{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:G72U46KJNUMEXRFEMUSA3GEDK7","short_pith_number":"pith:G72U46KJ","canonical_record":{"source":{"id":"2604.16126","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-17T15:00:21Z","cross_cats_sorted":[],"title_canon_sha256":"961fd3ba231626f3c46b837aab714a88d0f6b3a112c475e5fe395ce0708be729","abstract_canon_sha256":"7dbce172b152f2b5626b5f36c75760e928b6ff95c0fc6bfa66e81fa92e2e5e6f"},"schema_version":"1.0"},"canonical_sha256":"37f54e79496d184bc4a465240d988357ea2394d2a0abb042b35f12dabeb63d2a","source":{"kind":"arxiv","id":"2604.16126","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.16126","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"arxiv_version","alias_value":"2604.16126v3","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.16126","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"pith_short_12","alias_value":"G72U46KJNUME","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"pith_short_16","alias_value":"G72U46KJNUMEXRFE","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"pith_short_8","alias_value":"G72U46KJ","created_at":"2026-06-02T01:03:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:G72U46KJNUMEXRFEMUSA3GEDK7","target":"record","payload":{"canonical_record":{"source":{"id":"2604.16126","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-17T15:00:21Z","cross_cats_sorted":[],"title_canon_sha256":"961fd3ba231626f3c46b837aab714a88d0f6b3a112c475e5fe395ce0708be729","abstract_canon_sha256":"7dbce172b152f2b5626b5f36c75760e928b6ff95c0fc6bfa66e81fa92e2e5e6f"},"schema_version":"1.0"},"canonical_sha256":"37f54e79496d184bc4a465240d988357ea2394d2a0abb042b35f12dabeb63d2a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T01:03:47.152540Z","signature_b64":"BklHmRtjF9bBr6jYwdg/7q+LBxhihjB+sPx1m2wkLQs+heNi8WmrSBWKYj1GA9Gd4jwL8eX77GNjSoGBNBLkAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37f54e79496d184bc4a465240d988357ea2394d2a0abb042b35f12dabeb63d2a","last_reissued_at":"2026-06-02T01:03:47.152004Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T01:03:47.152004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.16126","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T01:03:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PiuNCSUDL4NNldVnHAN05apgQmLqYfJD+UBXzN80ON3KeoPOchBsgOjjKDT4BSt+GmZK3l9eN5cuRxatrDN4BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T06:02:02.838332Z"},"content_sha256":"4c144e49a84398c9c1dfe609c916ee78a8a2e090a9e4ea6dfbbfa368e5d7c4b1","schema_version":"1.0","event_id":"sha256:4c144e49a84398c9c1dfe609c916ee78a8a2e090a9e4ea6dfbbfa368e5d7c4b1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:G72U46KJNUMEXRFEMUSA3GEDK7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cells, convexity and contractibility in general categories","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Categories obeying basic axioms admit convex contractible cells whose maps reconstruct homology and homotopy.","cross_cats":[],"primary_cat":"math.CT","authors_text":"Suddhasattwa Das","submitted_at":"2026-04-17T15:00:21Z","abstract_excerpt":"The two pillars of Algebraic topology - homology and homotopy theory rely on the availability of basic building blocks called cells. Cells take the form of simplexes, and have properties such as faces, sub-cells, convexity and contractibility. The first two cells, namely the line and point lead to the concept of homotopy. The collection of maps from the cells and the redundancies among them determine the homology of objects. This article presents a procedure by which such cells can be built in general categories satisfying some simple axioms. The cells satisfy the categorical analogs of convex"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"In any category satisfying a short list of axioms, one can construct cells that are convex and contractible in the categorical sense, and the maps from objects into these cells determine the homology and homotopy of the category.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The category obeys a small collection of axioms that are sufficient to guarantee the existence of the required cells and the reconstruction of homology and homotopy from maps into them.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Cells with convexity and contractibility can be built inside any category satisfying a short list of axioms, and the resulting cell data suffice to define homology and homotopy.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Categories obeying basic axioms admit convex contractible cells whose maps reconstruct homology and homotopy.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ceb1b2509cf17ad4d9227e0e377a68878b6305666d8cf3bd49283d4a9f061215"},"source":{"id":"2604.16126","kind":"arxiv","version":3},"verdict":{"id":"30d84e3d-5064-47a7-8daf-85d39de25be9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T00:50:31.155583Z","strongest_claim":"In any category satisfying a short list of axioms, one can construct cells that are convex and contractible in the categorical sense, and the maps from objects into these cells determine the homology and homotopy of the category.","one_line_summary":"Cells with convexity and contractibility can be built inside any category satisfying a short list of axioms, and the resulting cell data suffice to define homology and homotopy.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The category obeys a small collection of axioms that are sufficient to guarantee the existence of the required cells and the reconstruction of homology and homotopy from maps into them.","pith_extraction_headline":"Categories obeying basic axioms admit convex contractible cells whose maps reconstruct homology and homotopy."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.16126/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":3,"snapshot_sha256":"46a10c2b586edb4c16a893db3199ea9c6aab989cda233236194ad673746403ef"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"30d84e3d-5064-47a7-8daf-85d39de25be9"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T01:03:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pa+b/ay2G6cttnx7rsxA1UBzxwueLSUUdp13VfsdEkHDUigJDhEqfDZW4ZSdkkZSBseYG2hvLLxB0feX5NNxCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T06:02:02.838826Z"},"content_sha256":"66518d4603d43cd2cb5c063e22f5d424588992e641a899293cb9fb6b80f74059","schema_version":"1.0","event_id":"sha256:66518d4603d43cd2cb5c063e22f5d424588992e641a899293cb9fb6b80f74059"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G72U46KJNUMEXRFEMUSA3GEDK7/bundle.json","state_url":"https://pith.science/pith/G72U46KJNUMEXRFEMUSA3GEDK7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G72U46KJNUMEXRFEMUSA3GEDK7/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":"2026-07-04T06:02:02Z","links":{"resolver":"https://pith.science/pith/G72U46KJNUMEXRFEMUSA3GEDK7","bundle":"https://pith.science/pith/G72U46KJNUMEXRFEMUSA3GEDK7/bundle.json","state":"https://pith.science/pith/G72U46KJNUMEXRFEMUSA3GEDK7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G72U46KJNUMEXRFEMUSA3GEDK7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:G72U46KJNUMEXRFEMUSA3GEDK7","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":"7dbce172b152f2b5626b5f36c75760e928b6ff95c0fc6bfa66e81fa92e2e5e6f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-17T15:00:21Z","title_canon_sha256":"961fd3ba231626f3c46b837aab714a88d0f6b3a112c475e5fe395ce0708be729"},"schema_version":"1.0","source":{"id":"2604.16126","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.16126","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"arxiv_version","alias_value":"2604.16126v3","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.16126","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"pith_short_12","alias_value":"G72U46KJNUME","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"pith_short_16","alias_value":"G72U46KJNUMEXRFE","created_at":"2026-06-02T01:03:47Z"},{"alias_kind":"pith_short_8","alias_value":"G72U46KJ","created_at":"2026-06-02T01:03:47Z"}],"graph_snapshots":[{"event_id":"sha256:66518d4603d43cd2cb5c063e22f5d424588992e641a899293cb9fb6b80f74059","target":"graph","created_at":"2026-06-02T01:03:47Z","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":"In any category satisfying a short list of axioms, one can construct cells that are convex and contractible in the categorical sense, and the maps from objects into these cells determine the homology and homotopy of the category."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The category obeys a small collection of axioms that are sufficient to guarantee the existence of the required cells and the reconstruction of homology and homotopy from maps into them."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Cells with convexity and contractibility can be built inside any category satisfying a short list of axioms, and the resulting cell data suffice to define homology and homotopy."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Categories obeying basic axioms admit convex contractible cells whose maps reconstruct homology and homotopy."}],"snapshot_sha256":"ceb1b2509cf17ad4d9227e0e377a68878b6305666d8cf3bd49283d4a9f061215"},"formal_canon":{"evidence_count":3,"snapshot_sha256":"46a10c2b586edb4c16a893db3199ea9c6aab989cda233236194ad673746403ef"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.16126/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The two pillars of Algebraic topology - homology and homotopy theory rely on the availability of basic building blocks called cells. Cells take the form of simplexes, and have properties such as faces, sub-cells, convexity and contractibility. The first two cells, namely the line and point lead to the concept of homotopy. The collection of maps from the cells and the redundancies among them determine the homology of objects. This article presents a procedure by which such cells can be built in general categories satisfying some simple axioms. The cells satisfy the categorical analogs of convex","authors_text":"Suddhasattwa Das","cross_cats":[],"headline":"Categories obeying basic axioms admit convex contractible cells whose maps reconstruct homology and homotopy.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-17T15:00:21Z","title":"Cells, convexity and contractibility in general categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.16126","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-12T00:50:31.155583Z","id":"30d84e3d-5064-47a7-8daf-85d39de25be9","model_set":{"reader":"grok-4.3"},"one_line_summary":"Cells with convexity and contractibility can be built inside any category satisfying a short list of axioms, and the resulting cell data suffice to define homology and homotopy.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Categories obeying basic axioms admit convex contractible cells whose maps reconstruct homology and homotopy.","strongest_claim":"In any category satisfying a short list of axioms, one can construct cells that are convex and contractible in the categorical sense, and the maps from objects into these cells determine the homology and homotopy of the category.","weakest_assumption":"The category obeys a small collection of axioms that are sufficient to guarantee the existence of the required cells and the reconstruction of homology and homotopy from maps into them."}},"verdict_id":"30d84e3d-5064-47a7-8daf-85d39de25be9"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4c144e49a84398c9c1dfe609c916ee78a8a2e090a9e4ea6dfbbfa368e5d7c4b1","target":"record","created_at":"2026-06-02T01:03:47Z","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":"7dbce172b152f2b5626b5f36c75760e928b6ff95c0fc6bfa66e81fa92e2e5e6f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-17T15:00:21Z","title_canon_sha256":"961fd3ba231626f3c46b837aab714a88d0f6b3a112c475e5fe395ce0708be729"},"schema_version":"1.0","source":{"id":"2604.16126","kind":"arxiv","version":3}},"canonical_sha256":"37f54e79496d184bc4a465240d988357ea2394d2a0abb042b35f12dabeb63d2a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37f54e79496d184bc4a465240d988357ea2394d2a0abb042b35f12dabeb63d2a","first_computed_at":"2026-06-02T01:03:47.152004Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T01:03:47.152004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BklHmRtjF9bBr6jYwdg/7q+LBxhihjB+sPx1m2wkLQs+heNi8WmrSBWKYj1GA9Gd4jwL8eX77GNjSoGBNBLkAg==","signature_status":"signed_v1","signed_at":"2026-06-02T01:03:47.152540Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.16126","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c144e49a84398c9c1dfe609c916ee78a8a2e090a9e4ea6dfbbfa368e5d7c4b1","sha256:66518d4603d43cd2cb5c063e22f5d424588992e641a899293cb9fb6b80f74059"],"state_sha256":"1533bc317674e7f4ed28a0bf5f1b0b520f6e092f04c89ebacf7fd70de37d497a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F3TSOxYFj0Nvi/ofIasvVWtLH2FnYvqF/RMhnOfz1en6i7aIBkhewTSbAukD23VXW7o6bZ7PwVR83aBWSxm+Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T06:02:02.841458Z","bundle_sha256":"25aa7dfa07c3d76083dcb1d0383a9c417b0bef25072ce3e42dd037de49ad703a"}}