{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:UNPHYMOO5T4ZSP7ZFMUUSKUCVH","short_pith_number":"pith:UNPHYMOO","schema_version":"1.0","canonical_sha256":"a35e7c31ceecf9993ff92b29492a82a9f4cb9a05a91e791a606a78587fdbbdf5","source":{"kind":"arxiv","id":"2605.18487","version":1},"attestation_state":"computed","paper":{"title":"The number of realisations of a random graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anthony Nixon, Ben Smith, Sean Dewar","submitted_at":"2026-05-18T14:39:31Z","abstract_excerpt":"Determining the number of realisations of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove that the $d$-dimensional realisation number of an Erd\\H{o}s-Renyi random graph is either infinity or a power of 2 with exponent computable in polynomial time. We also determine a similar formula for the number of complex solutions to the generic rank-$d$ PSD matrix completion problem with randomly-selected non-diagonal unknown entries."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.18487","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-18T14:39:31Z","cross_cats_sorted":[],"title_canon_sha256":"f33fea2ab2139f67cc21462cddf6e988889100ee4c99906d9cec835bf42caaa1","abstract_canon_sha256":"0ef350d45388bfda83ca8b6ec6a81b0550ead1f0bdf8bbce6bd320faf1a545ea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:06:03.816555Z","signature_b64":"kWh755VBiWaH742/Q1gHx1iYkweYOnI6zlVGowkjwoOE+mkH3Z9tVLQT32WZ1z+uxiJNb6EOi4Oaa7hNsnaVDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a35e7c31ceecf9993ff92b29492a82a9f4cb9a05a91e791a606a78587fdbbdf5","last_reissued_at":"2026-05-20T00:06:03.815989Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:06:03.815989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The number of realisations of a random graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anthony Nixon, Ben Smith, Sean Dewar","submitted_at":"2026-05-18T14:39:31Z","abstract_excerpt":"Determining the number of realisations of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove that the $d$-dimensional realisation number of an Erd\\H{o}s-Renyi random graph is either infinity or a power of 2 with exponent computable in polynomial time. We also determine a similar formula for the number of complex solutions to the generic rank-$d$ PSD matrix completion problem with randomly-selected non-diagonal unknown entries."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.18487","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18487/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.18487","created_at":"2026-05-20T00:06:03.816079+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.18487v1","created_at":"2026-05-20T00:06:03.816079+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.18487","created_at":"2026-05-20T00:06:03.816079+00:00"},{"alias_kind":"pith_short_12","alias_value":"UNPHYMOO5T4Z","created_at":"2026-05-20T00:06:03.816079+00:00"},{"alias_kind":"pith_short_16","alias_value":"UNPHYMOO5T4ZSP7Z","created_at":"2026-05-20T00:06:03.816079+00:00"},{"alias_kind":"pith_short_8","alias_value":"UNPHYMOO","created_at":"2026-05-20T00:06:03.816079+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UNPHYMOO5T4ZSP7ZFMUUSKUCVH","json":"https://pith.science/pith/UNPHYMOO5T4ZSP7ZFMUUSKUCVH.json","graph_json":"https://pith.science/api/pith-number/UNPHYMOO5T4ZSP7ZFMUUSKUCVH/graph.json","events_json":"https://pith.science/api/pith-number/UNPHYMOO5T4ZSP7ZFMUUSKUCVH/events.json","paper":"https://pith.science/paper/UNPHYMOO"},"agent_actions":{"view_html":"https://pith.science/pith/UNPHYMOO5T4ZSP7ZFMUUSKUCVH","download_json":"https://pith.science/pith/UNPHYMOO5T4ZSP7ZFMUUSKUCVH.json","view_paper":"https://pith.science/paper/UNPHYMOO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.18487&json=true","fetch_graph":"https://pith.science/api/pith-number/UNPHYMOO5T4ZSP7ZFMUUSKUCVH/graph.json","fetch_events":"https://pith.science/api/pith-number/UNPHYMOO5T4ZSP7ZFMUUSKUCVH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UNPHYMOO5T4ZSP7ZFMUUSKUCVH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UNPHYMOO5T4ZSP7ZFMUUSKUCVH/action/storage_attestation","attest_author":"https://pith.science/pith/UNPHYMOO5T4ZSP7ZFMUUSKUCVH/action/author_attestation","sign_citation":"https://pith.science/pith/UNPHYMOO5T4ZSP7ZFMUUSKUCVH/action/citation_signature","submit_replication":"https://pith.science/pith/UNPHYMOO5T4ZSP7ZFMUUSKUCVH/action/replication_record"}},"created_at":"2026-05-20T00:06:03.816079+00:00","updated_at":"2026-05-20T00:06:03.816079+00:00"}