The Critical Level of Distribution for Integers Representable as Sums of Two Squares in Short

We prove that the indicator of integers representable as a sum of two squares attains the critical level of distribution 1/2 in short intervals. For H = x^{1/2}, we obtain Σ(x, H) ≪ H^{1/2 + o(1)}. The proof uses Ramanujan sums c_q(h), a divisor-square expansion to enforce even multiplicity for primes p ≡ 3 (mod 4), and a linear kernel with a structured-remainder decomposition. A lossless bridge connects the dispersion functional to the autocorrelation average, achieving the final bound.