Exact Two-Branch Classification in a Prime-Square Divisor-Parametric Slice

This non-peer-reviewed preprint studies the divisor parametrization of theErdos-Straus equation in the fixed slice n = p^2, a = p for primes p congruentto 1 modulo 4. Writing N = bp + 1, g = gcd(b,p), and M = 4b/g, every divisorparameter k dividing pN is uniquely k = d or k = pd with d dividing N. Themain theorem proves that admissibility is equivalent, respectively, tod congruent to -p^2 modulo M or d congruent to -p modulo M. The paper recordsp-adic and denominator-swap orbit corollaries and a credited b = 2 application.The main theorem was independently rederived and supported by a computationalaudit of 4,680,578 branch certificates with zero mismatches. The manuscriptdoes not claim a proof of the Erdos-Straus conjecture, and its historicalpriority language is explicitly qualified.