{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:FINGO7VYFHSCJD4JZOVPV3T625","short_pith_number":"pith:FINGO7VY","canonical_record":{"source":{"id":"1705.06310","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-17T18:49:21Z","cross_cats_sorted":[],"title_canon_sha256":"175559eefecb5912e490903810762af96ea0ffcae527a4e97d8dc5878c030331","abstract_canon_sha256":"398ca9496b9326fe79f7461748c3b9b1c718a7a157178253aba509ee33595469"},"schema_version":"1.0"},"canonical_sha256":"2a1a677eb829e4248f89cbaafaee7ed74a9157b03aa8cdb4701510427904b48a","source":{"kind":"arxiv","id":"1705.06310","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.06310","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"arxiv_version","alias_value":"1705.06310v3","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.06310","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"pith_short_12","alias_value":"FINGO7VYFHSC","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"pith_short_16","alias_value":"FINGO7VYFHSCJD4J","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"pith_short_8","alias_value":"FINGO7VY","created_at":"2026-07-05T00:07:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:FINGO7VYFHSCJD4JZOVPV3T625","target":"record","payload":{"canonical_record":{"source":{"id":"1705.06310","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-17T18:49:21Z","cross_cats_sorted":[],"title_canon_sha256":"175559eefecb5912e490903810762af96ea0ffcae527a4e97d8dc5878c030331","abstract_canon_sha256":"398ca9496b9326fe79f7461748c3b9b1c718a7a157178253aba509ee33595469"},"schema_version":"1.0"},"canonical_sha256":"2a1a677eb829e4248f89cbaafaee7ed74a9157b03aa8cdb4701510427904b48a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:07:43.098651Z","signature_b64":"6nQvjbNHh+4OQuLJsnjfMwB9X5FPCZc36opyX9UTy85BMx+ugw/45TCA8/+GiwLVT9/pz8xKWugwR11lEpVHCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a1a677eb829e4248f89cbaafaee7ed74a9157b03aa8cdb4701510427904b48a","last_reissued_at":"2026-07-05T00:07:43.098232Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:07:43.098232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.06310","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:07:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nnOgKINWxy0MhEQcR2lfxqq/1Ps/ciZIh9OBMB5FcmwVv+wOLXRBbdDUM7k3pUihlyyh3O4iKGNPlbmkSh3GAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T14:22:33.404668Z"},"content_sha256":"265071c6e5f2a2f65ff3db1e78466f3b5ce07041e3932b2c47caf79825214793","schema_version":"1.0","event_id":"sha256:265071c6e5f2a2f65ff3db1e78466f3b5ce07041e3932b2c47caf79825214793"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:FINGO7VYFHSCJD4JZOVPV3T625","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Functions on Antipower Prefix Lengths of the Thue-Morse Word","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shyam Narayanan","submitted_at":"2017-05-17T18:49:21Z","abstract_excerpt":"We say that a word $w$ of length $kn$ is a $k$-\\textit{antipower} if it can be written in the form $w_1 \\cdots w_k$, where each $w_i$ is a distinct word of length $n$. We analyze prefixes of the Thue-Morse word $\\textbf{t}$ and lengths of antipowers occurring in them. Define $\\Gamma(k)$ to be the largest odd $n$ such that the prefix of $\\textbf{t}$ of length $kn$ is not a $k$-antipower, and $\\gamma(k)$ to be the smallest odd $n$ such that the corresponding prefix is a $k$-antipower. We provide strong bounds on the asymptotic values of $\\gamma(k)$ and $\\Gamma(k)-\\gamma(k)$. Our bounds on $\\gamm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.06310","kind":"arxiv","version":3},"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/1705.06310/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":"2026-07-05T00:07:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"npvzOEw7sL9+lZg6vCGIeazGVawUwEZFn1DgHwmlcchRowSf0kXfxKWBg87uC5RivpDELCy9KOh5xnqVPRX+DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T14:22:33.405183Z"},"content_sha256":"daa67474703c1b8bcf16445e0c0496676697de7cd2cc94c71fa98faa3db85da4","schema_version":"1.0","event_id":"sha256:daa67474703c1b8bcf16445e0c0496676697de7cd2cc94c71fa98faa3db85da4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FINGO7VYFHSCJD4JZOVPV3T625/bundle.json","state_url":"https://pith.science/pith/FINGO7VYFHSCJD4JZOVPV3T625/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FINGO7VYFHSCJD4JZOVPV3T625/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":"2026-08-16T14:22:33Z","links":{"resolver":"https://pith.science/pith/FINGO7VYFHSCJD4JZOVPV3T625","bundle":"https://pith.science/pith/FINGO7VYFHSCJD4JZOVPV3T625/bundle.json","state":"https://pith.science/pith/FINGO7VYFHSCJD4JZOVPV3T625/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FINGO7VYFHSCJD4JZOVPV3T625/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:FINGO7VYFHSCJD4JZOVPV3T625","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":"398ca9496b9326fe79f7461748c3b9b1c718a7a157178253aba509ee33595469","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-17T18:49:21Z","title_canon_sha256":"175559eefecb5912e490903810762af96ea0ffcae527a4e97d8dc5878c030331"},"schema_version":"1.0","source":{"id":"1705.06310","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.06310","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"arxiv_version","alias_value":"1705.06310v3","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.06310","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"pith_short_12","alias_value":"FINGO7VYFHSC","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"pith_short_16","alias_value":"FINGO7VYFHSCJD4J","created_at":"2026-07-05T00:07:43Z"},{"alias_kind":"pith_short_8","alias_value":"FINGO7VY","created_at":"2026-07-05T00:07:43Z"}],"graph_snapshots":[{"event_id":"sha256:daa67474703c1b8bcf16445e0c0496676697de7cd2cc94c71fa98faa3db85da4","target":"graph","created_at":"2026-07-05T00:07:43Z","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/1705.06310/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We say that a word $w$ of length $kn$ is a $k$-\\textit{antipower} if it can be written in the form $w_1 \\cdots w_k$, where each $w_i$ is a distinct word of length $n$. We analyze prefixes of the Thue-Morse word $\\textbf{t}$ and lengths of antipowers occurring in them. Define $\\Gamma(k)$ to be the largest odd $n$ such that the prefix of $\\textbf{t}$ of length $kn$ is not a $k$-antipower, and $\\gamma(k)$ to be the smallest odd $n$ such that the corresponding prefix is a $k$-antipower. We provide strong bounds on the asymptotic values of $\\gamma(k)$ and $\\Gamma(k)-\\gamma(k)$. Our bounds on $\\gamm","authors_text":"Shyam Narayanan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-17T18:49:21Z","title":"Functions on Antipower Prefix Lengths of the Thue-Morse Word"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.06310","kind":"arxiv","version":3},"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":"sha256:265071c6e5f2a2f65ff3db1e78466f3b5ce07041e3932b2c47caf79825214793","target":"record","created_at":"2026-07-05T00:07:43Z","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":"398ca9496b9326fe79f7461748c3b9b1c718a7a157178253aba509ee33595469","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-17T18:49:21Z","title_canon_sha256":"175559eefecb5912e490903810762af96ea0ffcae527a4e97d8dc5878c030331"},"schema_version":"1.0","source":{"id":"1705.06310","kind":"arxiv","version":3}},"canonical_sha256":"2a1a677eb829e4248f89cbaafaee7ed74a9157b03aa8cdb4701510427904b48a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2a1a677eb829e4248f89cbaafaee7ed74a9157b03aa8cdb4701510427904b48a","first_computed_at":"2026-07-05T00:07:43.098232Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:07:43.098232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6nQvjbNHh+4OQuLJsnjfMwB9X5FPCZc36opyX9UTy85BMx+ugw/45TCA8/+GiwLVT9/pz8xKWugwR11lEpVHCw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:07:43.098651Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.06310","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:265071c6e5f2a2f65ff3db1e78466f3b5ce07041e3932b2c47caf79825214793","sha256:daa67474703c1b8bcf16445e0c0496676697de7cd2cc94c71fa98faa3db85da4"],"state_sha256":"603d2337b2fff94db5ba4ef3df92626e7294289bcadb5a087e799946bcdb1726"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8URf8KlGZZ5r0UV5kvCPAA0fHekOa4ZQhSRky/Co51Kiwv5rWVgyzrJQWrRDJ1MMR33JWDQIY5QS6sdXD4QbBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T14:22:33.408920Z","bundle_sha256":"0ff1392b5129721dd74f159ac29fde464e402203f26c612a6fc78fd4cdc5843a"}}