VI- A CONSTRAINT-MATRIX COVERAGE FRAMEWORK FOR THE ERDŐS–STRAUS CONJECTURE

Abstract The Erdős–Straus conjecture asks whether, for every integer n ≥ 2, there exist positive integers x, y, and z such that: 4/n = 1/x + 1/y + 1/z. (1) Despite extensive theoretical study and reported computational verification through 10¹⁸, the conjecture continues to be regarded as open. Recent work has further clarified its modular structure, developed exact parametric solution families, and established modular characterizations of the underlying Diophantine equation. A constraint-matrix coverage framework organizes constructive approaches to the Erdős–Straus conjecture. For a selected modulus M, rows of a binary incidence matrix represent residue classes of n, while columns represent verified constructive identities or modular solution families. An entry records whether a family certifies the Erdős–Straus equation throughout the corresponding residue class. The resulting coverage matrix exposes uncovered classes and supports quantitative analysis of coverage multiplicity, overlap, redundancy, residue refinement, and minimal-cover formulations. A second component converts these uncovered classes into targets for constructive search. After selecting a first denominator x and defining A = 4x − n, the remaining two-unit-fraction equation admits the factorization: [(4x - n)y - nx] [(4x - n)z - nx] = n^2 x^2. (2) This factorization converts the determination of y and z into divisibility and congruence conditions and provides a systematic mechanism for searching for parametric identities tailored to uncovered residue classes. No proof of the conjecture follows from the framework alone. Its role is to integrate established modular coverage, matrix-based structural analysis, discrete optimization, and targeted symbolic search into a reproducible method for identifying where existing constructive families apply, where they overlap, and where additional identities are required.