Prime-shift operators on Björner's complex of squarefree integers: an exact decomposition of the Mertens invariant, exact kernel asymptotics, rigorous recovery bounds for the Guinand–Weil pipeline, and a structural obstruction to the spectral modification

Let Δ_n denote Björner's simplicial complex on the squarefree integers m ≤ n and let H_n = Σ_p≤n c_p(T_p + T_p^*) be the self-adjoint operator obtained from the prime-shift operators T_p with bounded weights c_p. This paper is the sixth revision of [9], superseding the first five. It contains corrections of errors, new exact results, an exact kernel decomposition of the Mertens residual with an explicit critical exponent, rigorous stability bounds for the Toeplitz–Hankel pipeline, and a structural obstruction to the spectral modification conjectured in the fourth revision.

Corrections. (i) The bipartite block A : C_even → C_odd of H_n is the full sum A = A_↑ + A_↓. (ii) The maximal elements lie in ker A_↑|_Ceven, not in ker A. The earlier statement ker A = {0} is false: at n = 10⁴, dim ker A = 889. (iii) The values c_* reported in earlier revisions correspond to A_↑, not to A: at n = 10⁴, rank(A_↑) = 1125, rank(A) = 2141, so c_* = 0.185 while the full density is c = 0.352. (iv) The kernel of H_n therefore has density 1 − 2c ≈ 0.296.

New exact results. (v) We prove that the maximal elements of [1, n] are exactly m = P_km′ with P_k = p₁…p_k the k-th primorial, m′ squarefree coprime to P_k, and m′ ∈ (n/(p_k+1P_k), n/(p_kP_k)]. This yields the exact formula for M_ℳ(n). (vi) We prove that the correct decomposition of the Mertens invariant is M(n) = M_ℳ(n) + dim R_n^↑ − dim R_n^↑,T.

Exact kernel decomposition. (vii) We introduce the exact kernel c_R^exact(ρ, n) of the residual ℛ(n) := M(n) − M_ℳ(n), prove that it converges to the asymptotic kernel of the fifth revision as n → ∞, and establish numerically that the convergence is power-law, |c_R^exact(ρ_k, n) − c_R^asym(ρ_k)| ∼ n^−α_k, over the first 175 zeros of ζ. The fitted exponents have sample mean 0.4918 and sample standard deviation 0.0506. A block bootstrap with B = 8 contiguous blocks gives SE_boot = 0.0078, so the deviation of the sample mean from 1/2 is approximately 1.0 bootstrap standard error and the hypothesis α = 1/2 cannot be rejected. A logarithmic-correction hypothesis is rejected by the data.

Rigorous recovery. (viii) We prove truncation bounds for the prime-power sum and the archimedean integral in the explicit formula, both of order O(e^−c²/4) in the truncation parameter. (ix) We prove a stability theorem for ESPRIT: the recovery error scales as ε e^σ²γ_M², with the exponential amplification being intrinsic to the inverse problem. (x) We prove a bound on the Vandermonde conditioning that is exponential in M, identifying the fundamental numerical bottleneck of the pipeline.

Structural obstruction. (xi) We prove two independent obstructions showing that the spectral modification H̃_n = (I − P_n)H_n(I − P_n) conjectured in the fourth revision cannot converge to the zeros of ζ. (I) For s > 1/2, ‖H_n‖_op ≤ 2P(s), so the spectral measure is compactly supported. (II) The second moment of H_n is finite, while the second moment of the prime term in the explicit formula diverges: H_n misses all prime powers p^j, j ≥ 2. A third, heuristic obstruction (III) is that the density of eigenvalues of H_n in any fixed interval is non-logarithmic, while the density of zeros is logarithmic. We also correct an error of the fourth revision: the operators γ_p = T_p + T_p^* do not commute in the truncated complex.

No proof of the Riemann Hypothesis is claimed.