{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:BJJQRNCI3KDWG3FUZQYASVCKXM","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":"1a57b9d801083758556cc918cd8c4f40919088010fa89bccf2f8f696060c519e","cross_cats_sorted":["math.GT","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-22T01:49:23Z","title_canon_sha256":"aa42de47ff6f34e7da25836382d8459d9042341c3ecbfb461f058dfc00787cf0"},"schema_version":"1.0","source":{"id":"1608.06877","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.06877","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"arxiv_version","alias_value":"1608.06877v1","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.06877","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"pith_short_12","alias_value":"BJJQRNCI3KDW","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"BJJQRNCI3KDWG3FU","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"BJJQRNCI","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:0c9d84f97dd89494e2017e4e0d967ce249bbe87ab0bcb511646bc38de5d10460","target":"graph","created_at":"2026-05-18T01:07:57Z","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":"The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the Poincar\\'e-Hopf indices at critical points correspond to the values of the Moebius function. The Morse inequalities link number theoretical quantities like the prime counting functions relevant for the distribution of primes with cohom","authors_text":"Oliver Knill","cross_cats":["math.GT","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-22T01:49:23Z","title":"On Primes, Graphs and Cohomology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06877","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:b065ef10239dabf9e5af6aebd7069785aa5e20e7b8385b1680790cb61516dcf7","target":"record","created_at":"2026-05-18T01:07:57Z","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":"1a57b9d801083758556cc918cd8c4f40919088010fa89bccf2f8f696060c519e","cross_cats_sorted":["math.GT","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-22T01:49:23Z","title_canon_sha256":"aa42de47ff6f34e7da25836382d8459d9042341c3ecbfb461f058dfc00787cf0"},"schema_version":"1.0","source":{"id":"1608.06877","kind":"arxiv","version":1}},"canonical_sha256":"0a5308b448da87636cb4cc3009544abb34c32c9ab312e874a61ba02f67165147","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a5308b448da87636cb4cc3009544abb34c32c9ab312e874a61ba02f67165147","first_computed_at":"2026-05-18T01:07:57.343232Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:57.343232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IdnT3W9Vwbrq0Yodx5hPBtPjFcWi9IHeq3SsvZ9MW/0SIR5+5JKyP3KtX6UaZEa2b74RCv+pOJlJwgXOPoplCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:57.343744Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.06877","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b065ef10239dabf9e5af6aebd7069785aa5e20e7b8385b1680790cb61516dcf7","sha256:0c9d84f97dd89494e2017e4e0d967ce249bbe87ab0bcb511646bc38de5d10460"],"state_sha256":"096b1d894f1e251d08950e560a35b0cc66ac41763823944337e93a624ddbf579"}