{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:K32VPB6CHZNZMQFH62UDNYOK3U","short_pith_number":"pith:K32VPB6C","canonical_record":{"source":{"id":"2509.04540","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-09-04T08:58:37Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"81726bb1e7f51b1ca42cf2a5d3af7318179c80890f723072222bc034c51c7fff","abstract_canon_sha256":"a301f8185f5f691ed3c1435022e31b2b3b58a75292086afb27141f5d44eff0f0"},"schema_version":"1.0"},"canonical_sha256":"56f55787c23e5b9640a7f6a836e1cadd0fd30205862a8787ede4e95d1765176a","source":{"kind":"arxiv","id":"2509.04540","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.04540","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"arxiv_version","alias_value":"2509.04540v4","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.04540","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"pith_short_12","alias_value":"K32VPB6CHZNZ","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"pith_short_16","alias_value":"K32VPB6CHZNZMQFH","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"pith_short_8","alias_value":"K32VPB6C","created_at":"2026-07-22T01:23:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:K32VPB6CHZNZMQFH62UDNYOK3U","target":"record","payload":{"canonical_record":{"source":{"id":"2509.04540","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-09-04T08:58:37Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"81726bb1e7f51b1ca42cf2a5d3af7318179c80890f723072222bc034c51c7fff","abstract_canon_sha256":"a301f8185f5f691ed3c1435022e31b2b3b58a75292086afb27141f5d44eff0f0"},"schema_version":"1.0"},"canonical_sha256":"56f55787c23e5b9640a7f6a836e1cadd0fd30205862a8787ede4e95d1765176a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-22T01:23:31.637617Z","signature_b64":"EgX81NPolIAzDV8rMRG1s91TtSUD/JWpsqDN+90KI6SZDPMgtD4/nYxOOJYsKoU7ZhhWCclUUcgtE/sxxgW1DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56f55787c23e5b9640a7f6a836e1cadd0fd30205862a8787ede4e95d1765176a","last_reissued_at":"2026-07-22T01:23:31.636595Z","signature_status":"signed_v1","first_computed_at":"2026-07-22T01:23:31.636595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.04540","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-22T01:23:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GqofVvU5Ey1NTr02BiUwLFywsZ6m8tPI2S7ZxddgcD0xpBoIfcrK7yyOeZsl1ot3NX30LXRVnSbGYi0WSLwIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T18:43:07.009419Z"},"content_sha256":"29ec41067829a33c259414d0a92ae304a28e2c7f93a0b9ad38636c8b2e421da2","schema_version":"1.0","event_id":"sha256:29ec41067829a33c259414d0a92ae304a28e2c7f93a0b9ad38636c8b2e421da2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:K32VPB6CHZNZMQFH62UDNYOK3U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Trace-Path Integral Formula over Function Fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"An arithmetic path integral over Jacobian ℓ-torsion equals the trace of Frobenius on a Heisenberg group representation, up to sign.","cross_cats":["math-ph","math.MP"],"primary_cat":"math.NT","authors_text":"Yan Yau Cheng","submitted_at":"2025-09-04T08:58:37Z","abstract_excerpt":"We show that an arithmetic path integral over the $\\ell$-torsion of a Jacobian $J[\\ell]$ is equal to the trace of the Frobenius action on a representation of the Heisenberg group $H(J[\\ell])$, up to an explicitly determined sign. This is an arithmetic analogue of trace--path integral formulae which arise in quantum field theory, where path integrals over a space of sections of a fibration over a circle can be expressed as the trace of the monodromy action on a Hilbert space."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"an arithmetic path integral over the ℓ-torsion of a Jacobian J[ℓ] is equal to the trace of the Frobenius action on a representation of the Heisenberg group H(J[ℓ]), up to an explicitly determined sign.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The arithmetic path integral over J[ℓ] is well-defined and the Heisenberg group representation is constructed so that the trace-Frobenius equality holds in the function-field setting.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An equality is shown between an arithmetic path integral over Jacobian ℓ-torsion and the trace of Frobenius on a Heisenberg group representation.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"An arithmetic path integral over Jacobian ℓ-torsion equals the trace of Frobenius on a Heisenberg group representation, up to sign.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"dd285c565520cff609f85c50d512058b60b4179504a25117b2614296320768a0"},"source":{"id":"2509.04540","kind":"arxiv","version":4},"verdict":{"id":"1a22a740-975a-4321-a82a-2067a284bc98","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T19:44:19.606333Z","strongest_claim":"an arithmetic path integral over the ℓ-torsion of a Jacobian J[ℓ] is equal to the trace of the Frobenius action on a representation of the Heisenberg group H(J[ℓ]), up to an explicitly determined sign.","one_line_summary":"An equality is shown between an arithmetic path integral over Jacobian ℓ-torsion and the trace of Frobenius on a Heisenberg group representation.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The arithmetic path integral over J[ℓ] is well-defined and the Heisenberg group representation is constructed so that the trace-Frobenius equality holds in the function-field setting.","pith_extraction_headline":"An arithmetic path integral over Jacobian ℓ-torsion equals the trace of Frobenius on a Heisenberg group representation, up to sign."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.04540/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":2,"snapshot_sha256":"ca1bee62fc5ba556e7bcf2bfaafac8947cb09cd3770235c619f609fd6949653b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"1a22a740-975a-4321-a82a-2067a284bc98"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-22T01:23:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NXL6MgJMjik8t5KeRmk7RkHUMvoCFOqE+l/mJf0izVwlM5/6VJ+VCXg31R8Hh/dQc1gRTzOEAHOG4RI19SsQCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T18:43:07.011717Z"},"content_sha256":"262970767ef6a74d17300dcf355c9230a2c5330ef1a8c586497d859cfaaeace9","schema_version":"1.0","event_id":"sha256:262970767ef6a74d17300dcf355c9230a2c5330ef1a8c586497d859cfaaeace9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K32VPB6CHZNZMQFH62UDNYOK3U/bundle.json","state_url":"https://pith.science/pith/K32VPB6CHZNZMQFH62UDNYOK3U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K32VPB6CHZNZMQFH62UDNYOK3U/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":"2026-08-14T18:43:07Z","links":{"resolver":"https://pith.science/pith/K32VPB6CHZNZMQFH62UDNYOK3U","bundle":"https://pith.science/pith/K32VPB6CHZNZMQFH62UDNYOK3U/bundle.json","state":"https://pith.science/pith/K32VPB6CHZNZMQFH62UDNYOK3U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K32VPB6CHZNZMQFH62UDNYOK3U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:K32VPB6CHZNZMQFH62UDNYOK3U","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":"a301f8185f5f691ed3c1435022e31b2b3b58a75292086afb27141f5d44eff0f0","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-09-04T08:58:37Z","title_canon_sha256":"81726bb1e7f51b1ca42cf2a5d3af7318179c80890f723072222bc034c51c7fff"},"schema_version":"1.0","source":{"id":"2509.04540","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.04540","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"arxiv_version","alias_value":"2509.04540v4","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.04540","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"pith_short_12","alias_value":"K32VPB6CHZNZ","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"pith_short_16","alias_value":"K32VPB6CHZNZMQFH","created_at":"2026-07-22T01:23:31Z"},{"alias_kind":"pith_short_8","alias_value":"K32VPB6C","created_at":"2026-07-22T01:23:31Z"}],"graph_snapshots":[{"event_id":"sha256:262970767ef6a74d17300dcf355c9230a2c5330ef1a8c586497d859cfaaeace9","target":"graph","created_at":"2026-07-22T01:23:31Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"an arithmetic path integral over the ℓ-torsion of a Jacobian J[ℓ] is equal to the trace of the Frobenius action on a representation of the Heisenberg group H(J[ℓ]), up to an explicitly determined sign."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The arithmetic path integral over J[ℓ] is well-defined and the Heisenberg group representation is constructed so that the trace-Frobenius equality holds in the function-field setting."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"An equality is shown between an arithmetic path integral over Jacobian ℓ-torsion and the trace of Frobenius on a Heisenberg group representation."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"An arithmetic path integral over Jacobian ℓ-torsion equals the trace of Frobenius on a Heisenberg group representation, up to sign."}],"snapshot_sha256":"dd285c565520cff609f85c50d512058b60b4179504a25117b2614296320768a0"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ca1bee62fc5ba556e7bcf2bfaafac8947cb09cd3770235c619f609fd6949653b"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2509.04540/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that an arithmetic path integral over the $\\ell$-torsion of a Jacobian $J[\\ell]$ is equal to the trace of the Frobenius action on a representation of the Heisenberg group $H(J[\\ell])$, up to an explicitly determined sign. This is an arithmetic analogue of trace--path integral formulae which arise in quantum field theory, where path integrals over a space of sections of a fibration over a circle can be expressed as the trace of the monodromy action on a Hilbert space.","authors_text":"Yan Yau Cheng","cross_cats":["math-ph","math.MP"],"headline":"An arithmetic path integral over Jacobian ℓ-torsion equals the trace of Frobenius on a Heisenberg group representation, up to sign.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-09-04T08:58:37Z","title":"A Trace-Path Integral Formula over Function Fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.04540","kind":"arxiv","version":4},"verdict":{"created_at":"2026-05-18T19:44:19.606333Z","id":"1a22a740-975a-4321-a82a-2067a284bc98","model_set":{"reader":"grok-4.3"},"one_line_summary":"An equality is shown between an arithmetic path integral over Jacobian ℓ-torsion and the trace of Frobenius on a Heisenberg group representation.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"An arithmetic path integral over Jacobian ℓ-torsion equals the trace of Frobenius on a Heisenberg group representation, up to sign.","strongest_claim":"an arithmetic path integral over the ℓ-torsion of a Jacobian J[ℓ] is equal to the trace of the Frobenius action on a representation of the Heisenberg group H(J[ℓ]), up to an explicitly determined sign.","weakest_assumption":"The arithmetic path integral over J[ℓ] is well-defined and the Heisenberg group representation is constructed so that the trace-Frobenius equality holds in the function-field setting."}},"verdict_id":"1a22a740-975a-4321-a82a-2067a284bc98"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:29ec41067829a33c259414d0a92ae304a28e2c7f93a0b9ad38636c8b2e421da2","target":"record","created_at":"2026-07-22T01:23:31Z","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":"a301f8185f5f691ed3c1435022e31b2b3b58a75292086afb27141f5d44eff0f0","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-09-04T08:58:37Z","title_canon_sha256":"81726bb1e7f51b1ca42cf2a5d3af7318179c80890f723072222bc034c51c7fff"},"schema_version":"1.0","source":{"id":"2509.04540","kind":"arxiv","version":4}},"canonical_sha256":"56f55787c23e5b9640a7f6a836e1cadd0fd30205862a8787ede4e95d1765176a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"56f55787c23e5b9640a7f6a836e1cadd0fd30205862a8787ede4e95d1765176a","first_computed_at":"2026-07-22T01:23:31.636595Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-22T01:23:31.636595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EgX81NPolIAzDV8rMRG1s91TtSUD/JWpsqDN+90KI6SZDPMgtD4/nYxOOJYsKoU7ZhhWCclUUcgtE/sxxgW1DQ==","signature_status":"signed_v1","signed_at":"2026-07-22T01:23:31.637617Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.04540","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29ec41067829a33c259414d0a92ae304a28e2c7f93a0b9ad38636c8b2e421da2","sha256:262970767ef6a74d17300dcf355c9230a2c5330ef1a8c586497d859cfaaeace9"],"state_sha256":"2e471ab3479d3272874713bf49e368043b82ee8af7923df39d60d12bd1005382"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eyFpt8w/L/coOLtOp6hJd8Z8Z9e34LkpJKbqkiP/KHOl6cWojWKMWyXmbu3S0vNytnmnKpfT1AMH5Q56VhNZBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T18:43:07.018109Z","bundle_sha256":"3d1951d2800c5cc7bdb6833e780486d5395d48357d2c4043b718d9573f06eec5"}}