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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","authors_text":"Arnav Garg","cross_cats":[],"headline":"P-positions in 4 by n Chomp exhibit unique fourth-row extensions, converging ratios, period-112 structure, and linear cone geometry.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-04-24T14:43:31Z","title":"Structural Results for 4 x n Chomp: Unique Extension, Bimodal Asymptotic Structure, and Period-112 Geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.25952","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-08T08:54:53.125901Z","id":"0efc6603-5c05-48bd-a397-46822c5fe103","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"P-positions in 4 by n Chomp exhibit unique fourth-row extensions, converging ratios, period-112 structure, and linear cone geometry.","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.","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."}},"verdict_id":"0efc6603-5c05-48bd-a397-46822c5fe103"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5de36fb791cfb4ea8fca75d78da9f33e0a09ff7480afe84b22cddb567329e518","target":"record","created_at":"2026-05-22T02:04:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5a71be7afec30f4e1dbe200af9b4f6bf8d942bf471eac12085bb28ce338a24a2","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-04-24T14:43:31Z","title_canon_sha256":"e6a9096e5b80e3c7d33ff9d0c2052fad4cc6e9ab6ada7af4e147975ca2ee2970"},"schema_version":"1.0","source":{"id":"2604.25952","kind":"arxiv","version":2}},"canonical_sha256":"e80844b475d3000949cd6b86ea41f3b777cd429ac063c47c01ceb823ab379068","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e80844b475d3000949cd6b86ea41f3b777cd429ac063c47c01ceb823ab379068","first_computed_at":"2026-05-22T02:04:41.360117Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T02:04:41.360117Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+ug7Rs1X4TqPAn7p4XniQVLnDzlJlp+3rKfpJSTYkslVlaYs+VPYtDrDs/feUbxl1dKLCuzltm8d3Y7tmAeZDA==","signature_status":"signed_v1","signed_at":"2026-05-22T02:04:41.361064Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.25952","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5de36fb791cfb4ea8fca75d78da9f33e0a09ff7480afe84b22cddb567329e518","sha256:c3e54602c81e9771520882c8503be59eb492361bdbd44002bf9a7344eef51a97"],"state_sha256":"e757551ae44a710f2e4a9e136faa9b6da7bc189b2f2cb4c73d20dba68cf3dd7c"}