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

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. Positive semidefiniteness of every Mₙ 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 Aₙ⁽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 ≥1:λₘᵢₙ(Aq-1⁽²⁾)≥η\ is syndetic. Directed MPFR interval arithmetic using rigorous enclosures of the first two ordinates certifies the explicit values L=111, η>2.8241555129529352103096443463274×10⁻⁶. 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≤ n≤13,895,119,949,858 contains at least one n for which Mₙ0. 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.