Topological Quantization and Spectral Filtration: A Superdeterministic Framework for a Prime-Attentive Neural Network (PANN)

This paper addresses the geometric crisis in discrete computation by establishing a unified framework that synthesizes arithmetic topology, non-linear dynamics, and thermodynamic optimality. We posit that prime numbers behave as irreducible topological knots within a three-dimensional state space, a structure traditionally obscured by the stochastic nature of standard factoring algorithms. To validate this, we introduce the prime-attentive neural network (PANN), an architecture governed by a stochastic Reynolds-filtered strange loop (S-RFSL). This system utilizes a local-deterministic update rule to resonate with arithmetic invariants, effectively transforming prime factorization from a search problem into a spectral analysis problem. Our methodology employs a dual-track simulation protocol to verify both mathematical rigor and engineering feasibility: a 4th-order Runge-Kutta solver in a noiseless environment; and a massive CMOS energy penalty ($1000\times$ Landauer limit) to simulate physical hardware. Despite adversarial conditions, the system successfully factors the composite number $15$ by locking onto spectral modes. The realistic efficiency demonstrates that a topological approach maintains a net-positive utility over brute-force digital methods even in “dirty” physical environments. These findings support a superdeterministic interpretation of quantum-like correlations and suggest a viable path toward prime-attentive silicon that operates near the thermodynamic limits of computation.