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

Let ξ(s)=12s(s-1)π-s/2Γ(s/2)ζ(s), ξ'/ξ\!(11-x)=∑m≥0fₘxᵐ, and define gᵢⱼ=f|i-j|-fᵢ₊ⱼ₊₁+δᵢⱼf₀, Mₙ=pmatrixgₙₙ&gn,n+1n,n+1&gn+1,n+1pmatrix. The condition Mₙ0 for every n≥0 is a previously established criterion equivalent to the Riemann hypothesis. Writing $q=n+1$, a critical-line zero 12+iγ contributes a rank-one atom whose exact phase is x=2q12γ. We study two fixed-width phase windows placed on opposite sides of an integer node x₀(q) q^β. Their inverse images have height q1-β and ordinate width q1-2β. Endpoint subtraction from an explicit Riemann--von Mangoldt estimate therefore exhibits a transition at β=1/2: below the transition the windows contain q1-2βlog q zeros, at the transition they contain 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 q2β-3log q(0≤β<1/2), q⁻²log q(β=1/2). We explicitly separate this conditional frame consequence from the unconditional zero-counting transition. For the critical choice β=1/2, we construct phase-locked windows around √ q/(6π)π. 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 Mₙ0(0≤ n≤52,039,046). 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.