Prime Detection via Loop Holonomy and Quasimodular Defects

We propose a geometric framework linking partition-theoretic prime detection, quasimodular forms, and gauge-theoretic holonomy. Building on recent results that characterize primes via weighted partition functions Ma (n) Mₐ (n) Ma (n), we interpret integer partitions as loop configurations with associated winding multiplicities, and regard the generating functions of these quantities as quasimodular objects. In this setting, the quasimodular correction term—exemplified by the Eisenstein series E2 (τ) E₂ () E2 (τ) —is interpreted as a local defect field on the upper half-plane. We construct an su (2) su (2) su (2) -valued connection whose curvature encodes the deviation from modular symmetry, and define loop holonomy as the geometric observable associated with these defects. Under this interpretation, composite integers correspond to configurations with nontrivial loop interactions and nonvanishing holonomy, while primes arise as modes where interaction terms vanish and the associated holonomy becomes trivial. We further propose that quasimodular forms admit a decomposition into a modular component and a defect term expressed as a linear combination of partition-generating functions. This suggests a unified picture in which modularity corresponds to flat connections (vanishing curvature), and quasimodularity reflects the presence of residual geometric defects. While several components of this framework rely on established results in partition theory and quasimodular forms, the loop–holonomy correspondence and its geometric interpretation are presented as a conceptual model aimed at connecting arithmetic structure with noncommutative geometry. This perspective provides a new lens on prime detection and suggests further connections between number theory, topology, and gauge theory.