Journal of Modern Classical Physics & Quantum Neuroscience

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-temperature endpoint values, while a finite-register energy-topology contrast obeys a provable asymptotic null limit at fixed k. Residue compression evaluates 78,498 prime modes, representing 3,080,928,753 implicit unordered pairs, without pair enumeration. The construction therefore provides a solvable arithmetic-topological complex system in which prime multiplication acts simultaneously at the levels of composition, energy, and topology. It is an engineered programmable model rather than a microscopic theory of matter, and it supplies falsifiable finite-size and thermal signatures for future tests of arithmetic organization in physical systems.