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 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 has 31 unit five-dimensional cubes and 256 pyramidal keys. Its compactness, tiling existence and absence of every nonzero translation period are proved in Lean 4.19 and Lean 4.35.0-rc2. 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. Proof source: https://github.com/jconorgrogan/palomar-chair-verification/blob/9008dec3afc006a62d3d2fd6b49b12f189248f33/SparseMonotiles/T5Monotile.lean#L13-L19Repository: https://github.com/jconorgrogan/palomar-chair-verification The paper and its figures are licensed CC-BY-4.0. The linked project-owned proof, checker, replay and build code is licensed Apache-2.0. Existing third-party licenses and notices remain applicable. The record is a preprint and makes no claim of human peer review or priority. The exact CLARK theorem has the retained local Lean 4.19 and Lean 4.35.0-rc2 checks described in the paper; complete T7 and uniform Lean verification remain open. A failed first external preflight is retained separately; it is not independent-kernel or registry acceptance. 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. At preparation of this version, the official independent Palomar verification retry was pending. No Palomar registration or acceptance is claimed. Verification run: https://github.com/jconorgrogan/palomar-chair-verification/actions/runs/37520435323