{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:2INOGF7A2XVFK6HOEQRZXXCBX6","short_pith_number":"pith:2INOGF7A","canonical_record":{"source":{"id":"2302.06525","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AT","submitted_at":"2023-02-13T17:10:49Z","cross_cats_sorted":["math.CO","math.CT","math.RT"],"title_canon_sha256":"1ec7ef931002ae17025ac3e269cb5bb2b4b318488b0c80b2c7a45818814b1d97","abstract_canon_sha256":"35b25e47db7e7899daf043efac094d2d4da50c34941a11af86569a71ffebd2de"},"schema_version":"1.0"},"canonical_sha256":"d21ae317e0d5ea5578ee24239bdc41bf9d09612dc3d86bf4bb51702bb2314773","source":{"kind":"arxiv","id":"2302.06525","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2302.06525","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"arxiv_version","alias_value":"2302.06525v2","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2302.06525","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"pith_short_12","alias_value":"2INOGF7A2XVF","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"2INOGF7A2XVFK6HO","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"2INOGF7A","created_at":"2026-07-05T09:32:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:2INOGF7A2XVFK6HOEQRZXXCBX6","target":"record","payload":{"canonical_record":{"source":{"id":"2302.06525","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AT","submitted_at":"2023-02-13T17:10:49Z","cross_cats_sorted":["math.CO","math.CT","math.RT"],"title_canon_sha256":"1ec7ef931002ae17025ac3e269cb5bb2b4b318488b0c80b2c7a45818814b1d97","abstract_canon_sha256":"35b25e47db7e7899daf043efac094d2d4da50c34941a11af86569a71ffebd2de"},"schema_version":"1.0"},"canonical_sha256":"d21ae317e0d5ea5578ee24239bdc41bf9d09612dc3d86bf4bb51702bb2314773","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:32:02.396460Z","signature_b64":"7nZCS4wML5Ap8kTKCbRewQsJo9WpsT6r/4uHGGEZQtRusgHJuUvpRYydTNI5SSfhBPMNUoc8p61XR/ymKkUmBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d21ae317e0d5ea5578ee24239bdc41bf9d09612dc3d86bf4bb51702bb2314773","last_reissued_at":"2026-07-05T09:32:02.396042Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:32:02.396042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2302.06525","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":"2026-07-05T09:32:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sgm5AjSThEvJx9t0NWmxQ83K8lTW9JzAEGY3rJzW0Iu+3dmRxac+JwgZw35R/o6aAmEF2nQ8qlkNLAEKhRabDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T12:24:51.293018Z"},"content_sha256":"0b2adefa15b017bcdac3075240862491945f0ed3e2117653e1dfe4282df0e7e4","schema_version":"1.0","event_id":"sha256:0b2adefa15b017bcdac3075240862491945f0ed3e2117653e1dfe4282df0e7e4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:2INOGF7A2XVFK6HOEQRZXXCBX6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On finite generation in magnitude (co)homology, and its torsion","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.CO","math.CT","math.RT"],"primary_cat":"math.AT","authors_text":"Carlo Collari, Luigi Caputi","submitted_at":"2023-02-13T17:10:49Z","abstract_excerpt":"The aim of this paper is to apply the framework, which was developed by Sam and Snowden, to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of graphs introduced by Hepworth and Willerton. More precisely, for graphs of bounded genus, we prove that magnitude cohomology, in each homological degree, has rank which grows at most polynomially in the number of vertices, and that its torsion is bounded. As a consequence, we obtain analogous results for path homology of (undirected) graphs. We complement the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.06525","kind":"arxiv","version":2},"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/2302.06525/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":"2026-07-05T09:32:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jTIuNL4UNCYBGK1GrQ5ep0ZrxCswLWOwPSEId0ZW2/t5J6p0uw5qKYZEsPopbhTJ8Z69zBBEwQHAnLe0FOBtBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T12:24:51.293544Z"},"content_sha256":"9d318f23f7cf5418d8100ee79819d257f6430afbf68116f6ca2aef43d73670d5","schema_version":"1.0","event_id":"sha256:9d318f23f7cf5418d8100ee79819d257f6430afbf68116f6ca2aef43d73670d5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2INOGF7A2XVFK6HOEQRZXXCBX6/bundle.json","state_url":"https://pith.science/pith/2INOGF7A2XVFK6HOEQRZXXCBX6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2INOGF7A2XVFK6HOEQRZXXCBX6/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":"2026-08-04T12:24:51Z","links":{"resolver":"https://pith.science/pith/2INOGF7A2XVFK6HOEQRZXXCBX6","bundle":"https://pith.science/pith/2INOGF7A2XVFK6HOEQRZXXCBX6/bundle.json","state":"https://pith.science/pith/2INOGF7A2XVFK6HOEQRZXXCBX6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2INOGF7A2XVFK6HOEQRZXXCBX6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:2INOGF7A2XVFK6HOEQRZXXCBX6","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":"35b25e47db7e7899daf043efac094d2d4da50c34941a11af86569a71ffebd2de","cross_cats_sorted":["math.CO","math.CT","math.RT"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AT","submitted_at":"2023-02-13T17:10:49Z","title_canon_sha256":"1ec7ef931002ae17025ac3e269cb5bb2b4b318488b0c80b2c7a45818814b1d97"},"schema_version":"1.0","source":{"id":"2302.06525","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2302.06525","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"arxiv_version","alias_value":"2302.06525v2","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2302.06525","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"pith_short_12","alias_value":"2INOGF7A2XVF","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"2INOGF7A2XVFK6HO","created_at":"2026-07-05T09:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"2INOGF7A","created_at":"2026-07-05T09:32:02Z"}],"graph_snapshots":[{"event_id":"sha256:9d318f23f7cf5418d8100ee79819d257f6430afbf68116f6ca2aef43d73670d5","target":"graph","created_at":"2026-07-05T09:32:02Z","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/2302.06525/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The aim of this paper is to apply the framework, which was developed by Sam and Snowden, to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of graphs introduced by Hepworth and Willerton. More precisely, for graphs of bounded genus, we prove that magnitude cohomology, in each homological degree, has rank which grows at most polynomially in the number of vertices, and that its torsion is bounded. As a consequence, we obtain analogous results for path homology of (undirected) graphs. We complement the ","authors_text":"Carlo Collari, Luigi Caputi","cross_cats":["math.CO","math.CT","math.RT"],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AT","submitted_at":"2023-02-13T17:10:49Z","title":"On finite generation in magnitude (co)homology, and its torsion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.06525","kind":"arxiv","version":2},"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":"sha256:0b2adefa15b017bcdac3075240862491945f0ed3e2117653e1dfe4282df0e7e4","target":"record","created_at":"2026-07-05T09:32:02Z","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":"35b25e47db7e7899daf043efac094d2d4da50c34941a11af86569a71ffebd2de","cross_cats_sorted":["math.CO","math.CT","math.RT"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AT","submitted_at":"2023-02-13T17:10:49Z","title_canon_sha256":"1ec7ef931002ae17025ac3e269cb5bb2b4b318488b0c80b2c7a45818814b1d97"},"schema_version":"1.0","source":{"id":"2302.06525","kind":"arxiv","version":2}},"canonical_sha256":"d21ae317e0d5ea5578ee24239bdc41bf9d09612dc3d86bf4bb51702bb2314773","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d21ae317e0d5ea5578ee24239bdc41bf9d09612dc3d86bf4bb51702bb2314773","first_computed_at":"2026-07-05T09:32:02.396042Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:32:02.396042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7nZCS4wML5Ap8kTKCbRewQsJo9WpsT6r/4uHGGEZQtRusgHJuUvpRYydTNI5SSfhBPMNUoc8p61XR/ymKkUmBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:32:02.396460Z","signed_message":"canonical_sha256_bytes"},"source_id":"2302.06525","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b2adefa15b017bcdac3075240862491945f0ed3e2117653e1dfe4282df0e7e4","sha256:9d318f23f7cf5418d8100ee79819d257f6430afbf68116f6ca2aef43d73670d5"],"state_sha256":"4bc0119069f5b85ff9ef5deae02280e94cceab9f9f17f5397b0ee500bca57fba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eXx2VQ8tt8FXCMJ9wOOaLx4yUkWUO8RPZB31VjjyjV/9gd0DORycBc/UiXEbTiAPj8CGNCy40Fvjopc+mgkDCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T12:24:51.299228Z","bundle_sha256":"95eca425e0983c5f0e0e852902f4b40f1aa2da1350c106ed8b13584d7ae64da5"}}