Critical Phase-Locked Zeta-Zero Windows and a Support-Scale Transition for Local Toeplitz--Hankel Positivity

Let $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad$ $\frac{\xi'}{\xi}\!\left(\frac1{1-x}\right)=\sum_{m\ge0}f_mx^m,$ and define $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0,\qquad$ $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}.$ The condition $M_n\succeq0$ for every $n\ge0$ is a previously established criterion equivalent to the Riemann hypothesis. Writing $q=n+1$, a critical-line zero $\frac12+i\gamma$ contributes a rank-one atom whose exact phase is $x=2q\arctan\frac1{2\gamma}.$ We study two fixed-width phase windows placed on opposite sides of an integer node $x_0(q)\asymp q^\beta$. Their inverse images have height $\asymp q^{1-\beta}$ and ordinate width $\asymp q^{1-2\beta}$. Endpoint subtraction from an explicit Riemann--von Mangoldt estimate therefore exhibits a transition at $\beta=1/2$: below the transition the windows contain $\gg q^{1-2\beta}\log q$ zeros, at the transition they contain $\gg\log q$ zeros when the limiting width exceeds the explicit error threshold, and above the transition the same endpoint-subtraction certificate becomes asymptotically negative. If the zeros in the windows lie on the critical line, the associated Gram-frame lower bounds have scales $q^{2\beta-3}\log q\quad(0\le\beta<1/2),$ $\qquad q^{-2}\log q\quad(\beta=1/2).$ We explicitly separate this conditional frame consequence from the unconditional zero-counting transition. For the critical choice $\beta=1/2$, we construct phase-locked windows around $\lfloor\sqrt q/(6\pi)\rfloor\pi$. Throughout the required range these windows lie far below the Platt--Trudgian verification height, so their atoms are unconditionally positive semidefinite. Directed-rounding estimates then give $M_n\succ0\qquad(0\le n\le52{,}039{,}046).$ Thus the first $52{,}039{,}047$ local inequalities are proved unconditionally. The finite endpoint is a corollary of the support-scale analysis, not a proposed structural threshold. The result is not a proof of the Riemann hypothesis.