{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:ZQORPUIAFX4XO2O7QXJUIVNQND","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":"ba57fdcdb614f634c7b42521e5ddbd5905469195d857818873246caf9af72b11","cross_cats_sorted":["cs.DM","math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-05T15:20:03Z","title_canon_sha256":"76dbd0755c278a3f44e65d98e63292832a6c2f3b2160e5ff77df754809f7763e"},"schema_version":"1.0","source":{"id":"1708.01778","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.01778","created_at":"2026-05-18T00:38:33Z"},{"alias_kind":"arxiv_version","alias_value":"1708.01778v1","created_at":"2026-05-18T00:38:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.01778","created_at":"2026-05-18T00:38:33Z"},{"alias_kind":"pith_short_12","alias_value":"ZQORPUIAFX4X","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZQORPUIAFX4XO2O7","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZQORPUIA","created_at":"2026-05-18T12:31:59Z"}],"graph_snapshots":[{"event_id":"sha256:62283581430be6f3117c108a08c6b197ef243c7ec2a5bdb846d011053ee1b8f0","target":"graph","created_at":"2026-05-18T00:38:33Z","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"},"paper":{"abstract_excerpt":"We define a ring R of geometric objects G generated by finite abstract simplicial complexes. To every G belongs Hodge Laplacian H as the square of the Dirac operator determining its cohomology and a unimodular connection matrix L). The sum of the matrix entries of the inverse of L is the Euler characteristic. The spectra of H as well as inductive dimension add under multiplication while the spectra of L multiply. The nullity of the Hodge of H are the Betti numbers which can now be signed. The map assigning to G its Poincare polynomial is a ring homomorphism from R the polynomials. Especially t","authors_text":"Oliver Knill","cross_cats":["cs.DM","math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-05T15:20:03Z","title":"The strong ring of simplicial complexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01778","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:d741633c9a3502c38e046cfb9ad9ff864f82168a429b03778fc1a3fa56c5cf5b","target":"record","created_at":"2026-05-18T00:38:33Z","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":"ba57fdcdb614f634c7b42521e5ddbd5905469195d857818873246caf9af72b11","cross_cats_sorted":["cs.DM","math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-05T15:20:03Z","title_canon_sha256":"76dbd0755c278a3f44e65d98e63292832a6c2f3b2160e5ff77df754809f7763e"},"schema_version":"1.0","source":{"id":"1708.01778","kind":"arxiv","version":1}},"canonical_sha256":"cc1d17d1002df97769df85d34455b068eaac1e76481be85e2ac321de07705d47","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cc1d17d1002df97769df85d34455b068eaac1e76481be85e2ac321de07705d47","first_computed_at":"2026-05-18T00:38:33.183820Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:33.183820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ccAMY90B89AZfO48Qu9fM098naLUURQmQ67rYUlWphX6XVXEB+9OloUWcden6Nkm3BVJPFz5RTj9EvYIJqtMCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:33.184340Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.01778","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d741633c9a3502c38e046cfb9ad9ff864f82168a429b03778fc1a3fa56c5cf5b","sha256:62283581430be6f3117c108a08c6b197ef243c7ec2a5bdb846d011053ee1b8f0"],"state_sha256":"cbb5d0a546275b9e09a98d2b2015c964b896587e9991e87c7f9085807e04dbb0"}