{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:SPBLEBDPPLPU3TCTEFA2PRUICP","short_pith_number":"pith:SPBLEBDP","canonical_record":{"source":{"id":"1309.1631","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-06T13:17:54Z","cross_cats_sorted":[],"title_canon_sha256":"68a55020b560af841b399a330ad0c9308c1990236c342e8eb9653e81648621c5","abstract_canon_sha256":"e8233f3bbe4069e62f13f2b645a6068205a8a9c6144e2da3190e078ab90ccd03"},"schema_version":"1.0"},"canonical_sha256":"93c2b2046f7adf4dcc532141a7c68813ee78d2d2cc202704b807b9830310b613","source":{"kind":"arxiv","id":"1309.1631","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.1631","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"arxiv_version","alias_value":"1309.1631v1","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.1631","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"pith_short_12","alias_value":"SPBLEBDPPLPU","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"SPBLEBDPPLPU3TCT","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"SPBLEBDP","created_at":"2026-05-18T12:27:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:SPBLEBDPPLPU3TCTEFA2PRUICP","target":"record","payload":{"canonical_record":{"source":{"id":"1309.1631","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-06T13:17:54Z","cross_cats_sorted":[],"title_canon_sha256":"68a55020b560af841b399a330ad0c9308c1990236c342e8eb9653e81648621c5","abstract_canon_sha256":"e8233f3bbe4069e62f13f2b645a6068205a8a9c6144e2da3190e078ab90ccd03"},"schema_version":"1.0"},"canonical_sha256":"93c2b2046f7adf4dcc532141a7c68813ee78d2d2cc202704b807b9830310b613","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:58.974510Z","signature_b64":"1tkVz9OIZGHuzaR0dbjSkHrtBk5BK/YR+jZy/oaDxDYc2Y6aNXivJvn1reuIvNRgjfuXz9pPj98z2GvdVu0FCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93c2b2046f7adf4dcc532141a7c68813ee78d2d2cc202704b807b9830310b613","last_reissued_at":"2026-05-18T03:13:58.973845Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:58.973845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.1631","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:13:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wnbqe2GGHcWO1lMfhkNUJ7qEARW0ztYMZZYFi0Vz/i4MOyY+HucZ+T9hAsraGizeEERYtwYA8sUx9oJQ5CvPBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:14:53.983584Z"},"content_sha256":"c335b705feb7169fcec634f418f404ab05c8d4d9565bef92519e64c12ee2fe1b","schema_version":"1.0","event_id":"sha256:c335b705feb7169fcec634f418f404ab05c8d4d9565bef92519e64c12ee2fe1b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:SPBLEBDPPLPU3TCTEFA2PRUICP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Partizan Kayles and Misere Invertibility","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Rebecca Milley","submitted_at":"2013-09-06T13:17:54Z","abstract_excerpt":"The impartial combinatorial game Kayles is played on a row of pins, with players taking turns removing either a single pin or two adjacent pins. A natural partizan variation is to allow one player to remove only a single pin and the other only a pair of pins. This paper develops a complete solution for \"Partizan Kayles\" under misere play, including the misere monoid all possible sums of positions, and discusses its significance in the context of misere invertibility: the universe of Partizan Kayles contains a position whose additive inverse is not its negative, and, moreover, this position is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.1631","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:13:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7jYeIGsfMEhMR/dCdih+JBWnKAhuRkuTErNfrj9WzJXJiiV/q7bvcDwaDvG9dpn0LMLnsHwAMqFuFQcRJZRXCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:14:53.983932Z"},"content_sha256":"e4917eab21a2c6fde1bd32f5d868c5b79a78e2963fd542ac37d538f67b1dc6c4","schema_version":"1.0","event_id":"sha256:e4917eab21a2c6fde1bd32f5d868c5b79a78e2963fd542ac37d538f67b1dc6c4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SPBLEBDPPLPU3TCTEFA2PRUICP/bundle.json","state_url":"https://pith.science/pith/SPBLEBDPPLPU3TCTEFA2PRUICP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SPBLEBDPPLPU3TCTEFA2PRUICP/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T16:14:53Z","links":{"resolver":"https://pith.science/pith/SPBLEBDPPLPU3TCTEFA2PRUICP","bundle":"https://pith.science/pith/SPBLEBDPPLPU3TCTEFA2PRUICP/bundle.json","state":"https://pith.science/pith/SPBLEBDPPLPU3TCTEFA2PRUICP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SPBLEBDPPLPU3TCTEFA2PRUICP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:SPBLEBDPPLPU3TCTEFA2PRUICP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e8233f3bbe4069e62f13f2b645a6068205a8a9c6144e2da3190e078ab90ccd03","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-06T13:17:54Z","title_canon_sha256":"68a55020b560af841b399a330ad0c9308c1990236c342e8eb9653e81648621c5"},"schema_version":"1.0","source":{"id":"1309.1631","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.1631","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"arxiv_version","alias_value":"1309.1631v1","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.1631","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"pith_short_12","alias_value":"SPBLEBDPPLPU","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"SPBLEBDPPLPU3TCT","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"SPBLEBDP","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:e4917eab21a2c6fde1bd32f5d868c5b79a78e2963fd542ac37d538f67b1dc6c4","target":"graph","created_at":"2026-05-18T03:13:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The impartial combinatorial game Kayles is played on a row of pins, with players taking turns removing either a single pin or two adjacent pins. A natural partizan variation is to allow one player to remove only a single pin and the other only a pair of pins. This paper develops a complete solution for \"Partizan Kayles\" under misere play, including the misere monoid all possible sums of positions, and discusses its significance in the context of misere invertibility: the universe of Partizan Kayles contains a position whose additive inverse is not its negative, and, moreover, this position is ","authors_text":"Rebecca Milley","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-06T13:17:54Z","title":"Partizan Kayles and Misere Invertibility"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.1631","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c335b705feb7169fcec634f418f404ab05c8d4d9565bef92519e64c12ee2fe1b","target":"record","created_at":"2026-05-18T03:13:58Z","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":"e8233f3bbe4069e62f13f2b645a6068205a8a9c6144e2da3190e078ab90ccd03","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-06T13:17:54Z","title_canon_sha256":"68a55020b560af841b399a330ad0c9308c1990236c342e8eb9653e81648621c5"},"schema_version":"1.0","source":{"id":"1309.1631","kind":"arxiv","version":1}},"canonical_sha256":"93c2b2046f7adf4dcc532141a7c68813ee78d2d2cc202704b807b9830310b613","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"93c2b2046f7adf4dcc532141a7c68813ee78d2d2cc202704b807b9830310b613","first_computed_at":"2026-05-18T03:13:58.973845Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:58.973845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1tkVz9OIZGHuzaR0dbjSkHrtBk5BK/YR+jZy/oaDxDYc2Y6aNXivJvn1reuIvNRgjfuXz9pPj98z2GvdVu0FCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:58.974510Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.1631","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c335b705feb7169fcec634f418f404ab05c8d4d9565bef92519e64c12ee2fe1b","sha256:e4917eab21a2c6fde1bd32f5d868c5b79a78e2963fd542ac37d538f67b1dc6c4"],"state_sha256":"b608b40ba0c7cf0871797454e29a15cd05f1fdc2764cc9fdae921e7b14c46e2a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fwol8yo5BBn5A305chRI3fJW1SMWmR8pfOVSugIbc5ufo1dLUwNuzHDxqCXe3fgGIcvKqLW+uL9SDlyEVNlLCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T16:14:53.985992Z","bundle_sha256":"30e0ce1fa4febdecfb8faa7600b7a4f3fd8f6ff02bb19453295ee8e4fa3f7fc9"}}