Two molecules with four sticky ends can only fully bond into an aperiodic pattern, at least in simulation. But growing it by guessing hides provable dead ends

Programmable matter will have to answer a basic question: which local rules force which global order? This paper studies “autocrystals”, local pattern statistics that force a structure to have no repeating unit cell. It gives an exact recipe: two hexagonal building blocks, each free to rotate, with two complementary pairs of sticky ends. Every fully bonded assembly is a known aperiodic tiling, and that part is mathematically certified. The catch is growth. Attaching a piece only where exactly one state fits never leads to a dead end. But growth that has to guess can build patches that look perfect and provably can never be completed; we show one that completes to 12×12 but never to 14×14. To be clear, this is simulation and proof, not yet a lab result. DNA-origami tiles and DNA-coated nanoparticles look like the natural platforms to test it. Paper, code and certificates: doi.org/10.5281/zenodo.23541901