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Nowakowski, Regularity in the G–S equences of Octal Games with a Pass, Integers 3 (2003), #G1","work_id":"f6e67158-94f9-460d-ac96-6ed44c7420ff","year":2003}],"snapshot_sha256":"b42cd6cee05376e835ae01ae111ecda5c444e8b5471c83ce607ba26a8d7ff20f"},"source":{"id":"2605.14321","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T02:38:54.294616Z","id":"38d70c46-cda5-4637-8360-82f6799553d9","model_set":{"reader":"grok-4.3"},"one_line_summary":"Subtraction Nim with moves {2,4n,4n+2} (n≥3) and its one-time-pass variant both satisfy the reverse-mex property for Grundy numbers.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Adding a one-time pass to this subtraction Nim leaves its reverse-mex Grundy property unchanged.","strongest_claim":"We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available.","weakest_assumption":"The subtraction set must be exactly {2, 4n, 4n+2} for integer n ≥ 3; the reverse-mex property is stated not to hold for n=1 or n=2, and the proofs rely on this specific arithmetic form."}},"verdict_id":"38d70c46-cda5-4637-8360-82f6799553d9"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a426db491ba802ce5ddcaf41182280fd11e19e544fa315e29bc33fb7fdea9352","target":"record","created_at":"2026-05-17T23:39:09Z","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":"a2a42acba97399b2d184611aa46873dad439558909003856cb00ae9edd3444fc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T03:34:08Z","title_canon_sha256":"51162b8cfa560ebc9be84d3add6601169d47f85da43e3419ef31d1d51947f659"},"schema_version":"1.0","source":{"id":"2605.14321","kind":"arxiv","version":1}},"canonical_sha256":"0f0794f4da255b10a8640ddd4e2494bf622327aae8afc2ec77e8c53aae6610b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f0794f4da255b10a8640ddd4e2494bf622327aae8afc2ec77e8c53aae6610b2","first_computed_at":"2026-05-17T23:39:09.832341Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:09.832341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5rLJqANdBnrv4FQdyqLNr8KaydjmeJUQKSjLionF5xL9csnNZyltctHZCV7lQJBjlXafMy3baMotwhCeCmNCCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:09.834932Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14321","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a426db491ba802ce5ddcaf41182280fd11e19e544fa315e29bc33fb7fdea9352","sha256:e2f0d4f49852c37061b89898cdd4655e937d498b4234285ac5e015d8ce1f95b9"],"state_sha256":"d7540236f8748c1c836d0afeb3efa2c204269148decf4826848b6b25c4a39c52"}