Erdős-Straus Outside One Congruence Class

The Erdős-Straus conjecture asserts that for every integer $n \\ge 2$ the fraction $4/n$ is a sum of three unit fractions. It is open. What is settled is every congruence class but one. We give four parametric families, each verified as a polynomial identity after clearing denominators, covering $n$ even, $n \\equiv 3 \\pmod 4$, $n \\equiv 2 \\pmod 4$ and $n \\equiv 9 \\pmod{12}$. Together they dispose of every $n \\ge 2$ except those with $n \\equiv 1 \\pmod 4$ and $3 \\nmid n$, and we show that the conjecture is equivalent to that single class. A descent then narrows it again: a representable divisor represents its multiple, so the conjecture is equivalent to its restriction to the primes of that class, and a composite is settled the moment one of its prime factors is. The gap is thereby located rather than estimated: it is not a residue left over by the method but the place where the difficulty of the problem is concentrated.