Gevrey Corridor Compression for Mellin-Neutralized Arithmetic Quantum Variance: Broad-Gaussian Dual Closure, Full Cubic-Log Harmonic Asymptotic, and a Fast-Side Renormalized Law

This work develops a quantitative crossover theory for Mellin-neutralized arithmetic quantum variance beyond the fixed-observable persistence regime. Starting from the previously established logarithmic persistence law, it analyzes the mesoscopic covariance kernel on bounded and growing Mellin microshift scales and identifies where the missing order-one quantum-variance mass can and cannot occur. For a rapid spectral cutoff $w$, let $\mathcal C_R(x,y;w)$ denote the normalized Mellin covariance kernel and let $$\mathcal C_{\mathrm{diag}}(x,y;w) = 3\kappa_w \int_0^1 \sin(xu)\sin(yu) \, du$$ be the explicit sine-diagonal kernel obtained from the two-piece symmetric-square approximate functional equation and the reciprocal-adjoint Kuznetsov diagonal. The first main result proves bounded-microshift local diagonalization: $$\mathcal C_R(x,y;w) \longrightarrow \mathcal C_{\mathrm{diag}}(x,y;w)$$ locally uniformly in $(x,y)$. Thus the order-one correction required by the parent persistence plateau does not arise from a hidden complementary kernel on bounded microshift sets. The local rigidity extends to expanding boxes. If $$X_R \to \infty, \qquad \frac{X_R^2}{\log R} \to 0,$$ then $$\sup_{\vert{}x\vert{}+\vert{}y\vert{}\le X_R} \left\vert{} \mathcal C_R(x,y;w) - \mathcal C_{\mathrm{diag}}(x,y;w) \right\vert{} \longrightarrow 0.$$ For compact Gevrey profiles of order $\sigma > 1$, Fourier decay simultaneously gives an outer localization at scale $C(\log R)^\sigma$. Consequently, for every fixed $0 < \delta < 1/2$, the order-one discrepancy between the parent persistence plateau and the sine-diagonal prediction is asymptotically confined to the quantitative escape corridor $$(\log R)^{1/2-\delta} < \vert{}u\vert{}+\vert{}v\vert{} \le C(\log R)^\sigma.$$ For nonzero odd Gevrey profiles $q$ and certified small fixed $\tau$, the signed contribution of this corridor converges to $$\left( \frac{\pi}{48} - \frac32 \right) \kappa_w \vert{}q\vert{}_2^2.$$ The second main result resolves the first perturbative correction to the local kernel. Writing $L_R = \log R$, the reciprocal-adjoint arithmetic moment admits the one-log-lower expansion $$\mathscr S_{R,w}(x,y) = \frac{H_w}{\pi^2}\Phi(x,y) + \frac{1}{\pi^2 L_R} \left[ (J_w + 4A_2 H_w)\operatorname{sinc}(x)\operatorname{sinc}(y) + i d_0 H_w(y-x)\Phi(x,y) \right] + o(L_R^{-1}),$$ where $$H_w = \int_0^\infty w(r) \, dr, \qquad J_w = \int_0^\infty w(r)\log r \, dr.$$ After combining this arithmetic term with the one-log-lower archimedean symbol, the full covariance satisfies $$\mathcal C_R(x,y;w) = \mathcal C_{\mathrm{diag}}(x,y;w) + \frac{1}{L_R}\mathcal G_{1,w}(x,y) + o(L_R^{-1}),$$ with the explicit Hermitian correction $$\mathcal G_{1,w}(x,y) = \frac{D_*^2}{\pi^2} \left[ (J_w + 4A_2 H_w)\sin x\sin y + i H_w \Delta_\infty(x-y) \int_0^1 \sin(xu)\sin(yu) \, du \right],$$ where $$\Delta_\infty = \frac{\zeta'(1/2)}{\zeta(1/2)} - 2 - \frac{\pi}{2} - \log(2\pi) - 2\gamma - d_0 \approx -0.046191417932242\ldots.$$ A key structural consequence is $$\vert{}\mathcal G_{1,w}(x,y)\vert{} \ll_w 1 + \vert{}x-y\vert{}.$$ Hence the first $1/\log R$ correction has at most linear microshift growth. In particular, the scale $\sqrt{\log R}$ arising from the earlier expanding-box error budget is not generated by the first explicit asymptotic coefficient; it is a remainder-budget boundary rather than a first-coefficient onset scale. Version v0.9r1 additionally closes the quantitative rapid-cutoff transfer through one-log order. After subtracting the explicit leading and first-order terms from each dyadic spectral block, the total remainder satisfies $$\sup_{\vert{}x\vert{},\vert{}y\vert{}\le X_0} \sum_{\nu\in\mathbb Z} \vert{}\mathscr R_{R,\nu}(x,y)\vert{} = o((\log R)^{-1}),$$ for every fixed bounded microshift box. This completes the passage from compact spectral cutoffs to the full rapid parent cutoff at the precision required by the one-log expansion. The results therefore establish bounded and expanding microshift rigidity, a quantitative Gevrey escape corridor, and the complete first one-log-lower covariance coefficient. They do not determine the actual mass-carrying escape saddle or the full crossover law. Preliminary analysis indicates that the next natural asymptotic regime is the fixed-physical-shift scale $x=\xi\log R$, where the coalescing sine packet is replaced by a two-frequency polar packet and diagonal/anti-diagonal resonance layers may emerge. That regime is left for subsequent work. Version v0.9r1 is the technical freeze of the bounded/expanding-microshift and first-coefficient stages of the crossover program.