Pith Number

pith:U2BMAT36

pith:2026:U2BMAT36M5SUH4KFTFGUUXOL47

2 verified not anchored not stored refs pending

Matching Rules as Cocycle Conditions: Discrete Potentials on Penrose and Canonical Projection Tilings

Elshad Allahyarov, Jonathan Washburn, Sebastian Pardo-Guerra

Matching rules for aperiodic tilings are exactly equivalent to the existence of consistent integer height functions through closed 1-cochains.

arxiv:2603.13553 v1 · 2026-03-13 · math.CO · math.GT

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\pithnumber{U2BMAT36M5SUH4KFTFGUUXOL47}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

✓ Author claim 2 verified · sign in to claim

Elshad Allahyarov orcid verified · Jonathan Washburn orcid verified

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest 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.

C2weakest 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.

C3one 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.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:21.946209Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a682c04f7e676543f145994d4a5dcbe7f7d41159d01451652b5cdd8fcaabfeee

Aliases

arxiv: 2603.13553 · arxiv_version: 2603.13553v1 · doi: 10.48550/arxiv.2603.13553 · pith_short_12: U2BMAT36M5SU · pith_short_16: U2BMAT36M5SUH4KF · pith_short_8: U2BMAT36

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/U2BMAT36M5SUH4KFTFGUUXOL47 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: a682c04f7e676543f145994d4a5dcbe7f7d41159d01451652b5cdd8fcaabfeee

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b99045004b00190c7e32cbdbfded1c7e662952779f5643b58e00229c31880059",
    "cross_cats_sorted": [
      "math.GT"
    ],
    "license": "",
    "primary_cat": "math.CO",
    "submitted_at": "2026-03-13T19:42:52Z",
    "title_canon_sha256": "679f4cb13052ba296721cf0bdc0216fb18c1c134e60b024b24be19b4d5525d85"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.13553",
    "kind": "arxiv",
    "version": 1
  }
}