Sums of two squares with few prime factors, III: the dispersion obstacle, sieve barriers at the level 1/2, and short-interval mean squares

Part III of a series. Parts I (doi:10.5281/zenodo.23826176) and II (doi:10.5281/zenodo.23826297) proved the level of distribution 1/2 for the weighted sequence r(n) r(N − n) through the theta series θ²(mz) θ²(m'z) of quaternary quadratic forms, and with it Ω(n(N − n)) ≤ 7 and max(Ω(n), Ω(N − n)) ≤ 5 for every large N ≡ 2 (mod 4), with n and N − n sums of two squares. This part examines the three questions these results leave open and answers them to the extent that the present methods allow. (1) The symmetric remainder. The remainder R_d = Σd₁d₂=d R(d₁, d₂; N) used by the two-dimensional sieve is the coefficient of the projection of the cusp form Gt,d onto the joint (+1)-eigenspace of the Atkin–Lehner involutions W_p, p | d (an exact identity Σu|d Gtu,d/u = Πp|d (1 + W_p) Gt,d). Every Hecke component with η_p(f) = −1 for some p | (d, M₁) cancels, in particular every oldform of odd level in the terms of smallest level; numerically the cancellation is of square-root strength and no more, a dᵒ⁽¹⁾ gain. (2) The dispersion form. The Fourier transform of the circle x² + y² ≡ a modulo any odd c is a classical Kloosterman sum, with the character χ−4 cancelling exactly, and the remainder sum Σ λ_d λ'_e R(d, e; N) has as its leading term a trilinear form in the Kloosterman sums S(inv(16m)·N, |h|²; m'k) over the moduli c ≡ 0 (mod m'). The uniform bound of Part I is equivalent to square-root cancellation in the sum over k; square-root cancellation over m' would give the level 2/3. The single fixed argument N, carried by the unit inv(16m) modulo the modulus, is, in every assignment of Drappeau's bound and in every Cauchy–Schwarz and Poisson arrangement, the term that blocks a level beyond 1/2. By the Chinese remainder theorem the form factors into a bilinear form in the two moduli (k, m') in which the fixed argument enters as inv(16m)·N·inv(k) modulo m'; this is the shape in which a saving over the sieve modulus would have to be found. (3) Sieve barriers at the level 1/2. The weights r(n) r(N − n) cost the sieve dimension, 2 against 1. Selberg-type mollifiers truncated below the sifting limit are identically 1 on the sifted integers; mollifiers reaching the Richert range cost more level than they return; and a linear programme over all non-negative weight functions, separable or with a term for each pair of prime factors, finds Richert's linear weight optimal on the grid, so that the gap of 0.030 to (7,4) is structural. (4) Short intervals. In every interval (X, X + X^Δ] with Δ > 1/2, almost all N have the level of distribution Δ − ε, unconditionally. A short-interval mean square ΣX<n≤X+H |a_G(n)|² ≪ X^η (H + L) X ‖G‖²/L would raise it to (1 + Δ)/2 for every Δ > 0, and the same bound without the term L, for the theta forms alone, to 1 − ε; the latter interpolates between the individual conjecture of Part II at H = 1 and its mean-square theorem at H = X, and the data support it down to H = 16. Spectrally, the diagonal of the Hecke expansion satisfies the conjectured bound unconditionally for every H ≥ X^ε (Deligne, Shiu, Hoffstein–Lockhart and the Rankin–Selberg formula of Part II), so that the conjecture is exactly the cancellation of the off-diagonal; the term L is the coherence of the Poincaré series, which the theta forms do not share. The lattice-point form of the variance does not avoid this: the precision it requires lies below the floor of every circle-method error term. The deposit contains the paper (36 pages, 4 figures), the working notes, the Python scripts and their outputs. Source repository: https://github.com/Ruqing1963/two-squares-short-intervals (tag v1.0).