{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5STATZAPRHCXFIMFAUCXHTB33U","short_pith_number":"pith:5STATZAP","canonical_record":{"source":{"id":"2604.27531","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-04-30T07:34:00Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"45e6b244e48e0861fa5d368995bba53808f264c178d4ea93805cb7670ed5a48f","abstract_canon_sha256":"810a236a128a37c70d5804461ad6f2f65003e2ce5f55ff7fabda0e2ec26c58f6"},"schema_version":"1.0"},"canonical_sha256":"eca609e40f89c572a185050573cc3bdd2bd6360a1765e8bdc5e96cafcc70b5dd","source":{"kind":"arxiv","id":"2604.27531","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.27531","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"arxiv_version","alias_value":"2604.27531v2","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.27531","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"pith_short_12","alias_value":"5STATZAPRHCX","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"pith_short_16","alias_value":"5STATZAPRHCXFIMF","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"pith_short_8","alias_value":"5STATZAP","created_at":"2026-06-01T01:02:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5STATZAPRHCXFIMFAUCXHTB33U","target":"record","payload":{"canonical_record":{"source":{"id":"2604.27531","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-04-30T07:34:00Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"45e6b244e48e0861fa5d368995bba53808f264c178d4ea93805cb7670ed5a48f","abstract_canon_sha256":"810a236a128a37c70d5804461ad6f2f65003e2ce5f55ff7fabda0e2ec26c58f6"},"schema_version":"1.0"},"canonical_sha256":"eca609e40f89c572a185050573cc3bdd2bd6360a1765e8bdc5e96cafcc70b5dd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T01:02:40.848856Z","signature_b64":"xmbIJ2gGKHMy7JpW59EULc/pgrHmd0Zz7duFhyrQb7MNfpXxUaYIgyptvIUyFGJ4pEGYkUg7O6TIQoIrg+DJDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eca609e40f89c572a185050573cc3bdd2bd6360a1765e8bdc5e96cafcc70b5dd","last_reissued_at":"2026-06-01T01:02:40.848072Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T01:02:40.848072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.27531","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-01T01:02:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BkJakCR24pF0dDXcCsyTZs8NPw74RJ+TCGNk+JZnCGEuKBB16l8jHyRATzAu4CYsUKyBN6762oH5mGeZczDgBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T06:41:39.250058Z"},"content_sha256":"44a806fe830248351513699a020409f05ee75257d509a0109b73d39f7ca24ec1","schema_version":"1.0","event_id":"sha256:44a806fe830248351513699a020409f05ee75257d509a0109b73d39f7ca24ec1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5STATZAPRHCXFIMFAUCXHTB33U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\mathbb{K}$-framings and $\\mathbb{K}$-quadratic forms on surfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit.","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Nariya Kawazumi","submitted_at":"2026-04-30T07:34:00Z","abstract_excerpt":"We introduce the notions of $\\mathbb{K}$-framings, based $\\mathbb{K}$-framings and relative $\\mathbb{K}$-framings of a compact connected oriented surface $\\Sigma$ for any commutative ring $\\mathbb{K}$ with unit, and a map which maps a based loop on $\\Sigma$ to a homology class of its unit tangent bundle $U\\Sigma$, which recovers Johnson's lifting in the case $\\mathbb{K} = \\mathbb{Z}/2$. This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring $\\mathbb{K}$ with unit. If the genus of $\\Sigma$ is positive, we have a bijection"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring K with unit. If the genus of Σ is positive, we have a bijection between the set of K-framings and the set of some twisted cocycles of the mapping class group of the surface Σ.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The constructions of K-framings and the stated bijection with twisted cocycles of the mapping class group are assumed to hold for every commutative ring K with unit on any compact connected oriented surface of positive genus (and the relation to the extended first Johnson homomorphism when the boundary is non-empty and connected).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"K-framings generalize Johnson's quadratic form-spin structure correspondence to any commutative ring K, yielding bijections with twisted cocycles of the mapping class group for positive-genus surfaces.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a38e03d42dd77dc213c39393106b5056363ebde921ac22da28938a5d6d91072e"},"source":{"id":"2604.27531","kind":"arxiv","version":2},"verdict":{"id":"f7b4ea68-3718-407e-ae7f-8b72ffd8ec3d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T07:48:28.808013Z","strongest_claim":"This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring K with unit. If the genus of Σ is positive, we have a bijection between the set of K-framings and the set of some twisted cocycles of the mapping class group of the surface Σ.","one_line_summary":"K-framings generalize Johnson's quadratic form-spin structure correspondence to any commutative ring K, yielding bijections with twisted cocycles of the mapping class group for positive-genus surfaces.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The constructions of K-framings and the stated bijection with twisted cocycles of the mapping class group are assumed to hold for every commutative ring K with unit on any compact connected oriented surface of positive genus (and the relation to the extended first Johnson homomorphism when the boundary is non-empty and connected).","pith_extraction_headline":"K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.27531/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T22:34:05.035571Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:12:50.537610Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"6104c32cd57796fde7f31ee990d0ebb753b861b9fd9b331e55dc59514737ffd7"},"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":"f7b4ea68-3718-407e-ae7f-8b72ffd8ec3d"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-01T01:02:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h6yAhT/wuKArxrMZ3448F9bCMWnBfiJAeYhUnwgrh9EfNqPAT9KBgJvdey8KMgA+12kyeOAi4X0yP83WDFh3Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T06:41:39.251119Z"},"content_sha256":"084c43462398efb43fe7c2a5d1fccd1377b32164024f3274ad0745899d1767e2","schema_version":"1.0","event_id":"sha256:084c43462398efb43fe7c2a5d1fccd1377b32164024f3274ad0745899d1767e2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5STATZAPRHCXFIMFAUCXHTB33U/bundle.json","state_url":"https://pith.science/pith/5STATZAPRHCXFIMFAUCXHTB33U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5STATZAPRHCXFIMFAUCXHTB33U/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":"2026-06-06T06:41:39Z","links":{"resolver":"https://pith.science/pith/5STATZAPRHCXFIMFAUCXHTB33U","bundle":"https://pith.science/pith/5STATZAPRHCXFIMFAUCXHTB33U/bundle.json","state":"https://pith.science/pith/5STATZAPRHCXFIMFAUCXHTB33U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5STATZAPRHCXFIMFAUCXHTB33U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5STATZAPRHCXFIMFAUCXHTB33U","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":"810a236a128a37c70d5804461ad6f2f65003e2ce5f55ff7fabda0e2ec26c58f6","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-04-30T07:34:00Z","title_canon_sha256":"45e6b244e48e0861fa5d368995bba53808f264c178d4ea93805cb7670ed5a48f"},"schema_version":"1.0","source":{"id":"2604.27531","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.27531","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"arxiv_version","alias_value":"2604.27531v2","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.27531","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"pith_short_12","alias_value":"5STATZAPRHCX","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"pith_short_16","alias_value":"5STATZAPRHCXFIMF","created_at":"2026-06-01T01:02:40Z"},{"alias_kind":"pith_short_8","alias_value":"5STATZAP","created_at":"2026-06-01T01:02:40Z"}],"graph_snapshots":[{"event_id":"sha256:084c43462398efb43fe7c2a5d1fccd1377b32164024f3274ad0745899d1767e2","target":"graph","created_at":"2026-06-01T01:02:40Z","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":"This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring K with unit. If the genus of Σ is positive, we have a bijection between the set of K-framings and the set of some twisted cocycles of the mapping class group of the surface Σ."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The constructions of K-framings and the stated bijection with twisted cocycles of the mapping class group are assumed to hold for every commutative ring K with unit on any compact connected oriented surface of positive genus (and the relation to the extended first Johnson homomorphism when the boundary is non-empty and connected)."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"K-framings generalize Johnson's quadratic form-spin structure correspondence to any commutative ring K, yielding bijections with twisted cocycles of the mapping class group for positive-genus surfaces."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit."}],"snapshot_sha256":"a38e03d42dd77dc213c39393106b5056363ebde921ac22da28938a5d6d91072e"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-20T22:34:05.035571Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T19:12:50.537610Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.27531/integrity.json","findings":[],"snapshot_sha256":"6104c32cd57796fde7f31ee990d0ebb753b861b9fd9b331e55dc59514737ffd7","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce the notions of $\\mathbb{K}$-framings, based $\\mathbb{K}$-framings and relative $\\mathbb{K}$-framings of a compact connected oriented surface $\\Sigma$ for any commutative ring $\\mathbb{K}$ with unit, and a map which maps a based loop on $\\Sigma$ to a homology class of its unit tangent bundle $U\\Sigma$, which recovers Johnson's lifting in the case $\\mathbb{K} = \\mathbb{Z}/2$. This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring $\\mathbb{K}$ with unit. If the genus of $\\Sigma$ is positive, we have a bijection","authors_text":"Nariya Kawazumi","cross_cats":["math.AT"],"headline":"K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-04-30T07:34:00Z","title":"$\\mathbb{K}$-framings and $\\mathbb{K}$-quadratic forms on surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.27531","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-07T07:48:28.808013Z","id":"f7b4ea68-3718-407e-ae7f-8b72ffd8ec3d","model_set":{"reader":"grok-4.3"},"one_line_summary":"K-framings generalize Johnson's quadratic form-spin structure correspondence to any commutative ring K, yielding bijections with twisted cocycles of the mapping class group for positive-genus surfaces.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit.","strongest_claim":"This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring K with unit. If the genus of Σ is positive, we have a bijection between the set of K-framings and the set of some twisted cocycles of the mapping class group of the surface Σ.","weakest_assumption":"The constructions of K-framings and the stated bijection with twisted cocycles of the mapping class group are assumed to hold for every commutative ring K with unit on any compact connected oriented surface of positive genus (and the relation to the extended first Johnson homomorphism when the boundary is non-empty and connected)."}},"verdict_id":"f7b4ea68-3718-407e-ae7f-8b72ffd8ec3d"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:44a806fe830248351513699a020409f05ee75257d509a0109b73d39f7ca24ec1","target":"record","created_at":"2026-06-01T01:02:40Z","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":"810a236a128a37c70d5804461ad6f2f65003e2ce5f55ff7fabda0e2ec26c58f6","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-04-30T07:34:00Z","title_canon_sha256":"45e6b244e48e0861fa5d368995bba53808f264c178d4ea93805cb7670ed5a48f"},"schema_version":"1.0","source":{"id":"2604.27531","kind":"arxiv","version":2}},"canonical_sha256":"eca609e40f89c572a185050573cc3bdd2bd6360a1765e8bdc5e96cafcc70b5dd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eca609e40f89c572a185050573cc3bdd2bd6360a1765e8bdc5e96cafcc70b5dd","first_computed_at":"2026-06-01T01:02:40.848072Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-01T01:02:40.848072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xmbIJ2gGKHMy7JpW59EULc/pgrHmd0Zz7duFhyrQb7MNfpXxUaYIgyptvIUyFGJ4pEGYkUg7O6TIQoIrg+DJDA==","signature_status":"signed_v1","signed_at":"2026-06-01T01:02:40.848856Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.27531","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:44a806fe830248351513699a020409f05ee75257d509a0109b73d39f7ca24ec1","sha256:084c43462398efb43fe7c2a5d1fccd1377b32164024f3274ad0745899d1767e2"],"state_sha256":"4d6f0c44c438b928a158761393d5af2e067089b2a60ac8c3f7cf4345231a6e68"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qy6qxBt2r/ZjoCi/lWh3LSMyeG1kGwg8apc+nYMxTkDN6xNjxRg93zHakQEv/V1Pepin7jVXsjhnnv6UFt3nDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T06:41:39.255875Z","bundle_sha256":"e525f0136be4cf1c70aca48ab1438fdae20afd970b19ef2f725f39a932e2ef82"}}