Prime Multiplication as a Physical Organizing Principle: Topological Coarse-Graining, Thermal Fixed Points, and Finite-Size Scaling

Phan Thành Trung Independent Researcher Date Written: August 05, 2026 Abstract Multiplicative arithmetic provides a natural composition law, but its role as an organizing principle for physical observables is largely unexplored. We construct an exactly reducible quantum-statistical ensemble in which the same two distinct primes determine a composite excitation energy and a topological Wilson observable. A two-mode state (p,q) has energy ε ln(pq) and is assigned the torus knot T(p,q) in SU(2) Chern-Simons theory at level k. At fixed level k, the Wilson value depends only on p and q modulo m k = 2(k+2). This yields an exact coarse-graining from an unbounded prime-labelled state space to a finite congruence-sector alphabet and reduces the thermal expectation to a finite quadratic form in partial prime-zeta functions. The resulting nonlinear thermal observable has analytic high-and low-tempe