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

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 $v={\sum }_{k=1}^N{x}_k(v){\mathbf{e}}_k^{\ast }$, 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.