2026-10-23
Two rotating hexagons with four sticky ends fully bond only as the Walton-Whittaker aperiodic tiling, at 1:1. Forced growth of 2484 particles extends to a valid 2048×2048 grid.
A quasicrystal is ordered and has no repeating cell. On a lattice the question is narrow: can an energy that sees only a finite neighborhood, and is the same everywhere, make every ground state nonperiodic? Wang tiles answer with colored edges that must match. The smallest aperiodic set uses 11 tiles. This paper measures size by how many cells the window contains and how many symbols it uses. Slide the window across a configuration and the pattern frequencies form a convex body. A local energy is a linear function of those frequencies, so a ground state is the face a supporting plane cuts. If that face contains no frequency of a periodic tiling, the interaction has selected a quasicrystal in the sense used here.
An autocrystal, or w-quasicrystal, is weaker. The frequencies are an extreme point, and no periodic tiling realizes them. A complete autocrystal asks the same of every large enough window. An extreme point is not a mixture of two other attainable frequencies, so the local statistics are rigid. The name means the statistics pin themselves down. It does not mean a molecule assembles on its own. Shechtman's metallic phase took the 2011 chemistry Nobel, and physical quasicrystals are recognized by diffraction. This paper asks for something narrower than a sharp peak.
A window that lies inside a line never works, for any alphabet. Three cells with two symbols never work. From 3 to 10 symbols the three-cell window is still open; from 11 symbols on, every window that is not collinear works. With two symbols, the 3×3 square, the 3×4 rectangle and a 2×7 strip work with rewards 0 and −1, and one eight-site window works with integer weights. The smallest binary window therefore has between four and eight cells. The 2×2 square remains open, and it is the central question of the paper.
The 3×3 construction encodes the 16 Ammann-Labbé Wang tiles into small blocks. That tiling is uniquely ergodic, its frequencies involve √5, and it is measurably a rotation of a torus, so the diffraction is pure point. Of 512 patterns, 165 receive reward 0 and the other 347 are penalized. The certificate reruns: 51 positive source squares, 1133 completable 4×4 words, 280 live words after 14 rounds of pruning. The ground state is unique and nonperiodic.
The molecular target is the Walton-Whittaker hexagon. Two rigid bodies, six orientations each, twelve local states. The four recognition sequences are s0, s1 and their complements. Each component carries eight overhangs, and some edges carry both sequence types. Of 864 directed contacts, 360 are allowed, each through exactly one complementary pair. One pair is never enough, and one scalar sticky end on an edge is never enough. Species B is species A with the two strands exchanged, so no mirror image is required. With equal bond energies the fully bonded configurations are exactly these tilings, and each species has frequency 1/2.
One species leaves at least 1/4 unbound contact per site, with best energy −11/4 ε against −3ε for full bonding. With vacancies, chemical potential above −3ε makes the aperiodic tiling the unique ground state, and below −3ε the ground state is the empty lattice. Any chemical-potential difference between the species hands the ideal ground state to dyadic tori that carry two defects. At full occupancy, with both channel strengths positive and the mix locked at exact equimolar composition, the aperiodic law is the unique ground law. A difference in channel strength can cancel the bias.
Growth places a particle only when the neighbors leave a single state, and an extendible seed stays extendible. Guessing commits to dead ends. A contact-valid 4×4 completes to 12×12 and not to 14×14: a verifier replays 1284 deletions until a domain is empty. The central 24×24 of a guessed cluster of 1742 particles is refuted at width 18.
| Question | Baseline | Result |
| Smallest binary window | Any three cells | Never enough; under 0/−1 rewards, 165 of 512 patterns in the 3×3 are legal, and the ground state is unique and nonperiodic |
| Extreme points of the binary 2×2 | At most 9 positive tiles | A nonperiodic extreme point needs at least 10 positive tiles; existence is open |
| 22,000 random plaquette Hamiltonians | Box upper bound vs an explicit periodic energy | After rounding coefficients to multiples of 1/1024, all have periodic ground states; 21,983 are settled by the 113 vertices of the 3×3 box |
| Four sticky ends | One pair, or one species | Fully bonded assemblies are the tiling, at 1:1; one pair is impossible; one species leaves at least 1/4 broken contact per site |
| Forced growth | Heat bath and reversible attachment | 2484 particles from a 16×16 seed and zero bad contacts; a cooled heat bath still leaves about 1.6% incompatible contacts |
| Annealing the binary 3×3 rules | Ideal statistics on a 60×60 torus | After 10^4 sweeps, 0.085 defective windows per site, ℓ1 distance about 0.6 |
The patch of 2484 particles is the largest forced closure in the batch. Its orientation pattern appears 340 times in each rotation of the reference crop. With charges included it sits in a valid 2048×2048 configuration: 12,574,721 bonds and no violation, at least 897 sites from the boundary. At a substitution singularity, seeds of side 8, 16 and 48 force 511, 2047 and 8929 particles, spanning 31, 63 and 127 sites, in line with dyadic supertiles. The paper does not prove that the next level is a free choice. Orientation contrast produces Bragg peaks that grow like N². Species contrast is continuous, and the measured finite patches grow about like N^1.6 for N from 64 to 512. Among random texture stimuli that obey every conservation law, 98.2% are realizable, 1.1% fail an exact inequality, and 0.7% stay undecided.
DNA-tile work gets a contact table that can be checked. Eleven Wang tiles become two rigid bodies and four recognition sequences, and two independent routes agree on all 864 directed contacts. Hexagonal and triangular DNA origami are scaffold precedents, and neither implements this table. A rigid geometry, selective bonding and assembly kinetics are still missing.
The same convex body certifies a ground-state search. A linear program on a finite box is an upper bound, and a periodic tiling whose energy meets it proves that nothing lies lower. Of the 22,000 samples, 16,000 have Gaussian coefficients and 6,000 heavily penalize one or two tiles. None left a persistent gap. A search that does not halt does not prove aperiodicity.
The realizable body of the binary 2×2 window is not a polytope and not a semialgebraic set. One line of Hamiltonians passes through infinitely many periodic phases. The ten binary texture coordinates of vision science have this body as their attainable range, so every conservation law can hold and still no texture exists.
Sol, an AI research system, proved that latent autocrystals exist, that complete autocrystals have zero entropy, that a 4×5 window carries rational autocrystals past pure-point spectrum, and that a nonperiodic extreme point needs at least ten positive tiles. Claude produced the two-pair contact table and the exact rerun of the 22,000 Hamiltonians; Sol recovered the same table on its own. The 3×3 and 3×4 certificates were reimplemented independently. The integer tables for the eight-site window were not. Finite certificates rerun in the Python standard library and are not yet in a proof assistant.
No material performance is shown, and no aperiodic phase at positive temperature either. The model has discrete lattice states only. Continuous positions, vibrations and three-dimensional stacking lie outside it. Finite window frequencies cannot certify positive configurational entropy, because every compatible finite marginal has a zero-entropy realization.
The gap is small. At ℓ1 distance 1/5 it is at most 1/524288 of the penalty per site, and the temperature test requires kB T below about 7.7×10^{-7} J. Annealing the two-species contact model on a 48×48 triangular torus still leaves 0.069 unbound contacts per site after 10^5 sweeps, falling roughly as t to the power −0.2.
The 2048×2048 configuration remains conditional: the collar needs a separately proved infinite background, and a finite crop is not one. The most productive 16×16 seeds stop between 1611 and 2484 particles, well short of a 96×96 region. Hierarchical assembly is named as the next step and is not carried out.
Local kinetics stay far from this certificate. Reversible attachment at τ=2 leaves about 0.7 bad contacts per particle. At 2.85 that falls to about 0.05, while the time grows from about 5×10^4 to about 2×10^9 in units of 1/kf. Three bonds stall the runs near 40 particles. A heat bath with no look-ahead still leaves about 1.6% incompatible contacts after cooling. Arc consistency scans the whole frontier, and a molecule does not run that scan. In the freely rotating model the all-A state yields when the chemical-potential gap equals ε, with a composition jump of at least 3/16. If the mix cannot be locked at equimolar composition, the 1:1 result does not reach an experiment.