Crystals, quasicrystals and autocrystals. Aperiodic order from local rules: Markov-field types, self-assembly and a route to Turing-complete matter

Local rules can force aperiodic order. Window statistics can be extreme in the convex body of realizable pattern frequencies, so every realization shows exactly them, and yet no periodic configuration has them: these are autocrystals. As a class they are statically Turing-complete: one complete autocrystal encodes every finite computation of a universal Turing machine.

The paper brackets the smallest windows and alphabets that support them. With two symbols, three sites never suffice, while eight sites and the 3×3 square do; a binary 4×5 window gives rational frequencies beyond pure point. It also proves:

• latent autocrystals exist, which no interaction selects as unique ground state;
• complete autocrystals have zero entropy;
• the binary 2×2 body is not semialgebraic, and all its faces up to dimension four are classified.

The physical application is two hexagonal species with four sticky-end sequences, whose fully bonded assemblies are exactly the Walton–Whittaker aperiodic tiling. Its orientation contrast diffracts into Bragg peaks. Its species contrast diffracts purely singular continuously and multifractally, with correlation dimension exactly 6 − log2 λ = 0.79976…, where λ is a root of a degree-17 integer polynomial. The frequency module is (1/27)Z[1/2], and the decoupled species form finite clusters with exact size recurrences. Protected cells, fractal one-contact domains (walls of dimension log2 3), growth as decoding and the budgets of computing with a mole of cells are worked out together with their limits.

Version 13 adds full, independently audited proofs of the lifting and entropy results. Its open problems are posed as checkable puzzles for AI (AI_puzzles.md). All certificates are exact and reproducible from the supplement, which has 43 verification packages. Prepared with Claude (Anthropic); several central results are due to Sol, an AI research system.