{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:JNNRHCCQLVUF6VCB7JP67LMHWQ","short_pith_number":"pith:JNNRHCCQ","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"},"canonical_sha256":"4b5b1388505d685f5441fa5fefad87b40a88e1262a939da3c2f63c0c8c8c79c0","source":{"kind":"arxiv","id":"2608.01610","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.01610","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"arxiv_version","alias_value":"2608.01610v1","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01610","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"pith_short_12","alias_value":"JNNRHCCQLVUF","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"pith_short_16","alias_value":"JNNRHCCQLVUF6VCB","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"pith_short_8","alias_value":"JNNRHCCQ","created_at":"2026-08-04T02:06:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:JNNRHCCQLVUF6VCB7JP67LMHWQ","target":"record","payload":{"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"},"canonical_sha256":"4b5b1388505d685f5441fa5fefad87b40a88e1262a939da3c2f63c0c8c8c79c0","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"},"source_kind":"arxiv","source_id":"2608.01610","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-08-04T02:06:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mzIjg2pK8w55K15bu7JoxXwvObcZM253ni5+gGdZ2VqcfwXU/CGpTTE+OBb1cuwbQA5M30BrWIE6W+7IUok5Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T09:28:53.505160Z"},"content_sha256":"380feccd611f1278db6e26d2f48a3b5783b817c0e7c1459835c3d1b96727863b","schema_version":"1.0","event_id":"sha256:380feccd611f1278db6e26d2f48a3b5783b817c0e7c1459835c3d1b96727863b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:JNNRHCCQLVUF6VCB7JP67LMHWQ","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-08-04T02:06:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1tzV3sx2LgAQ4XE16K39wLw2BViNSMmGmO86r5htCIP0rRS9H2V7gj4oNeQYrR5pT1PkBjen/5w2CrX+WOcgAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T09:28:53.505678Z"},"content_sha256":"4e1e44366445341b4e998f1a8a154ac500da3c1c97ac69e5b57616548cf95400","schema_version":"1.0","event_id":"sha256:4e1e44366445341b4e998f1a8a154ac500da3c1c97ac69e5b57616548cf95400"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/bundle.json","state_url":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/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":"2026-08-05T09:28:53Z","links":{"resolver":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ","bundle":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/bundle.json","state":"https://pith.science/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JNNRHCCQLVUF6VCB7JP67LMHWQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:JNNRHCCQLVUF6VCB7JP67LMHWQ","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":"4e439790d9d911cff030c46b09405fc554548fca278fc7ccede7accda83d7058","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2026-08-03T02:35:51Z","title_canon_sha256":"0690bc98026d2b17f58f1543a75aeee283a049aecc95bbd81846ee014083ae38"},"schema_version":"1.0","source":{"id":"2608.01610","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.01610","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"arxiv_version","alias_value":"2608.01610v1","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01610","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"pith_short_12","alias_value":"JNNRHCCQLVUF","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"pith_short_16","alias_value":"JNNRHCCQLVUF6VCB","created_at":"2026-08-04T02:06:07Z"},{"alias_kind":"pith_short_8","alias_value":"JNNRHCCQ","created_at":"2026-08-04T02:06:07Z"}],"graph_snapshots":[{"event_id":"sha256:4e1e44366445341b4e998f1a8a154ac500da3c1c97ac69e5b57616548cf95400","target":"graph","created_at":"2026-08-04T02:06:07Z","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/2608.01610/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Matthew Fried","cross_cats":["math.GR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2026-08-03T02:35:51Z","title":"Conjugation invariants determine the metacommutation permutation only up to relabelling"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01610","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":"sha256:380feccd611f1278db6e26d2f48a3b5783b817c0e7c1459835c3d1b96727863b","target":"record","created_at":"2026-08-04T02:06:07Z","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":"4e439790d9d911cff030c46b09405fc554548fca278fc7ccede7accda83d7058","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2026-08-03T02:35:51Z","title_canon_sha256":"0690bc98026d2b17f58f1543a75aeee283a049aecc95bbd81846ee014083ae38"},"schema_version":"1.0","source":{"id":"2608.01610","kind":"arxiv","version":1}},"canonical_sha256":"4b5b1388505d685f5441fa5fefad87b40a88e1262a939da3c2f63c0c8c8c79c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b5b1388505d685f5441fa5fefad87b40a88e1262a939da3c2f63c0c8c8c79c0","first_computed_at":"2026-08-04T02:06:07.426068Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-04T02:06:07.426068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dmfz88aBQRn+oJyJYDJZ7hu59wttW4J1UjNuStfkn/oKR5hSgV4SNFxPDDLYuBQGh4tJd261BjG88M+WVskVCA==","signature_status":"signed_v1","signed_at":"2026-08-04T02:06:07.427905Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.01610","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:380feccd611f1278db6e26d2f48a3b5783b817c0e7c1459835c3d1b96727863b","sha256:4e1e44366445341b4e998f1a8a154ac500da3c1c97ac69e5b57616548cf95400"],"state_sha256":"7231f524ee179d867c3b58361fd2bc29cc52debfc5903f3fbed4af0b50611cb7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hBTn7Er42OlajURZ7CeHKt5AaW2SLyl4UehgGC2F6+/5jFteMxU7KazzGbJ/TdACDhVv+tv4eh5ZlZJZ8bt/Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T09:28:53.509182Z","bundle_sha256":"de181563ca742d6403c3bf731316cdf72b28dbd984be90f08021d652273c1a34"}}