Odd Perfect Number Part 1

This is the authors' abstract. We don't add key points for this paper.

This paper is the foundational first part of a four-part investigation into the odd perfect number (OPN) existence problem. We reformulate the classical condition σ(n) = 2n using the perfect number ratio function Γ(n) = S(n)/n = σ(n)/n - 1, where S(n) represents the sum of the proper divisors of n. We partition the set K of odd positive integers into three disjoint subsets: odd primes (P), odd perfect squares (R), and odd composite non-squares (D = K (P ∪ R)). Using divisor parity results, we demonstrate that no element of P or R can be perfect. Combining this with Euler's canonical form, we eliminate the residue class 4y + 3 and establish that any odd perfect number N must reside in D and satisfy N ≡ 1 4. This paper establishes the foundational search space D ∩ \{n ∈ N : n ≡ 1 4\} for subsequent analytical work in Parts II–IV.