{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:I5AYFWN433MMO3W7QUBS2TNXAG","short_pith_number":"pith:I5AYFWN4","canonical_record":{"source":{"id":"2607.24462","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2026-07-27T14:03:37Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"fcf0dc947ce7b8f4fe760ebd5dec409ebee17f5d303a123b0873a6533fb5fe2d","abstract_canon_sha256":"b0ea5dcc1988a9a094ade5ef78b9372cbec63e1f32a4afd23c9cdcdd4b462b9c"},"schema_version":"1.0"},"canonical_sha256":"474182d9bcded8c76edf85032d4db701b3b5641d416c08fc2ab2cc188593f33f","source":{"kind":"arxiv","id":"2607.24462","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.24462","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"arxiv_version","alias_value":"2607.24462v1","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.24462","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"pith_short_12","alias_value":"I5AYFWN433MM","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"pith_short_16","alias_value":"I5AYFWN433MMO3W7","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"pith_short_8","alias_value":"I5AYFWN4","created_at":"2026-07-28T02:24:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:I5AYFWN433MMO3W7QUBS2TNXAG","target":"record","payload":{"canonical_record":{"source":{"id":"2607.24462","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2026-07-27T14:03:37Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"fcf0dc947ce7b8f4fe760ebd5dec409ebee17f5d303a123b0873a6533fb5fe2d","abstract_canon_sha256":"b0ea5dcc1988a9a094ade5ef78b9372cbec63e1f32a4afd23c9cdcdd4b462b9c"},"schema_version":"1.0"},"canonical_sha256":"474182d9bcded8c76edf85032d4db701b3b5641d416c08fc2ab2cc188593f33f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T02:24:05.043641Z","signature_b64":"bRvHamHw/drC6Cws8geNjbAY+ha+M3tvahyac37sl6hBW+B119GvRbxcB7ukHSvh96xRlkb5vp67SM+PDDpUBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"474182d9bcded8c76edf85032d4db701b3b5641d416c08fc2ab2cc188593f33f","last_reissued_at":"2026-07-28T02:24:05.042807Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T02:24:05.042807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.24462","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":"2026-07-28T02:24:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8+MzgyfsQ5xqS4kDyh0QbdWmaNf5RANYQgBRFtnLjssLzOt0BBed/fetHW7JGFrVo/NUxBvKuCjRcNf9NCXrBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T08:08:22.928419Z"},"content_sha256":"386c1f3694ac17efe8c2e3869586cab0a1f541c5207c9b98b6b4ff42e2c45d0a","schema_version":"1.0","event_id":"sha256:386c1f3694ac17efe8c2e3869586cab0a1f541c5207c9b98b6b4ff42e2c45d0a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:I5AYFWN433MMO3W7QUBS2TNXAG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Realisability of Twisted Homology","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Baylee Schutte, Mark Grant, Michael Jung","submitted_at":"2026-07-27T14:03:37Z","abstract_excerpt":"We discuss the question of when a homology or cohomology class with twisted integer coefficients of a manifold $X$ is realised by a submanifold. While this question is classical in nature, providing an answer requires relatively modern techniques from parametrised homotopy theory. More specifically, we introduce cobordism classes twisted by a coefficient system and then define a twisted Thom space $\\operatorname{M^\\mathrm{tw}O}(n)$ over $\\operatorname{BO}(1)$, which serves as the classifying object for this cobordism theory under a twisted Pontryagin-Thom construction. As a result, a twisted h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.24462","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/2607.24462/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":"2026-07-28T02:24:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tkKtKqkxT5qc7d6fYp3GpOnqlavnUi1M2xq9s21febEGmfyv8/w7q9etf4nTDjKaWE4H3DEpzMJUSYhxQTDODw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T08:08:22.952900Z"},"content_sha256":"539c89d64decb5839da447023b18fda2141733d50c1421180f7e06276f8ba8c3","schema_version":"1.0","event_id":"sha256:539c89d64decb5839da447023b18fda2141733d50c1421180f7e06276f8ba8c3"},{"event_type":"integrity_finding","subject_pith_number":"pith:2026:I5AYFWN433MMO3W7QUBS2TNXAG","target":"integrity","payload":{"note":"DOI is split by whitespace or line breaks in the printed bibliography. Reconstructed DOI 10.1090/surv/132 resolves to 'Parametrized Homotopy Theory'. A reader following the printed text alone cannot reach it.","snippet":"J. P. May and J. Sigurdsson.Parametrized Homotopy Theory. Vol. 132. Mathematical Surveys and Monographs. American Mathematical Society, 2006.doi:10.1090/surv/ 132","arxiv_id":"2607.24462","detector":"doi_compliance","evidence":{"ref_index":22,"verdict_class":"incontrovertible","resolved_title":"Parametrized Homotopy Theory","printed_excerpt":"10.1090/surv/","reconstructed_doi":"10.1090/surv/132"},"severity":"advisory","ref_index":22,"audited_at":"2026-07-31T14:41:44.538298Z","event_type":"pith.integrity.v1","detected_doi":"10.1090/surv/132","detector_url":"https://pith.science/pith-integrity-protocol#doi_compliance","external_url":null,"finding_type":"recoverable_identifier","evidence_hash":"78cfb7ed17321d284e37de98185ee1ff51c2f9fc555af6425256143b45ba7615","paper_version":1,"verdict_class":"incontrovertible","resolved_title":"Parametrized Homotopy Theory","detector_version":"1.1.0","detected_arxiv_id":null,"integrity_event_id":14416,"payload_sha256":"f9260556c49b15998cd51d542bcf8e12c77ab566ce0c758537f3a17039a48aa5","signature_b64":"Nr0fQ+NWXG94wDi7fRydNQDrpcCdkaw6/6Wy5N/m/GaWtF4w9Pnt8vi69MzTWwixdi+8YZtDfgCrUPpJfbiDDQ==","signing_key_id":"pith-v1-2026-05"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-31T14:46:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OLEVD1dI+suo2FOF7/AxlZ9LRWfXDo3U0mwL6wu3z7Ky3WAlONkI283L4CojGJVrWrXkiDAohUM4TOStEosABw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T08:08:23.038468Z"},"content_sha256":"a4347f82641eacc5ed5dd651b89122d6468afbc86c7838e67569722a68bba264","schema_version":"1.0","event_id":"sha256:a4347f82641eacc5ed5dd651b89122d6468afbc86c7838e67569722a68bba264"},{"event_type":"integrity_finding","subject_pith_number":"pith:2026:I5AYFWN433MMO3W7QUBS2TNXAG","target":"integrity","payload":{"note":"DOI in the printed bibliography is fragmented by whitespace or line breaks. A longer candidate (10.4310/ACTA.2025.n235.n2.a1) was visible in the surrounding text but could not be confirmed against doi.org as printed.","snippet":"B. Calmès et al. “Hermitian K-theory for stable∞-categories. II: Cobordism categories and additivity”. In:Acta Mathematica235.2 (2025), pp. 149–400.doi: 10.4310/ACTA. 2025.n235.n2.a1","arxiv_id":"2607.24462","detector":"doi_compliance","evidence":{"ref_index":7,"verdict_class":"incontrovertible","resolved_title":null,"printed_excerpt":"10.4310/acta","reconstructed_doi":"10.4310/ACTA.2025.n235.n2.a1"},"severity":"advisory","ref_index":7,"audited_at":"2026-07-31T14:41:44.538298Z","event_type":"pith.integrity.v1","detected_doi":"10.4310/ACTA.2025.n235.n2.a1","detector_url":"https://pith.science/pith-integrity-protocol#doi_compliance","external_url":null,"finding_type":"recoverable_identifier","evidence_hash":"4978e613722b4107ceed7b56e9b6aea84d11068f5bd84c6c24dbb2cfaece3dc9","paper_version":1,"verdict_class":"incontrovertible","resolved_title":null,"detector_version":"1.1.0","detected_arxiv_id":null,"integrity_event_id":14415,"payload_sha256":"e1f5d584a2737572e00633b3d24658e7456efa77f3a68f8536f4e03b022c313a","signature_b64":"UtnOrcJTU0cV/pvgqpWRZucNkUogWqrfcH3Zed/zoAhH4FsKLy+Rp9dvCs7zCWR/FvLPO5iQDlOBG90MhnSGCA==","signing_key_id":"pith-v1-2026-05"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-31T14:46:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6dxjRJ+zUh0cMk2sdBLb2JS1On4O8QxJ9GKQ2vlAQCw8zOahcNdibvguuVdgDb73xl76Wkm3FOLXMdE8jomlCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T08:08:23.039484Z"},"content_sha256":"5064811aa2d58395e0b227f30dfb64ec46853d209842fe3746dce35bc313df41","schema_version":"1.0","event_id":"sha256:5064811aa2d58395e0b227f30dfb64ec46853d209842fe3746dce35bc313df41"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I5AYFWN433MMO3W7QUBS2TNXAG/bundle.json","state_url":"https://pith.science/pith/I5AYFWN433MMO3W7QUBS2TNXAG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I5AYFWN433MMO3W7QUBS2TNXAG/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":"2026-08-17T08:08:23Z","links":{"resolver":"https://pith.science/pith/I5AYFWN433MMO3W7QUBS2TNXAG","bundle":"https://pith.science/pith/I5AYFWN433MMO3W7QUBS2TNXAG/bundle.json","state":"https://pith.science/pith/I5AYFWN433MMO3W7QUBS2TNXAG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I5AYFWN433MMO3W7QUBS2TNXAG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:I5AYFWN433MMO3W7QUBS2TNXAG","merge_version":"pith-open-graph-merge-v1","event_count":4,"valid_event_count":4,"invalid_event_count":0,"equivocation_count":1,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b0ea5dcc1988a9a094ade5ef78b9372cbec63e1f32a4afd23c9cdcdd4b462b9c","cross_cats_sorted":["math.GT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2026-07-27T14:03:37Z","title_canon_sha256":"fcf0dc947ce7b8f4fe760ebd5dec409ebee17f5d303a123b0873a6533fb5fe2d"},"schema_version":"1.0","source":{"id":"2607.24462","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.24462","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"arxiv_version","alias_value":"2607.24462v1","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.24462","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"pith_short_12","alias_value":"I5AYFWN433MM","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"pith_short_16","alias_value":"I5AYFWN433MMO3W7","created_at":"2026-07-28T02:24:05Z"},{"alias_kind":"pith_short_8","alias_value":"I5AYFWN4","created_at":"2026-07-28T02:24:05Z"}],"graph_snapshots":[{"event_id":"sha256:539c89d64decb5839da447023b18fda2141733d50c1421180f7e06276f8ba8c3","target":"graph","created_at":"2026-07-28T02:24:05Z","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/2607.24462/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We discuss the question of when a homology or cohomology class with twisted integer coefficients of a manifold $X$ is realised by a submanifold. While this question is classical in nature, providing an answer requires relatively modern techniques from parametrised homotopy theory. More specifically, we introduce cobordism classes twisted by a coefficient system and then define a twisted Thom space $\\operatorname{M^\\mathrm{tw}O}(n)$ over $\\operatorname{BO}(1)$, which serves as the classifying object for this cobordism theory under a twisted Pontryagin-Thom construction. As a result, a twisted h","authors_text":"Baylee Schutte, Mark Grant, Michael Jung","cross_cats":["math.GT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2026-07-27T14:03:37Z","title":"On Realisability of Twisted Homology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.24462","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":"sha256:386c1f3694ac17efe8c2e3869586cab0a1f541c5207c9b98b6b4ff42e2c45d0a","target":"record","created_at":"2026-07-28T02:24:05Z","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":"b0ea5dcc1988a9a094ade5ef78b9372cbec63e1f32a4afd23c9cdcdd4b462b9c","cross_cats_sorted":["math.GT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2026-07-27T14:03:37Z","title_canon_sha256":"fcf0dc947ce7b8f4fe760ebd5dec409ebee17f5d303a123b0873a6533fb5fe2d"},"schema_version":"1.0","source":{"id":"2607.24462","kind":"arxiv","version":1}},"canonical_sha256":"474182d9bcded8c76edf85032d4db701b3b5641d416c08fc2ab2cc188593f33f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"474182d9bcded8c76edf85032d4db701b3b5641d416c08fc2ab2cc188593f33f","first_computed_at":"2026-07-28T02:24:05.042807Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-28T02:24:05.042807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bRvHamHw/drC6Cws8geNjbAY+ha+M3tvahyac37sl6hBW+B119GvRbxcB7ukHSvh96xRlkb5vp67SM+PDDpUBA==","signature_status":"signed_v1","signed_at":"2026-07-28T02:24:05.043641Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.24462","source_kind":"arxiv","source_version":1}}},"equivocations":[{"signer_id":"pith.science","event_type":"integrity_finding","target":"integrity","event_ids":["sha256:5064811aa2d58395e0b227f30dfb64ec46853d209842fe3746dce35bc313df41","sha256:a4347f82641eacc5ed5dd651b89122d6468afbc86c7838e67569722a68bba264"]}],"invalid_events":[],"applied_event_ids":["sha256:386c1f3694ac17efe8c2e3869586cab0a1f541c5207c9b98b6b4ff42e2c45d0a","sha256:539c89d64decb5839da447023b18fda2141733d50c1421180f7e06276f8ba8c3"],"state_sha256":"2c10b2cee6babb616cf319891f0ed5351757248e0c32b5c3f829565d080b69b6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OInso/rC/DsWQBXhAUv2Id2Lh4WScllS/kkdiblN2/QhCJ1T3UyzEHznF/t3ozHeX/MJb5ckrmY9bosJ5E+dDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T08:08:23.064328Z","bundle_sha256":"a6fb1c85b0a9334a29695c8af5375313cfade612bccbe92dbd096405d0ca10df"}}