{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:M2WMDTT4OLJUQ37JLW6XOASTW4","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":"d09cf68415cf3d0b560fbc2bdd0ddcc5bc3a800c9fc835e0681925a38591cdef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-05-06T22:22:26Z","title_canon_sha256":"b60acf7788cebe8da87ba14a051ba5983803961ed92cb62bdfafb959402923f7"},"schema_version":"1.0","source":{"id":"1105.1395","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.1395","created_at":"2026-05-18T04:22:39Z"},{"alias_kind":"arxiv_version","alias_value":"1105.1395v1","created_at":"2026-05-18T04:22:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.1395","created_at":"2026-05-18T04:22:39Z"},{"alias_kind":"pith_short_12","alias_value":"M2WMDTT4OLJU","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"M2WMDTT4OLJUQ37J","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"M2WMDTT4","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:22531158fbd6086623764ca3a24e932138f528d0b311f7de0bead6bc5c8531b8","target":"graph","created_at":"2026-05-18T04:22:39Z","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":"In his influential work Choquet systematically studied capacities on Boolean algebras in a topological space, and gave a probabilistic interpretation for completely monotone (and completely alternating) capacities. Beyond complete monotonicity we can view a capacity as a marginal condition for probability distribution over the distributive lattice of dual order ideals. In this paper we discuss a combinatorial approach when capacities are defined over a finite lattice, and investigate Fr\\'{e}chet bounds given the marginal condition, probabilistic interpretation of difference operators, and stoc","authors_text":"Motoya Machida","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-05-06T22:22:26Z","title":"Capacities on a finite lattice"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.1395","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:e1a00e0bfd129577c581fbd70bb8e70cb3c7c067f46f4457910f3eb3268647d8","target":"record","created_at":"2026-05-18T04:22:39Z","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":"d09cf68415cf3d0b560fbc2bdd0ddcc5bc3a800c9fc835e0681925a38591cdef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-05-06T22:22:26Z","title_canon_sha256":"b60acf7788cebe8da87ba14a051ba5983803961ed92cb62bdfafb959402923f7"},"schema_version":"1.0","source":{"id":"1105.1395","kind":"arxiv","version":1}},"canonical_sha256":"66acc1ce7c72d3486fe95dbd770253b73f1a925d0cdd39819e989e25cf4bb1b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66acc1ce7c72d3486fe95dbd770253b73f1a925d0cdd39819e989e25cf4bb1b2","first_computed_at":"2026-05-18T04:22:39.518755Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:22:39.518755Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vn6zI7hmrwYKiP8Vgle7TU+D402uuGOEFrHxH51xQcmtTSfo3Wgxr7CCSAisEddkfIdnA0VXyqXvnmzk/4MlAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:22:39.519240Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.1395","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e1a00e0bfd129577c581fbd70bb8e70cb3c7c067f46f4457910f3eb3268647d8","sha256:22531158fbd6086623764ca3a24e932138f528d0b311f7de0bead6bc5c8531b8"],"state_sha256":"295d8b6ce5628bffdb3601d219d7d13ce270ff7fac94e9437fe312ad5582471e"}