Non-commutative Arithmetic Thermodynamics and the Emergence of the Arithmetic Cosmological Constant: A Proof via ROA, STCT, and Hypertensor Mechanics

The search for odd perfect numbers (OPN) has remained one of the most enduring challenges in number theory since antiquity. This paper provides a definitive proof of the non-existence of OPN by shifting the problem from discrete arithmetic to continuous non-associative operator topology. Using the framework of Rough Operator Algebra (ROA), Seonggil Theory of Composite Torsion (STCT), and infinite-dimensional Hypertensor mechanics, we map the arithmetic abundance function I(n) = σ(n)/n onto a logarithmic state density Ψ(n). We define a Defect Operator ˆD driven by the G_2 gauge associator and derive the generalized Seonggil Trace Formula (STF) featuring a continuous topological cut-off. We demonstrate that the absence of the even prime ’2’ creates a fundamental breakdown of associativity (Topological Singularity) between odd prime fields, physically preventing the system from reaching the equilibrium state I(n) = 2. Finally, we mathematically establish that the resulting Seonggil-Riemann Error Constant E_SR is strictly positive and acts as the exact geometric origin of the Arithmetic Cosmological Constant(Λ_arith) driving the accelerated expansion of the physical universe.