{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:22WNUW7M2D5DM2MWMUNMCHL5CR","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":"0f433ac6e1e30655d090c8ff90cf942bd13903e4554cf9878bb6c513e8051ecb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-27T03:52:52Z","title_canon_sha256":"3f2df29a854835e5c93a7943c027f9657c279fb4fe3d8aa036969fd429727e60"},"schema_version":"1.0","source":{"id":"2606.28718","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.28718","created_at":"2026-06-30T01:16:49Z"},{"alias_kind":"arxiv_version","alias_value":"2606.28718v1","created_at":"2026-06-30T01:16:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.28718","created_at":"2026-06-30T01:16:49Z"},{"alias_kind":"pith_short_12","alias_value":"22WNUW7M2D5D","created_at":"2026-06-30T01:16:49Z"},{"alias_kind":"pith_short_16","alias_value":"22WNUW7M2D5DM2MW","created_at":"2026-06-30T01:16:49Z"},{"alias_kind":"pith_short_8","alias_value":"22WNUW7M","created_at":"2026-06-30T01:16:49Z"}],"graph_snapshots":[{"event_id":"sha256:441d47f310cb796378577e910eecdf5215fb113a1f6e0f17ac5bb62685799035","target":"graph","created_at":"2026-06-30T01:16:49Z","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/2606.28718/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $T(x)=\\prod_{k=0}^{\\infty}(1-x^{2^k})$ be the generating function of the Thue--Morse sequence, and write $T(x)^m=\\sum_{n\\geq 0}t_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: \\[ \\nu_2\\bigl(t_5(4n+j)\\bigr)\n  =4\\Bigl\\lceil\\tfrac{\\nu_2(n+1)}{2}\\Bigr\\rceil-\\bigl(\\nu_2(n+1)\\bmod 2\\bigr), \\quad j\\in\\{0,1,2,3\\}, \\] \\[ \\nu_2\\bigl(t_9(8n+j)\\bigr)\n  =5\\Bigl\\lceil\\tfrac{\\nu_2(n+1)}{2}\\Bigr\\rceil-2\\bigl(\\nu_2(n+1)\\bmod 2\\bigr), \\quad j\\in\\{0,1,\\ldots,7\\}. \\] These formulas confirm Conjecture~5.2 of Gawron--Miska--Ulas~\\cite{ga} for $m=5$ and $","authors_text":"Xinping Wang, Zhao Shen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-27T03:52:52Z","title":"2-adic Valuations of Coefficients of the Fifth and Ninth Powers of the Thue--Morse Generating Function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28718","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":"sha256:6f5f6648736b14f3b23b2844a31a1accb21cdc8aabb68f0370490ef3bacd173a","target":"record","created_at":"2026-06-30T01:16:49Z","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":"0f433ac6e1e30655d090c8ff90cf942bd13903e4554cf9878bb6c513e8051ecb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-27T03:52:52Z","title_canon_sha256":"3f2df29a854835e5c93a7943c027f9657c279fb4fe3d8aa036969fd429727e60"},"schema_version":"1.0","source":{"id":"2606.28718","kind":"arxiv","version":1}},"canonical_sha256":"d6acda5becd0fa366996651ac11d7d1448277dba3a22a7f43e769866d29d4ef2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6acda5becd0fa366996651ac11d7d1448277dba3a22a7f43e769866d29d4ef2","first_computed_at":"2026-06-30T01:16:49.106061Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-30T01:16:49.106061Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/vwwQknxzpFUMGVJ5XBWbUWvPrbxkQdn+w5vaFylZam28uV0X6Lecmr7UYE92l0wmyZZtfm4utPUK+nZXZD1BQ==","signature_status":"signed_v1","signed_at":"2026-06-30T01:16:49.106775Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.28718","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6f5f6648736b14f3b23b2844a31a1accb21cdc8aabb68f0370490ef3bacd173a","sha256:441d47f310cb796378577e910eecdf5215fb113a1f6e0f17ac5bb62685799035"],"state_sha256":"c34263c65ea6f13b8b4e7cf90f88ad4cea0de267609ba7cbb13caa124fd1d7b2"}