Let $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad$ $F(x)=\frac{\xi'}{\xi}\!\left(\frac1{1-x}\right)=\sum_{m\ge0}f_mx^m,$ and define the symmetric Toeplitz--Hankel coefficients $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0.$ The adjacent matrices $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}$ form the local family in a previously established criterion equivalent to the Riemann hypothesis. We prove an unconditional finite-range positivity theorem for this family. The zero-pair moment representation of $g_{ij}$ expresses $M_n$ as a sum of rank-one polynomial atoms. A finite verification of the Riemann hypothesis up to height $H$ then splits this sum into a positive-semidefinite verified part and an unrestricted high-zero tail. Two rigorously isolated low zeros provide a positive core; every zero above $H$ is controlled in operator norm by an explicit zero-counting estimate. For the polynomial recurrence $B_0(t)=1,\quad B_1(t)=3-t,\quad B_{m+1}(t)=(2-t)B_m(t)-B_{m-1}(t),$ we derive the exact oscillatory form $B_m(4\sin^2\alpha)=\frac{\sin((2m+1)\alpha)}{\sin\alpha}$ and a closed formula for its two-point Christoffel--Darboux kernel. Using the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$, the first two LMFDB/Platt zero intervals, and directed-rounding MPFR arithmetic, we certify the kernel away from zero at every integer level $0\le n\le99{,}999$. The resulting positive-core lower bound exceeds the adversarial high-zero tail bound by more than ten orders of magnitude. Consequently $M_n\succ0\qquad(0\le n\le99{,}999),$ or equivalently the first $100{,}000$ adjacent Toeplitz--Hankel determinants are strictly positive. No assumption is made about zeros above the verified height. To the best of our knowledge, this is the first finite-height transfer theorem for this adjacent Toeplitz--Hankel family and the first rigorous certification of its initial $100{,}000$ strict inequalities. This is not a proof of the Riemann hypothesis.
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. We prove an unconditional finite-range extension for this family without scanning individual zero ordinates or individual Christoffel--Darboux values. Writing $q=n+1$, a critical-line zero $\frac12+i\gamma$ contributes a rank-one atom generated by a two-dimensional polynomial vector. In diagonal and anti-diagonal coordinates its exact phase is controlled by $x=2q\arctan\frac1{2\gamma}.$ We use two level-dependent ordinate windows $(q,5q/4],\qquad (2q/5,q/2],$ whose phase slopes have opposite signs. Every cross-window pair has wedge at least $c_*/q^3$, while an explicit zero-counting estimate supplies at least $q\log q/100$ zeros in each window. The resulting moving Gram core satisfies $\lambda_{\min}(A_{q-1}^{\rm mov}) \ge \frac{c_*^2}{10400}\frac{\log q}{q^3}.$ Combining this with the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$ and an unrestricted high-zero tail estimate gives $M_n\succ0\qquad(0\le n\le2{,}030{,}956).$ Thus the first $2{,}030{,}957$ local inequalities are proved unconditionally. We also show that every fixed finite zero core has smallest eigenvalue with liminf zero, explaining why level adaptation is structurally necessary for this method. To the best of our knowledge, the moving-window frame transfer and this finite-range theorem are new. The result is not a proof of the Riemann hypothesis.