Local Divisor Obstructions and Complementary Parametric Constructions for the Erdős-Straus Equation: An Exploratory Computational and Theoretical Study

The Erdős-Straus conjecture asserts that $4/n = 1/x + 1/y + 1/z$ admits positive integer solutions for all n ≥ 2. By classical reductions, difficulty concentrates on primes p ≡ 1 840. This paper presents an exploratory computational and theoretical study of a family of divisor-based constructions (Fₖ) targeting this residual class. We derive from first principles a parametric family indexed by k, reduce its solvability to a single divisor-residue condition modulo $R(k)$, and prove a Local Valuation Theorem characterizing exactly which prime-power components obstruct success. We highlight a key 2-adic precision subtlety in this theorem. Additionally, we evaluate a restricted squarefree-k subfamily against an independently generated adversarial sample of fourteen primes. We observe strong empirical complementarity between constructions: primes highly resistant to one are resolved trivially by the other (all 14 adversarial primes resolve at k ≤ 12). Finally, we correct an auxiliary claim regarding local obstructions, demonstrating that ~21.6% of local failures require a bounded-exponent criterion rather than a subgroup non-membership obstruction alone. no proof or reduction of the full conjecture is claimed