Holonomy Obstruction and Primitive Selection in Cohomological Winding Systems

This work proposes a structural framework for understanding prime numbers from a topological and variational perspective. Instead of treating integers purely arithmetically, it interprets them as configurations of interacting loop-like structures whose complexity is determined by nearby factorisation patterns. In this setting, composite numbers correspond to configurations that can be decomposed into multiple interacting components, while primes arise as configurations that cannot be further decomposed. The framework combines a variational layer, which introduces an energy functional driving configurations toward simpler states, with a topological layer that provides the rigorous foundation. The key proven result is that primitive monodromy implies primality, establishing a precise link between topological irreducibility and prime numbers. Building on this, the paper introduces a holonomy obstruction framework, in which increasing structural complexity leads to noncommutative interactions and the emergence of nontrivial holonomy, effectively locking the configuration and preventing further decomposition. As a result, complex configurations undergo an irreversible reduction process in which interacting components are progressively eliminated until a single indecomposable structure remains. This final state is interpreted as a primitive configuration corresponding to a prime. The work does not claim to derive primes dynamically, but rather identifies the structural conditions under which the topological selection rule applies, offering a unified perspective connecting arithmetic complexity, geometric obstruction, and topological irreducibility.