{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:JBSL7TINUESHQKQXRB7AEL6GQS","merge_version":"pith-open-graph-merge-v1","event_count":3,"valid_event_count":3,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1f42b1ee7c18d20a59666669276946bb85606a5df3736315412b1eff20bd2789","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-11T06:34:26Z","title_canon_sha256":"d0684289127372946ef70fc478123102440f27339a0b4c8bb0e0fc2abd9cf090"},"schema_version":"1.0","source":{"id":"2607.10155","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.10155","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"arxiv_version","alias_value":"2607.10155v1","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10155","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"pith_short_12","alias_value":"JBSL7TINUESH","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"pith_short_16","alias_value":"JBSL7TINUESHQKQX","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"pith_short_8","alias_value":"JBSL7TIN","created_at":"2026-07-14T01:20:27Z"}],"graph_snapshots":[{"event_id":"sha256:6a1054f241f3e567a87c68291b5e58d3edadf262d2833acebeed7f828a5c7b61","target":"graph","created_at":"2026-07-14T01:20:27Z","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/2607.10155/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Combinatorial Discretizable Distance Geometry Problem combines a finite binary lateration process with additional distance constraints. When predecessor sets are not consecutive, these pruning constraints interact in ways that make the symmetry-based counting methods available for molecular instances insufficient. We establish an exact, dimension-uniform counting theorem for seed-fixed feasible realizations under strict discretization and a generic feasible framework assumption. Our approach identifies partial reflections through seed-free connected components of the lateration skeleton. T","authors_text":"Carlile Lavor, Michael Souza, Wagner da Rocha","cross_cats":["math.MG"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-11T06:34:26Z","title":"A Conditional Rank-Count Theory for the Combinatorial Discretizable Distance Geometry Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10155","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":"sha256:3ceb2e232866255e33430153ad679d77f69886d610b4ab6aafd03ffab4e930d0","target":"record","created_at":"2026-07-14T01:20:27Z","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":"1f42b1ee7c18d20a59666669276946bb85606a5df3736315412b1eff20bd2789","cross_cats_sorted":["math.MG"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-11T06:34:26Z","title_canon_sha256":"d0684289127372946ef70fc478123102440f27339a0b4c8bb0e0fc2abd9cf090"},"schema_version":"1.0","source":{"id":"2607.10155","kind":"arxiv","version":1}},"canonical_sha256":"4864bfcd0da124782a17887e022fc6848a655889c24931991563d199921c7513","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4864bfcd0da124782a17887e022fc6848a655889c24931991563d199921c7513","first_computed_at":"2026-07-14T01:20:27.929116Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-14T01:20:27.929116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m1DueQGcJorLRUIGuO9zkdgaPPHOxMlOSPsavDQTPpxY/YpDEY44yx97IrSeM/GmNeuW2Lgrmz92QhAjMcI9DA==","signature_status":"signed_v1","signed_at":"2026-07-14T01:20:27.929963Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.10155","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3ceb2e232866255e33430153ad679d77f69886d610b4ab6aafd03ffab4e930d0","sha256:6a1054f241f3e567a87c68291b5e58d3edadf262d2833acebeed7f828a5c7b61","sha256:0acec76507dcccde41617ac559c9b55869824855414cfa0f02f0fe33229e3f8f"],"state_sha256":"c8ba812c79612a41395e8e4fe174d3460259b8f35f8d32924c18d4b2272d06aa"}