An Arithmetic Family of Aperiodic Tilings in Odd Prime Dimensions

We give a uniform arithmetic construction of aperiodic chair matching rules in every odd prime dimension, yielding an explicit infinite family. In the general family, local matching rules for fully framed chairs on a common lattice force a hierarchy at arbitrarily large scales. We also construct an explicit five-dimensional strongly aperiodic monotile, CLARK (Chair with Local Asymmetric Registration Keys): one shape that tiles space but permits no periodically repeating tiling. Its boundary alone enforces nonperiodicity, even when arbitrary translations, rotations and reflections are allowed. A d-dimensional chair is a side-two cube with one corner unit cube omitted. The registered construction uses affine maps over the finite field F_p; its tilings exist, have unique hierarchical decompositions and have finite Euclidean symmetry groups. A separate dense rational decoration gives physical monotiles with finite full tiling symmetry groups for primes p congruent to 3 modulo 4. CLARK is built from a carrier of 31 unit five-dimensional cubes with 256 pyramidal keys. Its compactness, tiling existence and absence of every nonzero translation period are proved in Lean and have passed independent verification. Every tiling by CLARK has at most 3,840 full Euclidean symmetries. An exact seven-dimensional body with 127 cubes and 1,024 keys has a computer-assisted proof of tiling existence and absence of nonzero translation periods; its complete Lean verification remains open. The uniform results remain ordinary mathematical proofs. Verification: the original CLARK compactness, tiling-existence and translation-aperiodicity theorem passed the official full preflight using Comparator, Lean, NanoDa and verified con-ron. The strengthened theorem also proves that every tiling has a finite full Euclidean symmetry group of size at most 3,840. It has passed local Lean checks, an independent local audit and the full statement comparison; its separate official full verification is in progress. No Palomar registry acceptance is claimed. Original verified run: https://github.com/jconorgrogan/palomar-chair-verification/actions/runs/37520435323Stronger proof source: https://github.com/jconorgrogan/palomar-chair-verification/tree/5f5ceedee770b76c21e8f1105e5e4e3b84b8d676Stronger verification run: https://github.com/jconorgrogan/palomar-chair-verification/actions/runs/37544058631 This revision expands the mathematical proofs for human readers and includes the new five-dimensional section and key diagrams. The paper, figures and author-controlled prose/data are licensed CC-BY-4.0. Project-owned proof, checker, replay and build code is licensed Apache-2.0. Third-party materials retain their licenses and notices; the package records the per-file scope. Author contribution and AI assistanceThe research direction and initial conjectures originated with the author. AI assistants across multiple models contributed to the mathematical arguments, computations, Lean proof development and drafting. The author directed the work and takes responsibility for the claims. Formal verification is reported separately for each result.