{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:NGPUKSZBSVQHTFCEVJTZEU6VE7","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":"3d7ce26c763c5fa81e1b539ddbfc339511a3bcd41d5bf04ce82e1cad025baca6","cross_cats_sorted":["math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2003-09-12T21:36:58Z","title_canon_sha256":"3e19aefd0c28e6fe11e6c75712980e33dd8091de6198e58a69aef6f0fa6594f7"},"schema_version":"1.0","source":{"id":"math-ph/0309032","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0309032","created_at":"2026-07-04T14:33:57Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0309032v1","created_at":"2026-07-04T14:33:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0309032","created_at":"2026-07-04T14:33:57Z"},{"alias_kind":"pith_short_12","alias_value":"NGPUKSZBSVQH","created_at":"2026-07-04T14:33:57Z"},{"alias_kind":"pith_short_16","alias_value":"NGPUKSZBSVQHTFCE","created_at":"2026-07-04T14:33:57Z"},{"alias_kind":"pith_short_8","alias_value":"NGPUKSZB","created_at":"2026-07-04T14:33:57Z"}],"graph_snapshots":[{"event_id":"sha256:39030c5ede724bd73a95987212997226b78b49730b1911f15742fdccdec29d0b","target":"graph","created_at":"2026-07-04T14:33:57Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math-ph/0309032/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables. The integrated density of states of this Laplace operator is shown to have discontinuities provided that the distribution of arc lengths of the loops has a nontrivial pure point part. Some numerical illustrations are also presented.","authors_text":"Robert Schrader, Vadim Kostrykin","cross_cats":["math.MP"],"headline":"","license":"","primary_cat":"math-ph","submitted_at":"2003-09-12T21:36:58Z","title":"A Random Necklace Model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0309032","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":"sha256:94753e4db99bc5f105d9c7bf713deaa6d5eb8194b8d22c1a496eb51a4c845aa6","target":"record","created_at":"2026-07-04T14:33:57Z","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":"3d7ce26c763c5fa81e1b539ddbfc339511a3bcd41d5bf04ce82e1cad025baca6","cross_cats_sorted":["math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2003-09-12T21:36:58Z","title_canon_sha256":"3e19aefd0c28e6fe11e6c75712980e33dd8091de6198e58a69aef6f0fa6594f7"},"schema_version":"1.0","source":{"id":"math-ph/0309032","kind":"arxiv","version":1}},"canonical_sha256":"699f454b219560799444aa679253d527f65132a9f763cd9eda9d60f2ed28003f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"699f454b219560799444aa679253d527f65132a9f763cd9eda9d60f2ed28003f","first_computed_at":"2026-07-04T14:33:57.065270Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:33:57.065270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7s9NOD69xfcCxBMStrqskPwHJkqe/sbIyH/SPzFEJMirHW1vPQmbHybwovjQhFKpwe7rVtNJjphUcAJW+yA6Cg==","signature_status":"signed_v1","signed_at":"2026-07-04T14:33:57.065707Z","signed_message":"canonical_sha256_bytes"},"source_id":"math-ph/0309032","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94753e4db99bc5f105d9c7bf713deaa6d5eb8194b8d22c1a496eb51a4c845aa6","sha256:39030c5ede724bd73a95987212997226b78b49730b1911f15742fdccdec29d0b"],"state_sha256":"3c85edfcbb688adf964ae69758ad91f0604c916148c659d899f0402f58978593"}