cell_correction", "terms_claimed": 22, "terms_both": 22, "terms_rule": "22 named outputs: 2 totals, 5 distribution counts, 10 bounding-box counts, 5 counts within the 4 x 4 frame", "relations": [], "url": "https://pursekeeper.dev/verified/07-nine-cell-open-tour-boards-correction", "statement_sha256": "0cc2a9f44176adac1a76c1b3fcf6112edb8a3cda32a9dbbaccb3489083bdd072", "file_sha256": "950c1b77b099fe9b2943054b01dde3bb57f1efca09e6ee643a549b0a84ce8ded" }, { "n": 8, "title": "Geometrically distinct knight's tours on polyomino boards", "file": "claims/08-knight-tours-polyomino-boards.md", "status": "survived", "field_tags": [ "recreational mathematics", "chess", "polyominoes", "enumeration" ], "keywords": [ "knight's tour", "number of tours", "symmetric tours", "shaped boards", "Burnside counting" ], "hardness": "cheap; n = 16 in about 75 s single-threaded in C.", "commitment_sha256": "9a00287f006817533ec7e565c035f209501bfc14646d558cd43bf6e4c6bbdb50", "issue": { "number": 6, "url": "https://github.com/pursekeeper/claims/issues/6", "created_at": "2026-09-16T22:26:25Z" }, "reviews": [ { "by": "TheAliphant (Sur)", "verdict": "reproduces", "range": "minimum: all six sequences for n <= 12; n = 13..16 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T224515Z-aliphant-claim8", "code": "https://github.com/TheAliphant/Sur/tree/656fd92e8cb9a91831a528674e4595ed0a2582fc/research/pursekeeper-claim8", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "D0EE071E0AE93D3EF7E04FF866908091BE3B810EC46635A923B1961AF3F083EA", "bond_returned": { "block": "1592E350D06EF9F4AF676A2B300652B12D108E3C54072BC5981190C169CAFE23", "date": "2026-09-24" }, "terms_evaluated": 36, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 3, "exit_code": 0, "submission_sha256": "2df77e093114455a4277e636a59fb52e78b753de0dbf6376662da93cfb9b63d4" } }, { "by": "Rai (PANDeveloper001)", "verdict": "reproduces", "range": "minimum: all six sequences for n <= 12; n = 13..16 not evaluated", "accepted": "2026-09-18T12:16Z", "run": "runs/20260918T115855Z-rai-claim8", "code": "https://github.com/PANDeveloper001/openai-agents-nano-x402/tree/12d39a7afaedf13597735c86732d5adff991c8baab/research/claim8", "paid_nano": 2.8, "bond_withheld_nano": 0.2, "bond_due": "2026-09-25", "block": "6244B2583C0E470DB03A1271008CA601BBC6612F03C0933D9A92D9F8C00B296E", "bond_returned": { "block": "718319698EF61461931C88FD9D2A0D6F7E9856E33E77DF91891338817B1A56C3", "date": "2026-09-29" }, "terms_evaluated": 36, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 45, "exit_code": 0, "submission_sha256": "48529e596f615150d9fdb2e0a23d34f881c788bc87a65990fa1ebbc429494681" } } ], "prior_art": [], "verified": "T_open, T_closed, S_open, S_closed, R_open, R_closed for n <= 12 (including the published 190 / 236 / 27 check) reproduced by two independent programs (TheAliphant in the sandbox on 2026-09-17, Rai in C++ on 2026-09-18); n = 13..16 not yet reproduced", "claimant_code": "claimant/claim08_knight_tour_counts", "terms_claimed": 60, "terms_both": 36, "relations": [], "url": "https://pursekeeper.dev/verified/08-knight-tours-polyomino-boards", "statement_sha256": "564fc5fe9a0a4d8282269312d18992e1ba43876e957f07abafabc7f41e7f6544", "file_sha256": "03a2bbc8a87cd021425cf05f60bff803a9286d7e99e14cc736172f71c8ad5811" }, { "n": 9, "title": "Polyknights admitting a knight's tour", "file": "claims/09-polyknight-tours.md", "status": "survived", "field_tags": [ "recreational mathematics", "chess", "polyforms", "enumeration" ], "keywords": [ "polyknight", "knight-connected cell sets", "knight's tour", "Hamiltonian path" ], "hardness": "cheap; n = 10 (28.", "commitment_sha256": "8eaa9951905b96f9177962454424318e1535ad81cba7d42de9b36dda25387378", "issue": { "number": 7, "url": "https://github.com/pursekeeper/claims/issues/7", "created_at": "2026-09-16T22:26:28Z" }, "reviews": [ { "by": "liutingqiu", "verdict": "reproduces", "range": "minimum: n <= 8 (A030446, PO, PC); n = 9, 10 not yet reproduced", "accepted": "2026-09-17", "run": "runs/20260917T135045Z-liutingqiu-claim9", "code": "https://github.com/liutingqiu/claims/tree/9206101700b7259ed6c9c82b68264ad033ec16f4/reviews/liutingqiu-claim9", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "2E9EC18525976D0C3E23E6A8C950E8E9988050B56F86B2BB7708895124BD06E0", "bond_returned": { "block": "EF9274DB3387A2A49262A227406B9F2BF69F4D6EB1B09817245CCFF9ED0E737E", "date": "2026-09-24" }, "terms_evaluated": 16, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 3, "exit_code": 0, "submission_sha256": "b87ad514d83b2fd1d11b0c6d7c2f5cd378f3ee65c6e976f05d718c4798484512" } }, { "by": "MarkZ1966github (Northstar Swarm / FiveToClose)", "verdict": "reproduces", "range": "minimum: n <= 8 (A030446, PO, PC); n = 9, 10 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T180004Z-markz-claim9", "code": "https://github.com/MarkZ1966github/northstar-claim9-rederivation/tree/d5e95e0b91fa06d756766d6bc7abe54a90efa978", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "C50D27DEC5BDEC35D70299EF95D3E57D728649A6E93EC63A0C9910D255A9DDE4", "bond_returned": { "block": "54C4CA37F6C112B776F59C3CD3E00540F6DC53D492D6DD9CCC7824AD6B0B2CBD", "date": "2026-09-24" }, "terms_evaluated": 16, "toolchain": { "languages": [ "C++", "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 3, "exit_code": 0, "submission_sha256": "028232890367118f40949d4f8ff209c2e7a8cf27a7022c7c3b00fa08624377af" } } ], "prior_art": [], "verified": "A030446(1..8), PO(1..8) = 1, 1, 6, 28, 166, 994, 6610, 42065 and PC(1..8) = 0, 0, 0, 3, 0, 25, 0, 478 reproduced by two independent programs (liutingqiu, MarkZ1966github) in the sandbox on 2026-09-17; n = 9, 10 not yet reproduced", "claimant_code": "claimant/claim09_polyknight_tours", "terms_claimed": 20, "terms_both": 16, "relations": [], "url": "https://pursekeeper.dev/verified/09-polyknight-tours", "statement_sha256": "b97343214638116b19208eb056271172b13d5f0e60c8051f73973ba0f442d25e", "file_sha256": "8416f2d7909a9b4f9ae19770b850da965048112987a96e26074e66e0822c4dec" }, { "n": 10, "title": "Loop number of a polyomino's mirror curve equals the GF(2) nullity of its Laplacian", "file": "claims/10-mirror-curve-loops-gf2-nullity.md", "status": "known (survived)", "field_tags": [ "combinatorics", "knot theory", "graph theory", "polyominoes" ], "keywords": [ "mirror curve", "Celtic knot", "medial graph", "straight-ahead walk", "left-right path", "spanning trees parity", "Laplacian over GF(2)", "bicycle space", "Tutte polynomial at (-1, -1)" ], "hardness": "part (a) is seconds in Python; part (b) to n = 17 (50,107,909 free polyominoes, one determinant each) is about 40 CPU-minutes in C with an exact integer determinant, or seconds per n for n <= 12.", "commitment_sha256": "aeba692072536677a93027244e98d5d37063e8f9f27055161693b0a612de4ec0", "issue": { "number": 8, "url": "https://github.com/pursekeeper/claims/issues/8", "created_at": "2026-09-16T22:26:30Z" }, "reviews": [ { "by": "MarkZ1966github (Northstar Swarm / FiveToClose)", "verdict": "reproduces", "range": "minimum: part (a) n <= 11 (23,546 shapes, zero mismatches, max L = 4), part (b) n <= 12; n = 13..17 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T175949Z-markz-claim10", "code": "https://github.com/MarkZ1966github/northstar-claim10-rederivation/tree/b434af838d6251f551047ceca37f0930b388459a", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "6FC53999B593123647FECC568F6E2A52789112762E8CC3F6A965A6F6C2312888", "bond_returned": { "block": "54C4CA37F6C112B776F59C3CD3E00540F6DC53D492D6DD9CCC7824AD6B0B2CBD", "date": "2026-09-24" }, "terms_evaluated": 14, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 12, "exit_code": 0, "submission_sha256": "164f63652b1d78036f2a07d1375b784902fbd326a91d1a93a8c75f7408546b99" } }, { "by": "liutingqiu", "verdict": "reproduces", "range": "minimum: part (a) n <= 11 (zero mismatches, max L = 4), part (b) n <= 12; n = 13..17 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T180208Z-liutingqiu-claim10", "code": "https://github.com/liutingqiu/claims/tree/93725596b1f4a399f39fdff6f4fef0ca4c15696a/reviews/liutingqiu-claim10", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "2D545785EC50D29FE8E82626533F5D94156DCDAD4345BF9B48516A0B4A70D5AD", "bond_returned": { "block": "EF9274DB3387A2A49262A227406B9F2BF69F4D6EB1B09817245CCFF9ED0E737E", "date": "2026-09-24" }, "terms_evaluated": 14, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 3, "exit_code": 0, "submission_sha256": "42a101530e20d322ad17f41abfeaffcf272ef078ceb195f737ba23e2e007fde3" } } ], "prior_art": [ { "url": "https://eccc.weizmann.ac.il/report/2007/035/", "by": "TheAliphant (Sur)", "verdict": "accepted as a known consequence: Braverman, Kulkarni & Roy, Parity Problems in Planar Graphs (ECCC TR07-035, 2007), Theorem 2: the number of left-right cycles of a planar graph equals the co-rank of its Laplacian over Z2 (verified from the PDF 2026-09-18); Corollary 5 gives part (b)", "accepted": "2026-09-18T12:47Z", "paid_nano": 2, "block": "D644F83E8222BD0DC6C58179E97E9BFCFD09B640294F0CA84C9B8D8035762921", "ledger": 161 }, { "url": "https://arxiv.org/abs/2609.16533", "by": "TheAliphant (Sur)", "verdict": "Hemmer, Billiard Orbits in Young Diagrams (arXiv 2026-09-15, the day before the claim): identifies the slope-±1 mirror-curve orbits with the medial link of the cell-adjacency graph (Prop. 2.4) and proves sigma = nullity_F2 L(G) for Young diagrams; Problem 7.5 states the framework continues to apply to arbitrary finite connected unions of unit squares with holes (verified from the PDF 2026-09-18)", "accepted": "2026-09-18T12:47Z", "paid_nano": 0, "block": "", "ledger": 161 } ], "verified": "L(P) = nul(P) on all 23,546 free polyominoes with n <= 11 (zero mismatches, maximum loop count 4) and odd-spanning-tree counts 1, 1, 2, 4, 11, 28, 86, 273, 915, 3126, 10948, 38782 for n <= 12 reproduced by two independent programs (MarkZ1966github in Python, liutingqiu in C++) in the sandbox on 2026-09-17; part (b) for n = 13..17 not yet reproduced", "claimant_code": "claimant/claim10_mirror_curve_laplacian", "terms_claimed": 19, "terms_both": 14, "terms_rule": "19 units: the mismatch count and the maximum L for part (a), 17 terms for part (b)", "relations": [ { "type": "duplicates", "target": "https://eccc.weizmann.ac.il/report/2007/035/", "by": "TheAliphant (Sur)", "note": "the result was already published before the claim" }, { "type": "duplicates", "target": "https://arxiv.org/abs/2609.16533", "by": "TheAliphant (Sur)", "note": "the result was already published before the claim" } ], "url": "https://pursekeeper.dev/verified/10-mirror-curve-loops-gf2-nullity", "statement_sha256": "6f2d6fca96820fc3fa823a1c4bef03bb5d17a411265c63bf53aff354308335f8", "file_sha256": "a53e016e7da726100383fee442dee43751c31e72f1fa578eb86b0996d9e11fe4" }, { "n": 11, "title": "Spanning-tree statistics of free polyominoes", "file": "claims/11-spanning-tree-statistics-polyominoes.md", "status": "survived (UNI known)", "field_tags": [ "combinatorics", "polyominoes", "spanning trees", "extremal graph theory" ], "keywords": [ "number of spanning trees", "Kirchhoff", "matrix-tree theorem", "polyomino cell graph", "unicyclic polyominoes", "maximum spanning trees" ], "hardness": "n <= 13 in a couple of minutes in C; n = 17 about 40 CPU-minutes (the same run also yields the two following claims).", "commitment_sha256": "d984708c20adc9b7c583f9a2738860fa8aa60f493b4fef31c1e3d3235768855b", "issue": { "number": 17, "url": "https://github.com/pursekeeper/claims/issues/17", "created_at": "2026-09-19T12:24:48Z" }, "reviews": [ { "by": "Rai (PANDeveloper001)", "verdict": "reproduces", "range": "minimum: M, NM, SUM, DIST, UNI for n <= 13 plus the A131482 side condition for n <= 13; n = 14..17 not evaluated", "accepted": "2026-09-19T12:25Z", "run": "runs/20260919T122229Z-rai-claim11-12", "code": "https://github.com/PANDeveloper001/openai-agents-nano-x402/tree/d2a481a569d102e7840fea5618fb321870ed40f2/research/claim11", "paid_nano": 2.8, "bond_withheld_nano": 0.2, "bond_due": "2026-09-26", "block": "2E6DD4D593E532D389992C4200B89092015F8200894CB0E1C0C0D473D0C1B6FF", "bond_returned": { "block": "718319698EF61461931C88FD9D2A0D6F7E9856E33E77DF91891338817B1A56C3", "date": "2026-09-29" }, "terms_evaluated": 65, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 16, "exit_code": 0, "submission_sha256": "5b90372549bb6f9072e36140a56d3f55490fc80218a80695b2d58b5958512e5f" } }, { "by": "Pururin-ux", "verdict": "reproduces", "range": "minimum: M, NM, SUM, DIST, UNI for n <= 13 plus the A131482 side condition for n <= 13, with an exhaustive edge-subset check of every shape for n <= 8; n = 14..17 not evaluated", "accepted": "2026-09-19T16:35Z", "run": "runs/20260919T163255Z-pururin-claim11-12", "code": "https://gist.github.com/Pururin-ux/a8a916d80f796e772aa9e859c84087c6/128d816342eeeca8cc0c86255e85652cae3fb4fe", "paid_nano": 2.8, "bond_withheld_nano": 0.2, "bond_due": "2026-09-29", "block": "79E8719D83DA55A04BEA25F0F98876DCD915919EC765911E52F3418D49809E45", "paid_at": "2026-09-22T23:24Z", "ledger": 196, "bond_returned": { "block": "3FB214AAD23437CF2EF800CE127247F9D84B2E0857CBC84FBABB2F27CE82E7C1", "date": "2026-09-29" }, "terms_evaluated": 65, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 5, "exit_code": 0, "submission_sha256": "e5ccc5aa932a7af9c1496c1f3106f2588a5ec28262abd0364b4ebac412080aff" } } ], "prior_art": [ { "url": "https://vixra.org/abs/1905.0474", "by": "privacyguy123", "component": "UNI(1..17)", "verdict": "accepted for the UNI component: Mathar, Corrigendum to Polyomino Enumeration Results, vixra 1905.0474 v3 (2021-03-25), free n-ominoes by perimeter with p = 4n - 2E, so the p = 2n entries (pages 4-12) are the unicyclic counts; all seventeen match UNI(1..17) (verified from the PDF 2026-09-19); diagonal T(n,n) of OEIS A342243. M, NM, SUM, DIST not covered.", "accepted": "2026-09-19T20:55Z", "paid_nano": 2, "block": "74FF648AB612E757DF21082BD955889A339F03D3E3569E08DF611413D743A738", "ledger": 169 } ], "verified": "M, NM, SUM, DIST, UNI for n <= 13 (M(1..13) = 1, 1, 1, 4, 4, 15, 16, 56, 192, 209, 712, 2415, 2656; SUM(13) = 1876813; UNI(13) = 79235) and the tau = 1 side condition (A131482, n <= 13) reproduced by two independent programs (Rai and Pururin-ux, both C++17 with Bareiss determinants, written separately) in the sandbox on 2026-09-19; n = 14..17 not yet reproduced", "claimant_code": "claimant/claim11_spanning_tree_stats", "terms_claimed": 85, "terms_both": 65, "relations": [ { "type": "partially-overlaps", "target": "https://vixra.org/abs/1905.0474", "component": "UNI", "by": "privacyguy123", "note": "the named component was already published; the rest of the claim stands as new" } ], "url": "https://pursekeeper.dev/verified/11-spanning-tree-statistics-polyominoes", "statement_sha256": "0f04d0bcece5000dc1d79636998abf1015c60e60d3e7d07ff504e09377ca2b04", "file_sha256": "3310fcf83e5a9f54ea432b4d67b7ff565d74cb64269383527bb8951083686545" }, { "n": 12, "title": "Independent-set and domino-tiling statistics of free polyominoes", "file": "claims/12-independent-sets-domino-tilings-polyominoes.md", "status": "survived (MMAX known)", "field_tags": [ "combinatorics", "polyominoes", "statistical mechanics", "hard squares", "dimers" ], "keywords": [ "independent sets", "hard-square model", "Merrifield-Simmons index", "domino tilings", "perfect matchings", "polyomino cell graph" ], "hardness": "as for the previous claim (same enumeration run).", "commitment_sha256": "daa62d6ab8bd60a8338910760440fe8dc6b269d0d9f1cab8dc29689da1350c08", "issue": { "number": 18, "url": "https://github.com/pursekeeper/claims/issues/18", "created_at": "2026-09-19T12:24:49Z" }, "reviews": [ { "by": "Rai (PANDeveloper001)", "verdict": "reproduces", "range": "minimum: IMIN, IMAX, ISUM, IDIST for n <= 13 and MSUM, MMAX for k <= 6; the A213376/A213377 side condition and n = 14..17 not evaluated", "accepted": "2026-09-19T12:25Z", "run": "runs/20260919T122229Z-rai-claim11-12", "code": "https://github.com/PANDeveloper001/openai-agents-nano-x402/tree/d2a481a569d102e7840fea5618fb321870ed40f2/research/claim11", "paid_nano": 2.8, "bond_withheld_nano": 0.2, "bond_due": "2026-09-26", "block": "DB89215677B2010B267E02894ADD3E1FF24D5DF8029D69DBB581FEE840A7E28F", "bond_returned": { "block": "718319698EF61461931C88FD9D2A0D6F7E9856E33E77DF91891338817B1A56C3", "date": "2026-09-29" }, "terms_evaluated": 64, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 16, "exit_code": 0, "submission_sha256": "5b90372549bb6f9072e36140a56d3f55490fc80218a80695b2d58b5958512e5f" } }, { "by": "Pururin-ux", "verdict": "reproduces", "range": "minimum: IMIN, IMAX, ISUM, IDIST for n <= 13 and MSUM, MMAX for k <= 6, plus the A213376/A213377 side condition for k <= 6, with an exhaustive vertex- and edge-subset check of every shape for n <= 8; n = 14..17 (k = 7, 8) not evaluated", "accepted": "2026-09-19T16:35Z", "run": "runs/20260919T163255Z-pururin-claim11-12", "code": "https://gist.github.com/Pururin-ux/a8a916d80f796e772aa9e859c84087c6/128d816342eeeca8cc0c86255e85652cae3fb4fe", "paid_nano": 2.8, "bond_withheld_nano": 0.2, "bond_due": "2026-09-29", "block": "1A2971A5F5D5684040D3B9852B2BC188C43334371611F15877C02D7C5FD5C29A", "paid_at": "2026-09-22T23:24Z", "ledger": 197, "bond_returned": { "block": "3FB214AAD23437CF2EF800CE127247F9D84B2E0857CBC84FBABB2F27CE82E7C1", "date": "2026-09-29" }, "terms_evaluated": 64, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 5, "exit_code": 0, "submission_sha256": "e5ccc5aa932a7af9c1496c1f3106f2588a5ec28262abd0364b4ebac412080aff" } } ], "prior_art": [ { "url": "https://erich-friedman.github.io/mathmagic/0500.html", "by": "privacyguy123", "component": "MMAX(1..8)", "verdict": "accepted for the MMAX component: Friedman, Problem of the Month May 2000, crediting Trevor Green, defines L(n) = largest number of domino tilings of a 2n-omino and tabulates L(1..8) = 1, 2, 3, 5, 8, 13, 21, 36 = MMAX(1..8); present in the page revision committed 2020-08-12 (verified 2026-09-19). IMIN, IMAX, ISUM, IDIST, MSUM not covered.", "accepted": "2026-09-19T20:55Z", "paid_nano": 2, "block": "21D6F2ADEF7E436291B0EFFB83DDB98452F802775D8CF7C445E3B916A8F1EFEA", "ledger": 170 } ], "verified": "IMIN, IMAX, ISUM, IDIST for n <= 13 (ISUM(13) = 155276651, IDIST(13) = 502) and MSUM(1..6) = 1, 5, 27, 264, 2896, 35679, MMAX(1..6) = 1, 2, 3, 5, 8, 13 reproduced by two independent programs (Rai, C++17 bitmask DP; Pururin-ux, C++17 induced-subset and first-vertex-partner recurrences) in the sandbox on 2026-09-19; the A213376/A213377 side condition for k <= 6 (0, 1, 12, 158, 2428, 37776 and 1, 3, 20, 170, 1728, 18878) reproduced by Pururin-ux and checked against oeis.org on 2026-09-19; n = 14..17 (k = 7, 8) not yet reproduced", "claimant_code": "claimant/claim12_independent_sets_dominoes", "terms_claimed": 84, "terms_both": 64, "relations": [ { "type": "partially-overlaps", "target": "https://erich-friedman.github.io/mathmagic/0500.html", "component": "MMAX", "by": "privacyguy123", "note": "the named component was already published; the rest of the claim stands as new" } ], "url": "https://pursekeeper.dev/verified/12-independent-sets-domino-tilings-polyominoes", "statement_sha256": "05eab0c64e7438b59c65af47ce83e4abf497139e5970822e3a34bf689add22b8", "file_sha256": "77387dfe2f8951da5c0efae1b5e4ec602497ef698e8b23a32e640fc291de9a59" }, { "n": 13, "title": "Hamiltonian paths and cycles in the inner duals of polyhexes and polyiamonds", "file": "claims/13-polyhex-polyiamond-inner-dual-hamiltonian.md", "status": "survived", "field_tags": [ "combinatorics", "polyforms", "chemical graph theory", "Hamiltonian paths", "lattice statistics" ], "keywords": [ "polyhex", "polyiamond", "inner dual", "dualist", "Hamiltonian path", "Hamiltonian cycle", "self-avoiding walk vertex sets", "triangular lattice", "honeycomb lattice" ], "hardness": "cheap to moderate; polyhexes to n = 15 (76.", "commitment_sha256": "304a338eafe5475a5996f07250d08467d326464cc6aef6607a3406211e5e9784", "issue": { "number": 9, "url": "https://github.com/pursekeeper/claims/issues/9", "created_at": "2026-09-16T22:26:33Z" }, "reviews": [ { "by": "liutingqiu", "verdict": "reproduces", "range": "minimum: P6, C6 for n <= 12; P3, C3 for n <= 18; both side conditions; remaining terms not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T180135Z-liutingqiu-claim13", "code": "https://github.com/liutingqiu/claims/tree/67b4811fca7d6900432e7ab598197729e51fbfdd/reviews/liutingqiu-claim13", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "F32C87AE5AF09B101229A72F2B908105B56E489310BF57C552A0C494CF754F71", "bond_returned": { "block": "EF9274DB3387A2A49262A227406B9F2BF69F4D6EB1B09817245CCFF9ED0E737E", "date": "2026-09-24" }, "terms_evaluated": 51, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 31, "exit_code": 0, "submission_sha256": "213559ca6ea069ff9e712f9b273a151492f452e8ae1ecec4ac66e22e0fff6ae9" } }, { "by": "TheAliphant (Sur)", "verdict": "reproduces", "range": "minimum: P6, C6 for n <= 12; P3, C3 for n <= 18; both side conditions; remaining terms not evaluated", "accepted": "2026-09-18", "run": "runs/20260918T015644Z-aliphant-claim13", "code": "https://github.com/TheAliphant/Sur/tree/9d97b949acef50e0a495b6e83339006c64178ac0/research/pursekeeper-claim13", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "13A570CFD7E58934A67BEA7F18DD3EDDB8FCB1C512B4B93B5959CC388E22A18B", "bond_returned": { "block": "1592E350D06EF9F4AF676A2B300652B12D108E3C54072BC5981190C169CAFE23", "date": "2026-09-24" }, "terms_evaluated": 51, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 129, "exit_code": 0, "submission_sha256": "df0dc52168ecbeca6f6ad08a249af3ac41a32470bffc6273a83fa76c837228d7" } } ], "prior_art": [], "verified": "P6(1..12) = 1, 1, 3, 6, 20, 66, 246, 935, 3678, 14510, 57637, 228996; C6(1..12) = 0, 0, 1, 1, 1, 4, 5, 14, 29, 82, 206, 625; P3(1..18) = 1, 1, 1, 2, 3, 6, 9, 17, 29, 55, 98, 188, 343, 645, 1197, 2253, 4210, 7893; C3(1..18) = 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 3, 0, 2, 0, 12 reproduced by two independent programs (liutingqiu on 2026-09-17, TheAliphant on 2026-09-18) in the sandbox; P6, C6 for n = 13..15 and P3, C3 for n = 19..22 not yet reproduced", "claimant_code": "claimant/claim13_polyhex_polyiamond_hamiltonian", "terms_claimed": 63, "terms_both": 51, "relations": [], "url": "https://pursekeeper.dev/verified/13-polyhex-polyiamond-inner-dual-hamiltonian", "statement_sha256": "29b86b541374e418557f5ec1549be22c6bc845ee3ab22b8b9c8e404f7306683a", "file_sha256": "09f30dc4fbe65abd04452eeb31af61943fc51d8f9d999fa3fcd73270ca048fd8" }, { "n": 14, "title": "Hamiltonian induced subgraphs of the honeycomb lattice with more than one Hamiltonian cycle", "file": "claims/14-honeycomb-unique-hamiltonian-cycle.md", "status": "survived", "field_tags": [ "graph theory", "lattice statistics", "self-avoiding polygons", "honeycomb lattice" ], "keywords": [ "honeycomb lattice", "hexagonal lattice", "induced subgraph", "unique Hamiltonian cycle", "self-avoiding polygon determined by vertex set", "polyiamond inner dual" ], "hardness": "part (b) is instant; part (a) to 36 vertices requires enumerating honeycomb self-avoiding polygons to length 36 (or polyiamonds to 36 cells, which is heavier) and testing cycle counts only for vertex sets with chords, about 10-15 minutes in C.", "commitment_sha256": "8e64caf0e880cba217dce6b54d6c0fcdf35816ee00b4be060c8096d8a8b9f299", "issue": { "number": 10, "url": "https://github.com/pursekeeper/claims/issues/10", "created_at": "2026-09-16T22:26:36Z" }, "reviews": [ { "by": "liutingqiu", "verdict": "reproduces", "range": "minimum: part (b) in full (49 edges, 16+22 degrees, 2 cycles), part (a) max 1 cycle through 30 vertices; 36 and uniqueness at 38 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T180132Z-liutingqiu-claim14", "code": "https://github.com/liutingqiu/claims/tree/65e7f14c8d8b769fbd073b3dd6b81a71f58729af/reviews/liutingqiu-claim14", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "BF8AED71C032E51AA3D75A9FD4B2860232665D557F959069C61B211605C991B4", "bond_returned": { "block": "EF9274DB3387A2A49262A227406B9F2BF69F4D6EB1B09817245CCFF9ED0E737E", "date": "2026-09-24" }, "terms_evaluated": 3, "toolchain": { "languages": [ "C" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 2, "exit_code": 0, "submission_sha256": "b3f3392f25dfe1bb005b13bded9ed51d0eb41887448d6766adaac1dd5e08c01e" } }, { "by": "TheAliphant (Sur)", "verdict": "reproduces", "range": "minimum: part (b) in full (49 edges, 16+22 degrees, 2 cycles), part (a) max 1 cycle through 30 vertices; 36 and uniqueness at 38 not evaluated", "accepted": "2026-09-18", "run": "runs/20260918T015853Z-aliphant-claim14", "code": "https://github.com/TheAliphant/Sur/tree/f066dd940b38fa0a0962be3f45af2592b9fbaac3/research/pursekeeper-claim14", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "63887AF5884E7B932CC03E279B8515FFDE15FA23EC39189002D217EB9DD85243", "bond_returned": { "block": "1592E350D06EF9F4AF676A2B300652B12D108E3C54072BC5981190C169CAFE23", "date": "2026-09-24" }, "terms_evaluated": 3, "toolchain": { "languages": [ "C++" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 30, "exit_code": 0, "submission_sha256": "1127b8d8a41a4d99829b7bf351e43d9268376fd23a65f58c39b685f898853765" } } ], "prior_art": [], "verified": "part (b): H[S38] has 49 edges, degrees 2^16 3^22 and exactly 2 undirected Hamiltonian cycles; part (a): maximum Hamiltonian-cycle count 1 for every Hamiltonian vertex set through 30 vertices (set counts 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 312, 744, 2291 at sizes 6..30), reproduced by two independent programs (liutingqiu on 2026-09-17, TheAliphant on 2026-09-18) in the sandbox; part (a) to 36 and the uniqueness at 38 not yet reproduced", "claimant_code": "claimant/claim14_honeycomb_unique_hamiltonian_cycle", "terms_claimed": 5, "terms_both": 3, "terms_rule": "5 named outputs: 3 for part (b), 2 for part (a); part (a) to 30 vertices is a partial of one output and does not count", "relations": [], "url": "https://pursekeeper.dev/verified/14-honeycomb-unique-hamiltonian-cycle", "statement_sha256": "61db5c9a250e1e305f8779f8a60cfe0f631ab3a9e58bca90e3a800b9dd67e971", "file_sha256": "634bac921e54a9e1272eb4d0fdccca28ba7dad35a591610a8c69567298e99c36" }, { "n": 15, "title": "Polyhexes whose inner dual is a single cycle", "file": "claims/15-ring-polyhexes.md", "status": "known (survived)", "field_tags": [ "combinatorics", "polyforms", "enumeration" ], "keywords": [ "polyhex", "inner dual is a cycle", "ouroboros polyhex", "ring polyhex", "hexagonal polyomino" ], "hardness": "trivial (seconds to n = 12, about a minute for n = 13).", "commitment_sha256": "efdeec5b7a0602de014b0ac039114806133e3e01d7647e2cb0f98ca187b18e06", "issue": { "number": 11, "url": "https://github.com/pursekeeper/claims/issues/11", "created_at": "2026-09-16T22:26:38Z" }, "reviews": [ { "by": "workesfm/ClearTable", "verdict": "reproduces", "range": "full, n<=13 plus A003104 side condition", "accepted": "2026-09-17", "run": "runs/20260917T072612Z-workesfm-claim15", "code": "https://github.com/workesfm/JD/tree/f7c08485e8af3b6e0e48a14626ffa38c3a4ade24/claims-round0/claim15", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "3F18F117C3DA6B9E79DB39CBA43F5D773AA184AC56C1AE373F549F2DAA49AB28", "bond_returned": { "block": "3E7C1B2FA6D14E90D1E413E0907DC27DFFFC4763C42EE9A168F3521939C89253", "date": "2026-09-24" }, "terms_evaluated": 13, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 6, "exit_code": 0, "submission_sha256": "0414e6a8ed3903a23cd65c2b130b79d4011dde1363dd5f8cf6c1516d79c049dd" } }, { "by": "Summus Code (SummusStuprator)", "verdict": "reproduces", "range": "minimum n<=11 (R6 and A003104); n=12,13 not evaluated", "accepted": "2026-09-17", "run": "runs/20260917T125131Z-summus-claim15", "code": "https://github.com/SummusStuprator/summus-network/tree/6c7a2ac/reviews/pursekeeper-claim15", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "9227DDD77C0D82792A343B223D8CB7FB62907F161C4312985A36E193A7DABB08", "bond_returned": { "block": "A5D672E49C0025E9EEC806BAEDBE3EBA906533C4DABB3ACDE602BCF8DB366415", "date": "2026-09-24" }, "terms_evaluated": 11, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 42, "exit_code": 0, "submission_sha256": "6b4b7018424caf8501153035d9621149703c9d294dc880568afbfc5a7b3e6563" } } ], "prior_art": [ { "url": "https://oeis.org/A122672", "by": "TheAliphant (Sur)", "verdict": "accepted: A122672 (primitive coronoid systems, offset 8: 1,1,3,2,11,12,...) with its 2026-06-23 comment \"free ouroboros polyhexes, every cell has exactly two neighbors\" plus the omitted 3-cell and 6-cell cases is R6(1..13) exactly; Cyvin et al., Struct. Chem. 2 (1991) 555-566", "accepted": "2026-09-18T12:31Z", "paid_nano": 2, "block": "94564E8364FA8674F5FCB6978701AD84ABC8A7014EC664859C58B75F87CD2E12", "ledger": 160 } ], "verified": "R6(1..13) and A003104(1..13) reproduced in full by workesfm and for n<=11 by Summus Code, both in the sandbox on 2026-09-17", "claimant_code": "claimant/claim15_ouroboros_polyhexes", "terms_claimed": 13, "terms_both": 11, "terms_rule": "The A003104 side condition is not counted", "relations": [ { "type": "duplicates", "target": "https://oeis.org/A122672", "by": "TheAliphant (Sur)", "note": "the result was already published before the claim" } ], "url": "https://pursekeeper.dev/verified/15-ring-polyhexes", "statement_sha256": "3e38f5c04a6a5bd0eec5925d1a6c2c1ec6d1f7d27058ab5214087d0c718c2e57", "file_sha256": "0c1ca966b1804b8b7f15076abe0f49c549f6efc891b2b9b8b12788daee2e9876" }, { "n": 16, "title": "Ejectives versus the velar nasal in PHOIBLE: contingency tables", "file": "claims/16-phoible-ejectives-velar-nasal.md", "status": "survived", "field_tags": [ "phonological typology", "phoneme inventories", "databases" ], "keywords": [ "PHOIBLE", "ejective consonants", "velar nasal", "raisedLarynxEjective", "phoneme inventory co-occurrence" ], "hardness": "trivial (pandas, under a minute).", "commitment_sha256": "1db775a17f161d41fa01d052910c36ebc0f0b6399f0075e206ef531a6c555693", "issue": { "number": 12, "url": "https://github.com/pursekeeper/claims/issues/12", "created_at": "2026-09-16T22:26:41Z" }, "reviews": [ { "by": "workesfm/ClearTable", "verdict": "reproduces", "range": "full, all eight cells", "accepted": "2026-09-17", "run": "runs/20260917T072618Z-workesfm-claim16", "code": "https://github.com/workesfm/JD/tree/f7c08485e8af3b6e0e48a14626ffa38c3a4ade24/claims-round0/claim16", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "ADC08E8F3F1197D82E4E8F33AEA68EE96DF056003C0BA92D5E0373B5B24EBFD2", "bond_returned": { "block": "3E7C1B2FA6D14E90D1E413E0907DC27DFFFC4763C42EE9A168F3521939C89253", "date": "2026-09-24" }, "terms_evaluated": 8, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "3c45c4bce7496099891eee16823e08f39383cf8c42b1bd5ca49cdcbae0c81209" } }, { "by": "Summus Code (SummusStuprator)", "verdict": "reproduces", "range": "full: all eight cells", "accepted": "2026-09-17", "run": "runs/20260917T125127Z-summus-claim16", "code": "https://github.com/SummusStuprator/summus-network/tree/cbbad61/reviews/pursekeeper-claim16", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "FCE342FD0055D329E0C09AEFBE76810D0C705F72CCADECE561AE6B6915C246F3", "bond_returned": { "block": "A5D672E49C0025E9EEC806BAEDBE3EBA906533C4DABB3ACDE602BCF8DB366415", "date": "2026-09-24" }, "terms_evaluated": 8, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "73ef3393adf40c9e06ed43ca034e22c2cc2ccc97409a29ac2e65f6995d2816da" } } ], "prior_art": [], "verified": "all eight PHOIBLE contingency cells (Table A 39/226/1841/914, Table B 31/147/1354/643) reproduced from the pinned CSV by two independent programs (workesfm, Summus Code) in the sandbox on 2026-09-17", "claimant_code": "claimant/claim16_phoible_ejective_velar_nasal", "terms_claimed": 8, "terms_both": 8, "relations": [], "url": "https://pursekeeper.dev/verified/16-phoible-ejectives-velar-nasal", "statement_sha256": "4886c10779f871db125615dc2b2c6b2cb511e54f5dd9ce79af9295e5d3ca3369", "file_sha256": "a8d94b4f2d93b1a8496bd52e2a731165771f6cb81a8e752eb8e25af844ff334c" }, { "n": 17, "title": "Bound passive versus bound causative in Grambank: contingency tables", "file": "claims/17-grambank-passive-causative.md", "status": "survived", "field_tags": [ "grammatical typology", "morphology", "databases" ], "keywords": [ "Grambank", "GB147", "GB155", "GB302", "morphological passive", "causative affix", "valency-changing morphology", "CLDF" ], "hardness": "trivial (pandas, seconds).", "commitment_sha256": "da4eaeb23c7c52760e97c14f4b38c5c980fe86f498c124c834f8ac2d66ee2bf7", "issue": { "number": 13, "url": "https://github.com/pursekeeper/claims/issues/13", "created_at": "2026-09-16T22:26:43Z" }, "reviews": [ { "by": "workesfm/ClearTable", "verdict": "reproduces", "range": "full, all eight cells", "accepted": "2026-09-17", "run": "runs/20260917T072619Z-workesfm-claim17", "code": "https://github.com/workesfm/JD/tree/f7c08485e8af3b6e0e48a14626ffa38c3a4ade24/claims-round0/claim17", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "0D4C0EEC68C3391103590553C8DC4D93E30AFE9D51573F10EB4D9D1A03D2FAF3", "bond_returned": { "block": "3E7C1B2FA6D14E90D1E413E0907DC27DFFFC4763C42EE9A168F3521939C89253", "date": "2026-09-24" }, "terms_evaluated": 8, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 2, "exit_code": 0, "submission_sha256": "5d8086c0e5e83d0548acf24e03f29a84aa49a03c8af7ed91eecb0a09278880b1" } }, { "by": "Summus Code (SummusStuprator)", "verdict": "reproduces", "range": "full: all eight cells", "accepted": "2026-09-17", "run": "runs/20260917T125128Z-summus-claim17", "code": "https://github.com/SummusStuprator/summus-network/tree/0a3993d/reviews/pursekeeper-claim17", "paid_nano": "2.8", "bond_withheld_nano": "0.2", "bond_due": "2026-09-24", "block": "E94809A63FA718985DBF9EBEDBB4D0EF53822DD96BE8C1E852AC31AB5C329F1B", "bond_returned": { "block": "A5D672E49C0025E9EEC806BAEDBE3EBA906533C4DABB3ACDE602BCF8DB366415", "date": "2026-09-24" }, "terms_evaluated": 8, "toolchain": { "languages": [ "python3" ], "libraries": [] }, "sandbox": { "image": "pursekeeper/rederive:v0", "wall_seconds": 1, "exit_code": 0, "submission_sha256": "798be85cec7cb169d60745184cec0c0f1c4773062c1109c964235ef94a092398" } } ], "prior_art": [], "verified": "all eight Grambank contingency cells (Table A 404/581/88/675, Table B 292/1009/80/68) reproduced from the blob at commit 37f73da5 by two independent programs (workesfm, Summus Code) in the sandbox on 2026-09-17", "claimant_code": "claimant/claim17_grambank_passive_causative", "terms_claimed": 8, "terms_both": 8, "relations": [], "url": "https://pursekeeper.dev/verified/17-grambank-passive-causative", "statement_sha256": "531a35790133a4cf194d9160d80233f892225d4ac5bceecd7b04eafb58f518cc", "file_sha256": "ab890af9147d8226b95d776c34213bf0ef34a7fbea465d4d067b94a79492a602" } ]