{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:GNQCDO26WJKQL2YMHIQYBRMAII","short_pith_number":"pith:GNQCDO26","canonical_record":{"source":{"id":"2009.09510","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-09-20T19:52:31Z","cross_cats_sorted":[],"title_canon_sha256":"ebb17cc416a506fcc70f2038bc7fa1a75b965ba87506c604b66a0936fda8838a","abstract_canon_sha256":"aa26113e3ac1634a19d604fd284c9ea7819820bb54f4640c9a709308af3ce6f8"},"schema_version":"1.0"},"canonical_sha256":"336021bb5eb25505eb0c3a2180c58042094833efcb2376567a6b7cc4cd70f047","source":{"kind":"arxiv","id":"2009.09510","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.09510","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"arxiv_version","alias_value":"2009.09510v1","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.09510","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"pith_short_12","alias_value":"GNQCDO26WJKQ","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"pith_short_16","alias_value":"GNQCDO26WJKQL2YM","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"pith_short_8","alias_value":"GNQCDO26","created_at":"2026-07-05T01:36:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:GNQCDO26WJKQL2YMHIQYBRMAII","target":"record","payload":{"canonical_record":{"source":{"id":"2009.09510","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-09-20T19:52:31Z","cross_cats_sorted":[],"title_canon_sha256":"ebb17cc416a506fcc70f2038bc7fa1a75b965ba87506c604b66a0936fda8838a","abstract_canon_sha256":"aa26113e3ac1634a19d604fd284c9ea7819820bb54f4640c9a709308af3ce6f8"},"schema_version":"1.0"},"canonical_sha256":"336021bb5eb25505eb0c3a2180c58042094833efcb2376567a6b7cc4cd70f047","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:36:41.349225Z","signature_b64":"irl6LJMoyz7kwJSXSwhOYwXoIvY4DdoUJeOPMhhwrRqH4cCal/I2CTo3/Qr5JTKZ+YaOFqVm5O3wdRrIBR7NDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"336021bb5eb25505eb0c3a2180c58042094833efcb2376567a6b7cc4cd70f047","last_reissued_at":"2026-07-05T01:36:41.348850Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:36:41.348850Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2009.09510","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":"2026-07-05T01:36:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0tJFz87ob9uVxjx6OD6jRhxkB8FfAx25aIPqWMIAlX714iJx7Ctedb9W2UtDyse8uX/O0dMOA5LPutpLg4NpDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T19:51:19.341098Z"},"content_sha256":"15bfc86c2b48aab635a75d9d3893af760bba0a03839f6be8a3052629e4dbba38","schema_version":"1.0","event_id":"sha256:15bfc86c2b48aab635a75d9d3893af760bba0a03839f6be8a3052629e4dbba38"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:GNQCDO26WJKQL2YMHIQYBRMAII","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounds on Zeckendorf Games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aiden Dunkelberg, Anna Cusenza, Clayton Mizgerd, Dianhui Ke, Jingkai Ye, Kate Huffman, Micah McClatchey, Steven J. Miller, Vashisth Tiwari, Xiaoyan Zheng","submitted_at":"2020-09-20T19:52:31Z","abstract_excerpt":"Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this decomposition to construct a two-player game. Given a fixed integer $n$ and an initial decomposition of $n=n F_1$, the two players alternate by using moves related to the recurrence relation $F_{n+1}=F_n+F_{n-1}$, and whoever moves last wins. The game always terminates in the Zeckendorf decomposition; depending on the choice of moves the length of the game and the winner can vary, though for $n\\ge 2$ there is a non-constructive proof that Player 2 has a winning st"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.09510","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2009.09510/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T01:36:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qldIhnaYvA7Il3kXBB4pBL0ViHgL25smxpD7P045mOlicZ+b4aZW27PbXLU6o6JZ7sgMYI04jWboUP9OWNeTCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T19:51:19.341483Z"},"content_sha256":"2bddf80c6040ce49abc897b1036dbe6f66556aeb9f030f1a53f1356d73983666","schema_version":"1.0","event_id":"sha256:2bddf80c6040ce49abc897b1036dbe6f66556aeb9f030f1a53f1356d73983666"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GNQCDO26WJKQL2YMHIQYBRMAII/bundle.json","state_url":"https://pith.science/pith/GNQCDO26WJKQL2YMHIQYBRMAII/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GNQCDO26WJKQL2YMHIQYBRMAII/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":"2026-08-03T19:51:19Z","links":{"resolver":"https://pith.science/pith/GNQCDO26WJKQL2YMHIQYBRMAII","bundle":"https://pith.science/pith/GNQCDO26WJKQL2YMHIQYBRMAII/bundle.json","state":"https://pith.science/pith/GNQCDO26WJKQL2YMHIQYBRMAII/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GNQCDO26WJKQL2YMHIQYBRMAII/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:GNQCDO26WJKQL2YMHIQYBRMAII","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":"aa26113e3ac1634a19d604fd284c9ea7819820bb54f4640c9a709308af3ce6f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-09-20T19:52:31Z","title_canon_sha256":"ebb17cc416a506fcc70f2038bc7fa1a75b965ba87506c604b66a0936fda8838a"},"schema_version":"1.0","source":{"id":"2009.09510","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.09510","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"arxiv_version","alias_value":"2009.09510v1","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.09510","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"pith_short_12","alias_value":"GNQCDO26WJKQ","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"pith_short_16","alias_value":"GNQCDO26WJKQL2YM","created_at":"2026-07-05T01:36:41Z"},{"alias_kind":"pith_short_8","alias_value":"GNQCDO26","created_at":"2026-07-05T01:36:41Z"}],"graph_snapshots":[{"event_id":"sha256:2bddf80c6040ce49abc897b1036dbe6f66556aeb9f030f1a53f1356d73983666","target":"graph","created_at":"2026-07-05T01:36:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2009.09510/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this decomposition to construct a two-player game. Given a fixed integer $n$ and an initial decomposition of $n=n F_1$, the two players alternate by using moves related to the recurrence relation $F_{n+1}=F_n+F_{n-1}$, and whoever moves last wins. The game always terminates in the Zeckendorf decomposition; depending on the choice of moves the length of the game and the winner can vary, though for $n\\ge 2$ there is a non-constructive proof that Player 2 has a winning st","authors_text":"Aiden Dunkelberg, Anna Cusenza, Clayton Mizgerd, Dianhui Ke, Jingkai Ye, Kate Huffman, Micah McClatchey, Steven J. Miller, Vashisth Tiwari, Xiaoyan Zheng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-09-20T19:52:31Z","title":"Bounds on Zeckendorf Games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.09510","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":"sha256:15bfc86c2b48aab635a75d9d3893af760bba0a03839f6be8a3052629e4dbba38","target":"record","created_at":"2026-07-05T01:36:41Z","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":"aa26113e3ac1634a19d604fd284c9ea7819820bb54f4640c9a709308af3ce6f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-09-20T19:52:31Z","title_canon_sha256":"ebb17cc416a506fcc70f2038bc7fa1a75b965ba87506c604b66a0936fda8838a"},"schema_version":"1.0","source":{"id":"2009.09510","kind":"arxiv","version":1}},"canonical_sha256":"336021bb5eb25505eb0c3a2180c58042094833efcb2376567a6b7cc4cd70f047","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"336021bb5eb25505eb0c3a2180c58042094833efcb2376567a6b7cc4cd70f047","first_computed_at":"2026-07-05T01:36:41.348850Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:36:41.348850Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"irl6LJMoyz7kwJSXSwhOYwXoIvY4DdoUJeOPMhhwrRqH4cCal/I2CTo3/Qr5JTKZ+YaOFqVm5O3wdRrIBR7NDA==","signature_status":"signed_v1","signed_at":"2026-07-05T01:36:41.349225Z","signed_message":"canonical_sha256_bytes"},"source_id":"2009.09510","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:15bfc86c2b48aab635a75d9d3893af760bba0a03839f6be8a3052629e4dbba38","sha256:2bddf80c6040ce49abc897b1036dbe6f66556aeb9f030f1a53f1356d73983666"],"state_sha256":"e991cb15bb1c6fed8fd07c218196cb7b180d151579aeea016e05beaf53ab6086"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"140eLB8gNEspQg8yqlBrTGAKd1H8rAe4jGz7bf9OZ0vMJ8tvAqE3LSpaC0x8MszTuK/KFGX5CqFnq3/I5XbiAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T19:51:19.343731Z","bundle_sha256":"ddbb8a557f2259f61615ce5c65f10e4b1e60f75f688288be1f9bbea4f0bea384"}}