{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:WFH3IRBVBL24CO4AVNSFMK5PMY","short_pith_number":"pith:WFH3IRBV","canonical_record":{"source":{"id":"1208.2505","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-13T06:37:48Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"2a9ae58abe59ca480379571a7aa7778edd778095f7b14988546bf940bc05475f","abstract_canon_sha256":"c314efcea322528bb5a4b0f02a6d734886a1b4c65a9b08546fd4698b91590490"},"schema_version":"1.0"},"canonical_sha256":"b14fb444350af5c13b80ab64562baf662bebfd7ad215bcad9f26327cbfcb0e2e","source":{"kind":"arxiv","id":"1208.2505","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.2505","created_at":"2026-05-18T03:34:03Z"},{"alias_kind":"arxiv_version","alias_value":"1208.2505v2","created_at":"2026-05-18T03:34:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.2505","created_at":"2026-05-18T03:34:03Z"},{"alias_kind":"pith_short_12","alias_value":"WFH3IRBVBL24","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WFH3IRBVBL24CO4A","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WFH3IRBV","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:WFH3IRBVBL24CO4AVNSFMK5PMY","target":"record","payload":{"canonical_record":{"source":{"id":"1208.2505","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-13T06:37:48Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"2a9ae58abe59ca480379571a7aa7778edd778095f7b14988546bf940bc05475f","abstract_canon_sha256":"c314efcea322528bb5a4b0f02a6d734886a1b4c65a9b08546fd4698b91590490"},"schema_version":"1.0"},"canonical_sha256":"b14fb444350af5c13b80ab64562baf662bebfd7ad215bcad9f26327cbfcb0e2e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:03.087197Z","signature_b64":"eht6wJQJN6+bUkZ2vsWajpY8LMgB8Ia21IZDJbyHPsyvN+brv+mpOJPi9io8A/k29EjbhA83CU9PgHwmRjR0Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b14fb444350af5c13b80ab64562baf662bebfd7ad215bcad9f26327cbfcb0e2e","last_reissued_at":"2026-05-18T03:34:03.086373Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:03.086373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.2505","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:34:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"84CZ6l6hp2g+kExZ7dnMpj2H5s4s8IdRjzAZ2tp2xaMC1oBLeddc1yikTqpZJs45oQrUNWPXlXohakirW+CICw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T21:09:50.938775Z"},"content_sha256":"61e11d0cd67b17405a79a39c978f7340221fb64fcfc99501b695b14ea4e72316","schema_version":"1.0","event_id":"sha256:61e11d0cd67b17405a79a39c978f7340221fb64fcfc99501b695b14ea4e72316"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:WFH3IRBVBL24CO4AVNSFMK5PMY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proof of Brlek-Reutenauer conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Edita Pelantova, Lubomira Balkova, Stepan Starosta","submitted_at":"2012-08-13T06:37:48Z","abstract_excerpt":"Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u) = \\sum_{n=0}^{\\infty}T_u(n) in which D(u) denotes the defect of u and T_u(n) denotes C_u(n+1)-C_u(n) +2 - P_U(n+1) - P_u(n), where C_u and P_u are the factor and palindromic complexity of u, respectively. This conjecture was verified for periodic words by Brlek and Reutenauer themselves. Using their results for periodic words, we have recently proved the conjecture for uniformly recurrent words. In the present article we prove the conjecture in its general version by a new"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2505","kind":"arxiv","version":2},"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":"2026-05-18T03:34:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YCKxCv8ZvMAKdz/AF36APqUHLtBFE2CH3f202xEESVdHTcIZD6y9NnuKBwAzVaVS/0CYwRXwUSgm2hTambNfCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T21:09:50.939121Z"},"content_sha256":"b0ce0f5784763d6f3aa4da2349d649f013bffb8a54cbc75ad11229858e377c1a","schema_version":"1.0","event_id":"sha256:b0ce0f5784763d6f3aa4da2349d649f013bffb8a54cbc75ad11229858e377c1a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WFH3IRBVBL24CO4AVNSFMK5PMY/bundle.json","state_url":"https://pith.science/pith/WFH3IRBVBL24CO4AVNSFMK5PMY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WFH3IRBVBL24CO4AVNSFMK5PMY/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":"2026-07-01T21:09:50Z","links":{"resolver":"https://pith.science/pith/WFH3IRBVBL24CO4AVNSFMK5PMY","bundle":"https://pith.science/pith/WFH3IRBVBL24CO4AVNSFMK5PMY/bundle.json","state":"https://pith.science/pith/WFH3IRBVBL24CO4AVNSFMK5PMY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WFH3IRBVBL24CO4AVNSFMK5PMY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:WFH3IRBVBL24CO4AVNSFMK5PMY","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":"c314efcea322528bb5a4b0f02a6d734886a1b4c65a9b08546fd4698b91590490","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-13T06:37:48Z","title_canon_sha256":"2a9ae58abe59ca480379571a7aa7778edd778095f7b14988546bf940bc05475f"},"schema_version":"1.0","source":{"id":"1208.2505","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.2505","created_at":"2026-05-18T03:34:03Z"},{"alias_kind":"arxiv_version","alias_value":"1208.2505v2","created_at":"2026-05-18T03:34:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.2505","created_at":"2026-05-18T03:34:03Z"},{"alias_kind":"pith_short_12","alias_value":"WFH3IRBVBL24","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WFH3IRBVBL24CO4A","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WFH3IRBV","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:b0ce0f5784763d6f3aa4da2349d649f013bffb8a54cbc75ad11229858e377c1a","target":"graph","created_at":"2026-05-18T03:34:03Z","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":"Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u) = \\sum_{n=0}^{\\infty}T_u(n) in which D(u) denotes the defect of u and T_u(n) denotes C_u(n+1)-C_u(n) +2 - P_U(n+1) - P_u(n), where C_u and P_u are the factor and palindromic complexity of u, respectively. This conjecture was verified for periodic words by Brlek and Reutenauer themselves. Using their results for periodic words, we have recently proved the conjecture for uniformly recurrent words. In the present article we prove the conjecture in its general version by a new","authors_text":"Edita Pelantova, Lubomira Balkova, Stepan Starosta","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-13T06:37:48Z","title":"Proof of Brlek-Reutenauer conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2505","kind":"arxiv","version":2},"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":"sha256:61e11d0cd67b17405a79a39c978f7340221fb64fcfc99501b695b14ea4e72316","target":"record","created_at":"2026-05-18T03:34:03Z","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":"c314efcea322528bb5a4b0f02a6d734886a1b4c65a9b08546fd4698b91590490","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-13T06:37:48Z","title_canon_sha256":"2a9ae58abe59ca480379571a7aa7778edd778095f7b14988546bf940bc05475f"},"schema_version":"1.0","source":{"id":"1208.2505","kind":"arxiv","version":2}},"canonical_sha256":"b14fb444350af5c13b80ab64562baf662bebfd7ad215bcad9f26327cbfcb0e2e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b14fb444350af5c13b80ab64562baf662bebfd7ad215bcad9f26327cbfcb0e2e","first_computed_at":"2026-05-18T03:34:03.086373Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:03.086373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eht6wJQJN6+bUkZ2vsWajpY8LMgB8Ia21IZDJbyHPsyvN+brv+mpOJPi9io8A/k29EjbhA83CU9PgHwmRjR0Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:03.087197Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.2505","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:61e11d0cd67b17405a79a39c978f7340221fb64fcfc99501b695b14ea4e72316","sha256:b0ce0f5784763d6f3aa4da2349d649f013bffb8a54cbc75ad11229858e377c1a"],"state_sha256":"f9d8d673929b80ada72cba8e687c39b11a624ae156fc971e0be0bd045ab64e5a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tc/rRw7jP6KvMAm/8npn3wGSbZM2PbvHjDsV1IUkXWkZhwuE/ebfasuvCi48oyr93DZz/6LRdCMnnK7WzuGCBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T21:09:50.941161Z","bundle_sha256":"0b956fe14813c40b7dbe6c033f4910f0c0d3781583689a4d069ae781fb19cfac"}}