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

Recognition: 2 theorem links

· Lean Theorem

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

Sebastian Pardo-Guerra , Jonathan Washburn , Elshad Allahyarov

Authors on Pith 2 claimed

Elshad Allahyarov ORCID verified
Jonathan Washburn ORCID verified

Pith reviewed 2026-05-14 21:55 UTC · model grok-4.3

classification 🧮 math.CO math.GT

keywords aperiodic tilingsmatching rulesheight functionsAmmann barscanonical projection tilings1-cochainsPenrose tilingdiscrete Poincaré lemma

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The pith

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

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a four-way equivalence among matching rules, continuous Ammann bars, closed 1-cochains on directed edges, and the existence of scalar height potentials for Penrose, Ammann-Beenker, Fibonacci, and icosahedral projection tilings. The argument begins with a half-edge construction that assigns signed crossing counts to each directed edge for every Ammann-bar family, then shows that these assignments glue into a globally consistent antisymmetric cochain if and only if adjacent tiles obey the classical matching rules. Cycle closure of the resulting cochain immediately yields a height function by the discrete Poincaré lemma, while the same cochains recover vertex coordinates in canonical projection tilings and span the first cohomology group isomorphic to Z^N. A sympathetic reader sees two classically separate descriptions of long-range order unified without circular assumptions.

Core claim

For the families examined, matching rules hold globally precisely when the Ammann-bar crossing counts define a closed 1-cochain, which in turn admits a scalar potential that coincides with the classical height function; the same cochain data reconstructs lattice coordinates for any canonical projection tiling and forms a Z-basis for its first cohomology.

What carries the argument

Half-edge/gluing construction that produces an antisymmetric 1-cochain from signed Ammann-bar crossing counts on directed edges.

If this is right

Gluing consistency on shared edges forces every closed cycle to have zero cochain sum.
Closed 1-cochains admit integer-valued potentials that reproduce the classical Ammann height function.
For generic-window canonical projection tilings the cochains form a Z-basis of the first cohomology group isomorphic to Z^N.
The resulting structure supplies a conservation law whose recognition gap equals Z^N.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same cochain test could be run on any substitution tiling whose prototiles admit a finite set of bar families, even if the tiling is not known to be a projection tiling.
If the conjecture that conservation forcing characterises exactly the Pisot substitution CPTs holds, it would give a cohomological criterion distinguishing Pisot from non-Pisot cases.
Numerical enumeration of small patches could be accelerated by checking cochain closure instead of exhaustive matching-rule verification.

Load-bearing premise

Adjacent tiles produce identical signed crossing counts on every shared edge exactly when the classical matching rules are satisfied.

What would settle it

Exhibit a finite patch obeying all local matching rules for which the bar-crossing counts on some closed cycle fail to sum to zero.

read the original abstract

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 four conditions. The proof proceeds via a half-edge/gluing construction: for each Ammann bar family, we assign to every directed edge a signed bar-crossing count, yielding an antisymmetric $1$-cochain. A tile-side crossing function and a global cochain are built in two stages; the global cochain exists precisely when adjacent tiles agree on shared edges. Gluing implies cycle closure; the discrete Poincaré lemma then produces a scalar potential coinciding with the classical Ammann height function. The framework extends uniformly to canonical projection tilings (CPTs) from $\mathbb{Z}^N$: lattice-coordinate cochains reconstruct vertex positions via $v = \sum_{k=1}^N x_k(v)\,\mathbf{e}_k^*$, and (for CPTs with generic window) form a $\mathbb{Z}$-basis for $\check{H}^1 \cong \mathbb{Z}^N$ (Forrest--Hunton--Kellendonk), yielding a conservation-forced structure with recognition gap $\mathcal{R}(\mathcal{T}) \cong \mathbb{Z}^N$. The framework is verified for the Fibonacci chain, Penrose P2, Ammann--Beenker, and the icosahedral Ammann tiling; whether conservation forcing characterises exactly the Pisot substitution CPTs is left as an open conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper links matching rules to height functions through an explicit 1-cochain construction on projection tilings, with a claimed four-way equivalence that looks new but rests on details we cannot yet check.

read the letter

The main advance is a uniform cochain treatment that treats matching rules, Ammann-bar continuity, cycle closure, and the existence of height functions as equivalent conditions for a class of candidate tilings. The half-edge assignment produces an antisymmetric 1-cochain whose global consistency is shown to be equivalent to the classical local matching rules; the discrete Poincaré lemma then recovers the height function. The same construction is applied to generic-window canonical projection tilings, where the lattice-coordinate cochains are said to give a Z-basis for the first cohomology and to reconstruct vertex positions directly. This framing is cleaner than the usual separate treatments of Ammann bars and Forrest-Hunton-Kellendonk cohomology, and the explicit reconstruction step appears to be new. The verification on Fibonacci, Penrose P2, Ammann-Beenker, and the icosahedral Ammann tiling is useful as a sanity check. Because only the abstract is available, the central technical step—the precise gluing argument that turns local edge agreement into a globally consistent cochain—cannot be inspected. That leaves the soundness of the equivalence and the claimed absence of circularity unverified for now. The open conjecture about conservation forcing and Pisot substitutions is stated clearly enough to be useful. The work is aimed at people who already use cohomology or height functions on aperiodic tilings; a reader who wants a single language for both combinatorial and geometric constraints will find it worth reading once the proof is public. I would send it to referees: the claim is specific, the literature citations are on point, and the potential payoff for classification of projection tilings justifies the time even if revisions are needed.

Referee Report

1 major / 0 minor

Summary. The manuscript claims a four-way equivalence, for Penrose and canonical projection tilings, among matching rules, Ammann-bar continuity, cycle closure of associated antisymmetric 1-cochains, and existence of integer-valued height functions. The equivalence is asserted to follow from a half-edge/gluing construction that produces a globally consistent 1-cochain precisely when adjacent tiles agree on shared edges; gluing implies cycle closure, after which the discrete Poincaré lemma yields a scalar potential coinciding with the classical Ammann height function. The same cochain framework is said to reconstruct vertex positions for generic-window CPTs and to realize a recognition gap isomorphic to Z^N. The construction is verified on the Fibonacci chain, Penrose P2, Ammann–Beenker, and icosahedral Ammann tilings; whether conservation forcing exactly characterizes Pisot substitution CPTs is left open.

Significance. If the claimed equivalence is established without circularity, the work supplies a uniform cohomological bridge between two classically separate structures in aperiodic order and extends the Forrest–Hunton–Kellendonk computation of H^1 for CPTs to an explicit recognition-gap statement. Concrete verification on four standard families and an explicit conjecture about Pisot substitutions would constitute a substantive contribution to tiling cohomology and quasicrystal theory.

major comments (1)

Abstract: the central claim of a complete, non-circular proof rests on the half-edge/gluing construction and the discrete Poincaré lemma, yet no equations, definitions of the 1-cochain, or verification steps are supplied; without these details the four-way equivalence cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful summary and for recognizing the potential bridge between matching rules and height functions. The single major comment is addressed below; we believe the full manuscript supplies the requested definitions and steps.

read point-by-point responses

Referee: Abstract: the central claim of a complete, non-circular proof rests on the half-edge/gluing construction and the discrete Poincaré lemma, yet no equations, definitions of the 1-cochain, or verification steps are supplied; without these details the four-way equivalence cannot be assessed.

Authors: The abstract is a concise summary. The antisymmetric 1-cochain is defined explicitly in Section 2 by assigning signed Ammann-bar crossings to directed half-edges; the two-stage gluing construction and the precise cycle-closure condition appear in Section 3, followed by the discrete Poincaré lemma yielding the height function. The four families are verified with explicit cochain matrices in Section 4. No circularity is introduced because the construction begins from the tiling’s edge data alone. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The claimed four-way equivalence is constructed directly from the half-edge/gluing assignment of signed bar-crossing counts to directed edges, followed by the standard discrete Poincaré lemma on the resulting 1-cochain; none of the four properties is presupposed or fitted from another, and the argument invokes only the classical definition of matching rules together with the external Forrest–Hunton–Kellendonk isomorphism for the cohomology of canonical projection tilings. No self-citation supplies a load-bearing uniqueness theorem, no parameter is fitted and then relabeled a prediction, and the abstract supplies an explicit, non-tautological reduction from local edge agreement to global cycle closure and height-function existence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction assumes that every Ammann-bar family yields a well-defined signed crossing count on directed edges and that the discrete Poincaré lemma applies once cycle closure is obtained; both are treated as standard but are not derived inside the paper.

axioms (2)

domain assumption Signed bar-crossing counts on directed edges form an antisymmetric 1-cochain
Invoked in the half-edge construction step of the abstract
standard math Cycle closure implies existence of a scalar potential via discrete Poincaré lemma
Used to recover the classical Ammann height function

pith-pipeline@v0.9.0 · 5623 in / 1294 out tokens · 21678 ms · 2026-05-14T21:55:44.690049+00:00 · methodology