{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:U2BMAT36M5SUH4KFTFGUUXOL47","short_pith_number":"pith:U2BMAT36","schema_version":"1.0","canonical_sha256":"a682c04f7e676543f145994d4a5dcbe7f7d41159d01451652b5cdd8fcaabfeee","source":{"kind":"arxiv","id":"2603.13553","version":1},"attestation_state":"computed","paper":{"title":"Matching Rules as Cocycle Conditions: Discrete Potentials on Penrose and Canonical Projection Tilings","license":"","headline":"Matching rules for aperiodic tilings are exactly equivalent to the existence of consistent integer height functions through closed 1-cochains.","cross_cats":["math.GT"],"primary_cat":"math.CO","authors_text":"Elshad Allahyarov, Jonathan Washburn, Sebastian Pardo-Guerra","submitted_at":"2026-03-13T19:42:52Z","abstract_excerpt":"Aperiodic tilings support two classically studied but hitherto separately presented structures: matching rules, which enforce global order via local constraints, and height functions, which encode global geometry through integer-valued potentials. Their precise relationship has remained implicit in the literature. This paper bridges them via a cochain-first framework, establishing a four-way equivalence -- between matching rules, Ammann bar continuity, cycle closure of the associated $1$-cochains, and height-function existence -- proved for candidate tilings without presupposing any of the fou"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":true,"weak_author_claims":0,"strong_author_claims":2,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":true},"canonical_record":{"source":{"id":"2603.13553","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2026-03-13T19:42:52Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"679f4cb13052ba296721cf0bdc0216fb18c1c134e60b024b24be19b4d5525d85","abstract_canon_sha256":"b99045004b00190c7e32cbdbfded1c7e662952779f5643b58e00229c31880059"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:21.946893Z","signature_b64":"YnFqaCctze2/E5mbQsjf/t8v7GDnclzWNzhmG3GlvUjRftkzKU7RM90AP5wHNx9lH33HY+dDlSJTXhpmLFuADQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a682c04f7e676543f145994d4a5dcbe7f7d41159d01451652b5cdd8fcaabfeee","last_reissued_at":"2026-05-17T23:39:21.946209Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:21.946209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Matching Rules as Cocycle Conditions: Discrete Potentials on Penrose and Canonical Projection Tilings","license":"","headline":"Matching rules for aperiodic tilings are exactly equivalent to the existence of consistent integer height functions through closed 1-cochains.","cross_cats":["math.GT"],"primary_cat":"math.CO","authors_text":"Elshad Allahyarov, Jonathan Washburn, Sebastian Pardo-Guerra","submitted_at":"2026-03-13T19:42:52Z","abstract_excerpt":"Aperiodic tilings support two classically studied but hitherto separately presented structures: matching rules, which enforce global order via local constraints, and height functions, which encode global geometry through integer-valued potentials. Their precise relationship has remained implicit in the literature. This paper bridges them via a cochain-first framework, establishing a four-way equivalence -- between matching rules, Ammann bar continuity, cycle closure of the associated $1$-cochains, and height-function existence -- proved for candidate tilings without presupposing any of the fou"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A four-way equivalence holds between matching rules, Ammann bar continuity, cycle closure of the associated 1-cochains, and height-function existence, proved for candidate tilings without presupposing any of the four conditions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The half-edge/gluing construction produces a globally consistent antisymmetric 1-cochain precisely when adjacent tiles agree on shared edges, and this agreement is equivalent to the classical matching rules for the families considered.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Matching rules, Ammann-bar continuity, 1-cochain cycle closure, and height-function existence are equivalent via a half-edge gluing construction on Penrose and canonical-projection tilings.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Matching rules for aperiodic tilings are exactly equivalent to the existence of consistent integer height functions through closed 1-cochains.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a34e7c20152c1fae813e7796fa397b7b7578f41f2fe31429dadd05f24608c509"},"source":{"id":"2603.13553","kind":"arxiv","version":1},"verdict":{"id":"e01ee3b6-02f8-42e0-8060-864c88a486d8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T21:55:44.690049Z","strongest_claim":"A four-way equivalence holds between matching rules, Ammann bar continuity, cycle closure of the associated 1-cochains, and height-function existence, proved for candidate tilings without presupposing any of the four conditions.","one_line_summary":"Matching rules, Ammann-bar continuity, 1-cochain cycle closure, and height-function existence are equivalent via a half-edge gluing construction on Penrose and canonical-projection tilings.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The half-edge/gluing construction produces a globally consistent antisymmetric 1-cochain precisely when adjacent tiles agree on shared edges, and this agreement is equivalent to the classical matching rules for the families considered.","pith_extraction_headline":"Matching rules for aperiodic tilings are exactly equivalent to the existence of consistent integer height functions through closed 1-cochains."},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"e13d9e9afc3dade90288a12cdb2f4bb2970fbe09ea3fb46a56afd3e529e532e6"},"author_claims":{"count":2,"strong_count":2,"snapshot_sha256":"62a20c410a2029f9951a7d3633c3ffb7f40335f960ec59a9ae6c6978322f695d"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2603.13553","created_at":"2026-05-17T23:39:21.946301+00:00"},{"alias_kind":"arxiv_version","alias_value":"2603.13553v1","created_at":"2026-05-17T23:39:21.946301+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.13553","created_at":"2026-05-17T23:39:21.946301+00:00"},{"alias_kind":"pith_short_12","alias_value":"U2BMAT36M5SU","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"U2BMAT36M5SUH4KF","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"U2BMAT36","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[{"claim_id":"a6d2571d-b209-489d-a9d6-11b162e27438","handle":"elshad","display_name":"Elshad Allahyarov","verification":"orcid_verified","verification_note":"Auto-claimed from ORCID 0000-0001-7212-4713 at registration","role_label":null,"open_disputes":0},{"claim_id":"94e0c568-7a65-4509-98d9-e38f021a52f6","handle":"jonathanwashburn","display_name":"Jonathan Washburn","verification":"orcid_verified","verification_note":"Auto-claimed from ORCID 0009-0001-8868-7497 at registration","role_label":null,"open_disputes":0}],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47","json":"https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47.json","graph_json":"https://pith.science/api/pith-number/U2BMAT36M5SUH4KFTFGUUXOL47/graph.json","events_json":"https://pith.science/api/pith-number/U2BMAT36M5SUH4KFTFGUUXOL47/events.json","paper":"https://pith.science/paper/U2BMAT36"},"agent_actions":{"view_html":"https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47","download_json":"https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47.json","view_paper":"https://pith.science/paper/U2BMAT36","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2603.13553&json=true","fetch_graph":"https://pith.science/api/pith-number/U2BMAT36M5SUH4KFTFGUUXOL47/graph.json","fetch_events":"https://pith.science/api/pith-number/U2BMAT36M5SUH4KFTFGUUXOL47/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47/action/timestamp_anchor","attest_storage":"https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47/action/storage_attestation","attest_author":"https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47/action/author_attestation","sign_citation":"https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47/action/citation_signature","submit_replication":"https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47/action/replication_record"}},"created_at":"2026-05-17T23:39:21.946301+00:00","updated_at":"2026-05-17T23:39:21.946301+00:00"}