A Theta Series for the Penrose Tiling: Mock Modularity and a Physical Principle.

This paper introduces a groundbreaking discovery that forges a direct link between number theory and the physics of aperiodic matter. We construct a theta series for the Penrose tiling—a foundational model for quasicrystals—and prove that it is a mock modular form, a sophisticated mathematical object previously studied in abstract contexts. The work provides a new, fundamental symmetry principle to explain the structure of quasicrystals, validating the framework against experimental X-ray diffraction data with remarkable accuracy. Furthermore, the theory yields a novel, falsifiable prediction for a long-range correlation function within the tiling, offering a clear path to experimental verification. This research presents a new paradigm for understanding aperiodic order and highlights the deep connections between pure mathematics and the physical world.<br><br>