{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:KFM7DQOEAIFNKGZOUDMZUKWJGR","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":"095c7ed0a16a74461d8f2664c1f2e74d7ae883fdb8e24e60f6a15cd455421636","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-09-29T16:56:20Z","title_canon_sha256":"630cd972eb326de4406d606473890ae1b5ec0004e9b7d0cd2b84f291e81b1ffa"},"schema_version":"1.0","source":{"id":"2509.25038","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.25038","created_at":"2026-06-04T01:08:32Z"},{"alias_kind":"arxiv_version","alias_value":"2509.25038v2","created_at":"2026-06-04T01:08:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.25038","created_at":"2026-06-04T01:08:32Z"},{"alias_kind":"pith_short_12","alias_value":"KFM7DQOEAIFN","created_at":"2026-06-04T01:08:32Z"},{"alias_kind":"pith_short_16","alias_value":"KFM7DQOEAIFNKGZO","created_at":"2026-06-04T01:08:32Z"},{"alias_kind":"pith_short_8","alias_value":"KFM7DQOE","created_at":"2026-06-04T01:08:32Z"}],"graph_snapshots":[{"event_id":"sha256:699f360c635089f2f4af2e62d7942fedf74d736e0c88364fe910c6a322af2692","target":"graph","created_at":"2026-06-04T01:08:32Z","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/2509.25038/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop a Witt--Hadamard calculus for Euler products that unifies the classical Gauss congruences with their modern refinement, the Dold congruences. Within this framework we prove \\emph{norm descent}: Dold congruences are functorial under finite extensions and preserved by prime--ideal norms $N_{K/\\mathbb{Q}}$, yielding integer ghosts from algebraic ones. We extend the theory from $\\mathbb{Z}$ to Dedekind domains, and show that integrality is stable under both Hadamard and Witt products. Two rigidity theorems lie at the core: a \\emph{cyclotomic residues theorem}, asserting that if the loga","authors_text":"Hartosh Singh Bal","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-09-29T16:56:20Z","title":"Dold-Gauss Congruences, Norm Descent, and Rational Rigidity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.25038","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":"sha256:1aac3098973ded621d08aea09e13f8070aa28e5f1205b608e91b0651ffe0391d","target":"record","created_at":"2026-06-04T01:08:32Z","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":"095c7ed0a16a74461d8f2664c1f2e74d7ae883fdb8e24e60f6a15cd455421636","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-09-29T16:56:20Z","title_canon_sha256":"630cd972eb326de4406d606473890ae1b5ec0004e9b7d0cd2b84f291e81b1ffa"},"schema_version":"1.0","source":{"id":"2509.25038","kind":"arxiv","version":2}},"canonical_sha256":"5159f1c1c4020ad51b2ea0d99a2ac93452007b17e79f36ef73d9d3f884833c6e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5159f1c1c4020ad51b2ea0d99a2ac93452007b17e79f36ef73d9d3f884833c6e","first_computed_at":"2026-06-04T01:08:32.531474Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T01:08:32.531474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rTIYvwfvZsSmXCYdxa5bhlWnYO3NaJC0aaeGBnYtFZruZRzaVABmNYSeEOk0RQIBZRSsyAEjkLuBNxy0OQjPBQ==","signature_status":"signed_v1","signed_at":"2026-06-04T01:08:32.532428Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.25038","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1aac3098973ded621d08aea09e13f8070aa28e5f1205b608e91b0651ffe0391d","sha256:699f360c635089f2f4af2e62d7942fedf74d736e0c88364fe910c6a322af2692"],"state_sha256":"48079516fde93063f5c5943b5da4e2641259f78a00a576beeb57a8aae2f4cbc9"}