One-Step Fermat Factorization and Goldbach Representations with Almost-Equal Summands in Square-Centered Intervals

A semiprime N is one-step Fermat factorable if ceil(√N)² - N is a perfect square, so that Fermat's method splits N at its first trial. We show that the one-step semiprimes in the square-centered interval J_n = [4n²-n, 4n²+n] are exactly parametrized by Goldbach representations of 4n and 4n+2 with almost-equal summands: in the lower half by 4n = p+q with |p-2n| ≤ √n; in the upper half by 4n+2 = p+q with √(3n) < |p-(2n+1)| ≤ √(4n). The two halves occupy complementary residue classes of the prime gap modulo 4. Summing over n, the total one-step count reduces exactly to shifted-prime pair counts over a long range of shifts, and the averaged results of Matomäki–Radziwiłł–Tao (already at the classical range of Mikawa and Perelli–Pintz) yield an unconditional asymptotic: the number of one-step semiprimes in the union over n ≤ X of J_n is ~ (4/3)(3-√3) · X^{3/2} (log X)^{-2}. Pointwise, the theorem of Coppola–Laporta gives: for almost all n, semiprimes with midpoint 2n that split in O(n^{1/4+ε}) Fermat steps, with an asymptotic count. The natural boundary statement — almost all J_n contain a one-step semiprime — is governed by the binary Goldbach problem with almost-equal summands at window exponent 1/2; its cleanest sufficient form — a representation of 4n with both summands within √n of 2n, for almost all n — sits below the current record (7/12 for existence, 5/8 for counts) and is open even under GRH-type hypotheses. We state it as an explicit open problem with a precise barrier analysis. An exhaustive census to 10^8 supports the refined prediction: no J_n with 1,884,296 < n ≤ 10^8 lacks a one-step semiprime, and we conjecture that n = 1,884,296 is the last index for which J_n is one-step-empty. Stratified averages of the normalized per-index counts track the corresponding singular-series averages to four decimal places, and the zero statistics are well modeled by a moment-calibrated binomial void model with no additional fitted parameter.