On the Perfect Number Ratio Function and the Existence of an Odd Perfect Number: Part II — Divisor-Conjugate Pairing and Reduction to D

This paper formalizes a derivation that decomposes the sum of proper divisors of an odd composite non-square integer w into conjugate pairs {di, qi} satisfying diqi = w. We use this pairing to express the perfect number ratio function Γ(w) = S(w)/w as an explicit closed-form rational function parameterized by the independent variables h and yi. Through a complete algebraic derivation, we show that the open question of whether this ratio can equal 1 is strictly equivalent to the classical existence problem for an odd perfect number. We restrict the search space to non-prime, non-square odd integers and prove that this restriction is lossless. Finally, we establish the concrete arithmetic constraints and the Diophantine equation needed for the proofs given in the subsequent parts of this series.