Divisor-Ratio Arithmetic and Cross-Shift Constraints in the Erdős–Straus Problem

For primes p ≡ 1 ( mod 24), we study the equation 4/p = 1/x+1/y +1/z through theexact divisor tests associated with x = (p − 1)/4 + h. We develop explicit Type II residuefamilies covering 25651/26496 of hard-prime progressions at parameter depth twelve,exact Type I criteria at selected affine indices, a universal affine Jacobi-square identity,and a density-one theorem giving arbitrarily many distinct strict Type I witnesses atevery prescribed finite set of indices. Structural results give sublinear cutoffs in thezero-remainder family, terminal exclusion bands, finite-core classifications, and restrictedsieve bounds. New cross-shift theorems show that neither arbitrary coprimality norbounded prime support at finitely many shifts forces a solution. A nonresidue-factorcriterion supplies a positive sufficient condition at prime moduli. A fixed-characterreciprocity theorem gives a sharp pointwise supply of nonresidue shifts and a positiveprincipal–Jacobi Fourier budget; other characters may still cancel it. Exact computationsand two-dimensional figures distinguish the proved statements from finite diagnostics.The remaining universal cross-shift existence statement, equivalent to the hard-primecase of the Erdős–Straus conjecture, is not proved here.