Prime-shift operators on Björner's complex of squarefree integers: spectral obstruction, kernel structure, and a Toeplitz--Hankel pipeline for the Guinand--Weil explicit formula

This is the authors' abstract. We don't add key points for this paper.

Let Δₙ denote Bj\"orner's simplicial complex on the squarefree integers m≤ n and let Hₙ = ∑p≤ ncₚ(Tₚ + Tₚ*) be the self-adjoint operator obtained from the prime-shift operators Tₚ with bounded weights cₚ. We present a corrected and extended analysis of Hₙ and of the Toeplitz pencil constructed from the Guinand--Weil explicit formula. Five groups of results. First, we confirm the structural properties of Hₙ: self-adjointness, Z/2-symmetric spectrum, the trace formula Tr(Hₙ²) = 2∑p≤ ncₚ²σₚ(n/p), and a rigorous Carleman bound for the limiting measure when cₚ=p⁻ˢ with $s>1$. Second, we correct several statements of the earlier revisions: the mixed products TₚTq* and Tq*Tₚ do not commute once the complex is truncated at n; the Mertens invariant is the index of the bipartite block A, not the signature of Hₙ (which vanishes by symmetry); and the kernel of the up-shift part A↑ restricted to C₁ has dimension π(n)-π(n/2), whereas ( Hₙ∩ C₁) = π(n)-π(n/2)-1 for n≥ 15. Third, we establish a structural decomposition of the kernel of A into maximal elements and a residual subspace of relations, and we compute the density of the maximal part in closed form: c₁ = ∑{k ≥ 0} (p_k⁻¹ - p{k+1}⁻¹) ∏j ≤ k (p_j + 1)⁻¹ = 0.5675527… We then show numerically that the residual rate is not a constant: Rₙ grows super-linearly in the observed range n≤ 2× 10⁵, with Rₙ/n increasing from $0.0196$ at n=10⁴ to $0.0209$ at n=10⁵. We therefore retract the numerical estimates c*≈ 0.185 and ρ≈ 0.0624 of the previous revisions; the true values satisfy c*≤ 0.1817 and ρ≥ 0.0691 at the largest computed n, with no evidence of stabilization. We also exclude the attractive candidate c*=18/π⁴ as incompatible with the data. Fourth, we verify numerically the independence of rank(Aₙ(s)) from the weight s (Proposition 4.3 of the second revision) for s∈\0,1/2,1,3/2,2\, confirming that the rank is a combinatorial invariant of the incidence matrix, and we note the need for large prime moduli to detect the true rational rank for s≠ 0. Fifth, we revisit the Toeplitz pencil for the explicit formula, confirming the prefactor 1/(4σ√π) of the prime-power term for the Gaussian test function, verifying the identity numerically with relative error below 10⁻¹³, proving an exact-recovery theorem for the pencil, and discussing the Hankel--ESPRIT pipeline and its bottlenecks. No proof of the Riemann Hypothesis is claimed. We emphasize that the numerical evidence assembled here constrains and falsifies several of the structural conjectures advanced in the first two revisions; the corrected picture is more conservative but more reliable.