{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:JNNRHCCQLVUF6VCB7JP67LMHWQ","short_pith_number":"pith:JNNRHCCQ","schema_version":"1.0","canonical_sha256":"4b5b1388505d685f5441fa5fefad87b40a88e1262a939da3c2f63c0c8c8c79c0","source":{"kind":"arxiv","id":"2608.01610","version":1},"attestation_state":"computed","paper":{"title":"Conjugation invariants determine the metacommutation permutation only up to relabelling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"cs.GT","authors_text":"Matthew Fried","submitted_at":"2026-08-03T02:35:51Z","abstract_excerpt":"Let $\\mathcal{H}$ be the Hurwitz quaternions, $p$ an odd prime, and $Q \\in \\mathcal{H}$ a prime of norm $q \\neq p$. Metacommutation $PQ = Q'P'$ induces a permutation $\\pi_Q$ of the $p+1$ left-associate classes of primes of norm $p$. Cohn and Kumar compute its sign and fixed-point count, and Leite and Machiavelo its full cycle structure, by formulas depending only on conjugation-invariant data of $Q$ (namely $q$ and $\\mathrm{tr}\\,Q$). We prove this is exactly the boundary of what such invariants can carry: no quantity $I(Q)$ invariant under unit conjugation determines $\\pi_Q$ as a labelled perm"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2608.01610","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2026-08-03T02:35:51Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"0690bc98026d2b17f58f1543a75aeee283a049aecc95bbd81846ee014083ae38","abstract_canon_sha256":"4e439790d9d911cff030c46b09405fc554548fca278fc7ccede7accda83d7058"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:06:07.427905Z","signature_b64":"dmfz88aBQRn+oJyJYDJZ7hu59wttW4J1UjNuStfkn/oKR5hSgV4SNFxPDDLYuBQGh4tJd261BjG88M+WVskVCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b5b1388505d685f5441fa5fefad87b40a88e1262a939da3c2f63c0c8c8c79c0","last_reissued_at":"2026-08-04T02:06:07.426068Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:06:07.426068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Conjugation invariants determine the metacommutation permutation only up to relabelling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"cs.GT","authors_text":"Matthew Fried","submitted_at":"2026-08-03T02:35:51Z","abstract_excerpt":"Let $\\mathcal{H}$ be the Hurwitz quaternions, $p$ an odd prime, and $Q \\in \\mathcal{H}$ a prime of norm $q \\neq p$. Metacommutation $PQ = Q'P'$ induces a permutation $\\pi_Q$ of the $p+1$ left-associate classes of primes of norm $p$. Cohn and Kumar compute its sign and fixed-point count, and Leite and Machiavelo its full cycle structure, by formulas depending only on conjugation-invariant data of $Q$ (namely $q$ and $\\mathrm{tr}\\,Q$). We prove this is exactly the boundary of what such invariants can carry: no quantity $I(Q)$ invariant under unit conjugation determines $\\pi_Q$ as a labelled perm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01610","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/2608.01610/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2608.01610","created_at":"2026-08-04T02:06:07.427787+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.01610v1","created_at":"2026-08-04T02:06:07.427787+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01610","created_at":"2026-08-04T02:06:07.427787+00:00"},{"alias_kind":"pith_short_12","alias_value":"JNNRHCCQLVUF","created_at":"2026-08-04T02:06:07.427787+00:00"},{"alias_kind":"pith_short_16","alias_value":"JNNRHCCQLVUF6VCB","created_at":"2026-08-04T02:06:07.427787+00:00"},{"alias_kind":"pith_short_8","alias_value":"JNNRHCCQ","created_at":"2026-08-04T02:06:07.427787+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ","json":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ.json","graph_json":"https://pith.science/api/pith-number/JNNRHCCQLVUF6VCB7JP67LMHWQ/graph.json","events_json":"https://pith.science/api/pith-number/JNNRHCCQLVUF6VCB7JP67LMHWQ/events.json","paper":"https://pith.science/paper/JNNRHCCQ"},"agent_actions":{"view_html":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ","download_json":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ.json","view_paper":"https://pith.science/paper/JNNRHCCQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.01610&json=true","fetch_graph":"https://pith.science/api/pith-number/JNNRHCCQLVUF6VCB7JP67LMHWQ/graph.json","fetch_events":"https://pith.science/api/pith-number/JNNRHCCQLVUF6VCB7JP67LMHWQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/action/storage_attestation","attest_author":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/action/author_attestation","sign_citation":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/action/citation_signature","submit_replication":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/action/replication_record"}},"created_at":"2026-08-04T02:06:07.427787+00:00","updated_at":"2026-08-04T02:06:07.427787+00:00"}