Syndetic Two-Zero Frame Recovery and Bounded-Gap Positivity for a Riemann Xi Toeplitz--Hankel Family

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 $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0,\qquad$ $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}.$ Positive semidefiniteness of every $M_n$ is a previously established criterion equivalent to the Riemann hypothesis. We prove a bounded-gap unconditional positivity theorem for this family using only the first two positive zeta-zero ordinates and a rigorous high-zero tail bound. The two selected zeros generate a rank-two Gram core $A_n^{(2)}$. Its smallest eigenvalue has liminf zero, so no pointwise all-level lower bound is possible. Nevertheless, an exact four-frequency expansion of the squared wedge shows that the superlevel set $\{q\ge1:\lambda_{\min}(A_{q-1}^{(2)})\ge\eta\}$ is syndetic. Directed MPFR interval arithmetic using rigorous enclosures of the first two ordinates certifies the explicit values $L=111,\qquad \eta>2.8241555129529352103096443463274\times10^{-6}.$ Combining this recovery with the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$ and an unrestricted operator-norm estimate for all zero orbits above $H$ yields the following: every block of $111$ consecutive levels contained in $0\le n\le13{,}895{,}119{,}949{,}858$ contains at least one $n$ for which $M_n\succ0$. In particular, at least $125{,}181{,}260{,}809$ distinct levels in this range are unconditionally positive. We also give a separate qualitative proof of syndetic recovery through minimal rotation on a compact torus. The result is not a proof of the Riemann hypothesis and does not assert nonpositivity at any uncertified level.