{"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"}