On the Perfect Number Ratio Function and the Existence of an Odd Perfect Number: Part III — The Divisor-Sum Offset and the Divisibility of the Proper-Divisor Sum

This paper, On the Perfect Number Ratio Function and the Existence of an Odd Perfect Number: Part III — The Divisor-Sum Offset and the Divisibility of the Proper-Divisor Sum, develops further the arithmetic framework established in Parts I and II of this research series. Let (w=2h+1) be an odd composite integer that is not a perfect square, let (S(w)) denote the sum of its proper divisors, and define the perfect number ratio function by (\Gamma(w)=S(w)/w). The paper introduces the divisor-sum offset (M(w)=|S(w)-w|) and investigates whether (w) divides (S(w)), a question logically prior to determining whether (w) is an odd perfect number. The paper establishes unconditionally that (S(w)) is odd for every integer in the specified search space, thereby excluding all even integer quotients in an equation of the form (S(w)=Aw). It further proves that a deficient integer cannot satisfy this equation for any positive integer multiplier and derives an explicit upper bound on the possible odd multiplier (A=2v+1\geq3), expressed in terms of the least prime divisor of (w) and the structure of its divisor-conjugate pairs. To establish this bound rigorously, the paper develops a self-contained theory of divisor-conjugate pairing, derives a closed rational expression for the sum of each conjugate pair, proves a strict monotonicity property, and establishes upper bounds for the total conjugate-pair sum. It then reduces the resulting inequality to a quadratic polynomial and provides a detailed analysis of its monotonicity, discriminant, and sign conditions, including an equivalent formulation directly in terms of (w). The least element of the search space, (w=15), is examined as a worked example in which the derived threshold excludes every odd multiplier (A\geq3). The paper also formulates an explicit open problem: whether any integer in the specified search space can satisfy (S(w)=Aw) for an odd integer (A\geq3). Finally, the paper distinguishes rigorously between unconditional theorems and consequences that depend on an unresolved hypothesis. In particular, the claim that divisibility of (S(w)) by (w) necessarily implies that (w) is perfect, and the claim that the specified search space contains no multiperfect numbers, are established only conditionally on the exclusion of all odd multipliers greater than one. These stronger conclusions are not claimed as unconditional results. The work contributes an explicit arithmetic obstruction and a framework for further investigation into odd perfect numbers, divisor-sum divisibility, and odd multiperfect numbers, while identifying precisely the open question that remains unresolved.