Nonlocal Cancellation in a Theta-Kernel Decomposition of the Riemann \( \Xi \)-Growth Derivative: An Obstruction to Phase-Aligned Blockwise Positivity

Let $\Phi$ be the theta kernel of the Riemann \( \Xi \)-function and \( D(z)=\int_0^\infty\Phi(u)\cos(zu)\,du \). The Riemann hypothesis is equivalent to the positivity of \( \partial_y|D(x+iy)|^2$ for $y>0 \), and it is natural to seek that positivity by decomposing the growth derivative over the phase-aligned blocks \( J_m=[m\pi/x,(m+1)\pi/x] \) on which the oscillation completes a full period, and controlling each block separately. We derive the exact two-variable representation of \( \partial_y|D|^2 \), obtain the resulting longitudinal--transverse decomposition \( \partial_y |D|^2 = 2 \int_{0}^{\infty} \big[ Q_y(a)\sin(2xa)+\varepsilon_x(a;y) \big] da \), and establish a complete oscillatory hierarchy: the structural fact is that a phase-aligned block annihilates one Taylor order against a sine but two against a cosine. On compact phase regions where \( Q_y' \) is bounded away from zero, the longitudinal block is of order \( x^{-2} \) while the transverse residual is two powers smaller, with explicit leading coefficients. We then prove that the decomposition cannot localise the positivity. Writing \( S=\sum_mC^{(Q)}_m \) and \( E=\sum_{m}C_{m}^{(\varepsilon)} \), both sectors admit complete algebraic asymptotic expansions in \( 1/x \), and these are termwise negatives of one another: \( S+E \) is exponentially small while each sector is of size \( x^{-5} \), so that \( E/S\to-1 \)(Theorem 5). As a concrete consequence we obtain a representation-specific no-go theorem for universal phase-aligned block positivity: for every \( y>0 \) and all sufficiently large \( x \) at least one aligned block is negative (Theorem 6). The latter proof is fully analytic and unconditional, resting only on the expansion\( Q_y(a)=\frac{4}{3}y\,p'(0)\Phi(0)^2a^4+O(a^6) \) together with a rigorous proof that \( \Phi''(0)