Prime Numbers as Universal Optimization Primitives

The scholarly understanding of prime numbers is structured around four paradigms: quantum chaos, classical optimization, algorithmic information theory, and deterministic law. However, these paradigms are marked by unresolved tensions—dynamical, hierarchical, and methodological—that have hindered a unified theory. This work posits that the apparent complexity of primes is a representational artifact of the discrete integer manifold. We propose a new framework wherein prime numbers act as universal optimization primitives, whose fundamental property of indivisibility provides robust solutions to constraint-satisfaction problems across diverse physical and computational domains. The core contribution is a computationally verifiable, geometric embodiment of primality. We define a functor that lifts integers to the category of smooth manifolds, where an integer $n > 1$ is prime if and only if its associated manifold has exactly two critical points under a standard Morse function. This criterion is computationally tractable via homology, as the total homology rank is 2 for a prime and $2^k$ for a square-free composite with $k$ distinct prime factors. By transforming primality from an arithmetic property into a topological invariant, this framework formally resolves the methodological tension and provides a concrete, base-independent, and verifiable foundation for re-evaluating the disparate paradigms of prime number theory.