The Operator-Topological Impossibility of Odd Perfect Numbers: A Proof via Rough Operator Algebra (ROA) and Seonggil Theory of Composite Torsion (STCT)

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 operator topology. Using the framework of Rough Operator Algebra (ROA) and Seonggil Theory of Composite Torsion (STCT), we map the arithmetic abundance function I(n) = σ(n)/n onto a logarithmic state density Ψ(n) within a non-commutative Hilbert space. We define a Defect Operator ˆD and derive the Seonggil Trace Formula (STF), which relates the spectral density of the Hamiltonian to the distribution of Riemann zeta zeros. We demonstrate that the ’Relative Distance Phase’ between odd primes creates a topological singularity that prevents the system from reaching the equilibrium state I(n) = 2. Finally, we show that the Seonggil-Riemann Error Constant ESR is strictly positive, implying that the topological matching required for an odd perfect number is physically and geometrically impossible.