Residual Divisor Certificates for the Erdos--Straus Equation large A Six-Class Sieve Modulo 840 and a Jacobi-Symbol Barrier

For a prime p and an integer a > p/4, write R = 4a − p and N = pa. The Erdős–Straus completion problem

4/p = 1/a + 1/b + 1/c

with this fixed first denominator is equivalent to the existence of a divisor d ∣ N² satisfying a single congruence d ≡ −N (mod R). This note records the resulting residual-shell calculus. In the hard prime class p ≡ 1 (mod 24), every shell has R ≡ 3 (mod 4) and the divisor search reduces to three residue targets attached to the possible p-adic origins 1, p, p² of the divisor. We prove an unconditional modulo-840 sieve: if p ≡ 1 (mod 24) is not a simultaneous quadratic residue modulo 5 and modulo 7, then the equation is solved already at residual R = 3 or R = 7. Hence any hard-class prime requiring R ≥ 11 lies in the six unit square classes

p ≡ 1, 121, 169, 289, 361, 529 (mod 840).

We also formulate a Jacobi-symbol obstruction which gives a sufficient algebraic reason for an entire residual shell to fail. These results are reductions and obstruction criteria; they do not prove the full Erdős–Straus conjecture.