BINARY GOLDBACH VIA A LOCALLY CENTERED L² RESIDUE-CLASS VARIANCE IN SHORT INTERVALS

We prove that every sufficiently large even integer N is a sum of two primes. Write N = 2x and parametrise representations byp = x + u and p′ = x − u, with u ranging over a short interval of length L, where x1/3+ε ≤ L ≤ x1/2−ε. On this interval we consider a z–rough Goldbach profile and its locally centered Barban–Davenport–Halberstam type residue-class variance ∥KQvt∥22 modulo Q.For z ≍ x1/3 and Q ≤ z, every z–rough integer has at most two prime factors, so the profile splits into a prime–prime channel and three semiprime channels. A Selberg/GPY (Goldston–Pintz–Yıldırım) moment computation for a surrogate detector yields a macroscopic variance floor on at least one admissible shift, proportional to the Goldbach singular series S(N). The semiprime channels admit Type II bilinear decompositions and satisfy a power-saving mean-square equidistributionestimate, proved via dispersion, reduction to bilinear Kloosterman sums of level Q, and a Kuznetsov/Deshouillers–Iwaniec spectral bound (with the Kim–Sarnak gap). A deterministic triangle inequality then forces a nontrivial prime–prime contribution on a hot shift, yielding primes p, p′ with N = p + p′.