Bilinear Rank-2 Factorization of Integrated Wronskian Kernels, Pure-Square Completion, and the 1+2d > 1 Montgomery-Vaughan Exponent Jump in the Critical Strip

Paper VIII of the series on the Transverse Wronskian of Riemann's Xi-Function. Establishes the universal (mn)^{-d} <= 2^{-d} hyperbolic diagonal advantage, order-r generalized von Mangoldt scaling b_n^{(r)}(X) = Lambda_r(n)/(ln X)^r and Stirling-Dirichlet convolution laws, the exact bilinear rank-2 factorization w_{mn}^{int}(d,t) = 2(cosh(d*omega_m)cosh(d*omega_n)-1)/d^2 reducing O(N(t)^2) off-diagonal double sums to O(N(t)) one-dimensional prefix sums with an Euler-Maclaurin phase transition N(t)^{1-4d} at d = 1/4 (sigma = 3/4), the pure-square completion ||psi_1 + Psi_{>=2}||^2 that cancels the -1 boundary term identically pair by pair against the critical-line ground state and eliminates the 1 + sqrt(2) root penalty, and the 1 + 2d > 1 exponent jump in L^2 combined with Preissmann's sharp Montgomery-Vaughan Hilbert constant c_MV = (4/3)*sqrt(6) and Gallagher-Sobolev pointwise transfer up to t = 10^8.