Prime-shift operators on Bj\"orner's complex of squarefree integers: spectral obstruction, and an extended Toeplitz pencil for the explicit formula recovering 1839 nontrivial zeros of ζ

Let Δₙ denote Bj\"orner's simplicial complex on the squarefree integers m≤ n and let Hₙ = ∑p≤ n cₚ(Tₚ + Tₚ*) be the self-adjoint operator obtained from the prime-shift operators Tₚ with bounded weights cₚ. We prove three groups of results on Hₙ. First, Hₙ is self-adjoint with a Z/2-symmetric spectrum, its second moment admits the exact trace formula Tr(Hₙ²) = 2∑p≤ n cₚ²σₚ(n/p), and its moment sequence satisfies Carleman's criterion. Second, the kernel of Hₙ has positive density, with the exact dimension formula (Hₙ) = dₙ - 2rank(A) and the level-one dimension π(n) - π(n/2); the spectral measure of the positive part is heavy-tailed and mutually singular with the empirical measure of the nontrivial zeros of ζ, providing a structural obstruction to the Hilbert--P\'olya program on Δₙ. Third, we revisit the Toeplitz pencil constructed from the Guinand--Weil explicit formula. We show that the restriction Δₘₐₓ 0.3 imposed in an earlier version of this work is not intrinsic: the pole term (Δ/2)e^σ²/4 is an exact contribution of the explicit formula and can be retained without loss of numerical stability up to Δₘₐₓ~ 15 in double precision. With parameters σ = 2.5× 10⁻⁴, δ = 5× 10⁻⁴, K = 3× 10⁴, the pencil recovers $1839$ of the first 2000 nontrivial zeros of ζ with mean relative error $0.3249%$, median $0.4155%$, and maximum $0.5000%$. A reality test on the generalised eigenvalues---the deviation of |zₖ| from $1$---shows that $5576$ of $7000$ extracted modes lie on the unit circle within 10⁻⁴, with median deviation 3.75× 10⁻⁵, providing numerical evidence for the Riemann Hypothesis on this finite dataset. No proof of RH is claimed.