{"paper":{"title":"Structural Results for 4 x n Chomp: Unique Extension, Bimodal Asymptotic Structure, and Period-112 Geometry","license":"http://creativecommons.org/licenses/by/4.0/","headline":"P-positions in 4 by n Chomp exhibit unique fourth-row extensions, converging ratios, period-112 structure, and linear cone geometry.","cross_cats":[],"primary_cat":"math.GM","authors_text":"Arnav Garg","submitted_at":"2026-04-24T14:43:31Z","abstract_excerpt":"We present a complete computational tabulation of all 961,619,972 P-positions in 4xn Chomp for n <= 3000, obtained via a new O(n^4) shadow-array sieve that replaces the O(n^5) hash-set approach of prior work. Three structural results are reported. First, we prove the Unique Extension property: for any triple (a,b,c), there is at most one value of d such that (a,b,c,d) is a P-position. The proof is a short contradiction using the move structure of Chomp and generalizes immediately to all k-row Chomp. Second, the P-positions exhibit a persistent bimodal decomposition into two subfamilies, HIGH a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The analysis reveals four structural conjectures: (1) a Unique Extension property, stating that for any triple (a,b,c) of row lengths, there is at most one valid fourth-row length d completing a P-position; (2) convergence of row-length ratios to fixed asymptotic constants; (3) a period-112 modular structure governing the set of extendable triples; and (4) a linear cone geometry for the P-position set in (a,b,c)-space.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the four observed patterns in P-positions computed for n ≤ 500 will continue to hold for all larger n without exception or deviation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Large-scale computation of P-positions in 4xn Chomp yields conjectures on unique row-length extension, asymptotic ratio convergence, period-112 modularity, and linear cone geometry.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"P-positions in 4 by n Chomp exhibit unique fourth-row extensions, converging ratios, period-112 structure, and linear cone geometry.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2487f93a0edb29afb02f1e618d6579bfa278a76584a60507613832c944206808"},"source":{"id":"2604.25952","kind":"arxiv","version":2},"verdict":{"id":"0efc6603-5c05-48bd-a397-46822c5fe103","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T08:54:53.125901Z","strongest_claim":"The analysis reveals four structural conjectures: (1) a Unique Extension property, stating that for any triple (a,b,c) of row lengths, there is at most one valid fourth-row length d completing a P-position; (2) convergence of row-length ratios to fixed asymptotic constants; (3) a period-112 modular structure governing the set of extendable triples; and (4) a linear cone geometry for the P-position set in (a,b,c)-space.","one_line_summary":"Large-scale computation of P-positions in 4xn Chomp yields conjectures on unique row-length extension, asymptotic ratio convergence, period-112 modularity, and linear cone geometry.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the four observed patterns in P-positions computed for n ≤ 500 will continue to hold for all larger n without exception or deviation.","pith_extraction_headline":"P-positions in 4 by n Chomp exhibit unique fourth-row extensions, converging ratios, period-112 structure, and linear cone geometry."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25952/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T10:36:34.667077Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:47:33.960185Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"d31fbaaf3759dda85acf0111f550478801f64a0a6377cd5a40864c066a1bc932"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}