{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:DP3DWTVHKASI56QG7C665BD3KW","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":"74036889b1199cf00218d7182bfc74151059d22556ff9aff23a4a1bcdef5ac75","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2022-04-27T19:05:01Z","title_canon_sha256":"f5425f1b750866d93736d3f1c71404e5656124ff101d26ced1831022860c79c7"},"schema_version":"1.0","source":{"id":"2204.13149","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2204.13149","created_at":"2026-07-05T04:18:31Z"},{"alias_kind":"arxiv_version","alias_value":"2204.13149v1","created_at":"2026-07-05T04:18:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.13149","created_at":"2026-07-05T04:18:31Z"},{"alias_kind":"pith_short_12","alias_value":"DP3DWTVHKASI","created_at":"2026-07-05T04:18:31Z"},{"alias_kind":"pith_short_16","alias_value":"DP3DWTVHKASI56QG","created_at":"2026-07-05T04:18:31Z"},{"alias_kind":"pith_short_8","alias_value":"DP3DWTVH","created_at":"2026-07-05T04:18:31Z"}],"graph_snapshots":[{"event_id":"sha256:8805b90fa8fa1fc59339f770dd75dbe79458168dfe1528f2d8c5adec286520cb","target":"graph","created_at":"2026-07-05T04:18:31Z","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/2204.13149/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For several classical nonnegative integer functions, we investigate if they are members of the counting complexity class #P or not. We prove #P membership in surprising cases, and in other cases we prove non-membership, relying on standard complexity assumptions or on oracle separations.\n  We initiate the study of the polynomial closure properties of #P on affine varieties, i.e., if all problem instances satisfy algebraic constraints. This is directly linked to classical combinatorial proofs of algebraic identities and inequalities. We investigate #TFNP and obtain oracle separations that prove","authors_text":"Christian Ikenmeyer, Igor Pak","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2022-04-27T19:05:01Z","title":"What is in #P and what is not?"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.13149","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":"sha256:02b0edd441094b30b632901daf33edddafaa14fba9294024a9714bc13c51d2cc","target":"record","created_at":"2026-07-05T04:18:31Z","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":"74036889b1199cf00218d7182bfc74151059d22556ff9aff23a4a1bcdef5ac75","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2022-04-27T19:05:01Z","title_canon_sha256":"f5425f1b750866d93736d3f1c71404e5656124ff101d26ced1831022860c79c7"},"schema_version":"1.0","source":{"id":"2204.13149","kind":"arxiv","version":1}},"canonical_sha256":"1bf63b4ea750248efa06f8bdee847b55a9c175ae613628660ff8083c49ede20d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1bf63b4ea750248efa06f8bdee847b55a9c175ae613628660ff8083c49ede20d","first_computed_at":"2026-07-05T04:18:31.539307Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:18:31.539307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"agXxBT7hfqH1YbHBg60xM7kTcRJknoGzGTEOTYiozbqPT07OLubAG/DFtkAKXD4o61KYHMUibSQHShfQVo3aAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T04:18:31.539812Z","signed_message":"canonical_sha256_bytes"},"source_id":"2204.13149","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02b0edd441094b30b632901daf33edddafaa14fba9294024a9714bc13c51d2cc","sha256:8805b90fa8fa1fc59339f770dd75dbe79458168dfe1528f2d8c5adec286520cb"],"state_sha256":"d2bfcd2f7e01054430ff048ed526dac8402d5f04baf0205e07296fc5c00c9823"}