Prime-shift operators on Björner's complex of squarefree integers: an exact kernel decomposition of the Mertens residual with critical exponent 1/2, rigorous recovery bounds for the Guinand–Weil pipeline, and a structural obstruction to the spectral modification

Abstract

Let $\Delta_n$ denote Björner's simplicial complex on the squarefree integers $m \leq n$ and let $H_n = \sum_{p \leq n} c_p (T_p + T_p^*)$ 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 $A : C_{\mathrm{even}} \to C_{\mathrm{odd}}$ of $H_n$ is the full sum $A = A_\uparrow + A_\downarrow$, with $A e_m = \sum_{p \mid m, \, pm \leq n} c_p e_{pm} + \sum_{p \mid m} c_p e_{m/p}$. (ii) The maximal elements lie in $\ker A_\uparrow|_{C_{\mathrm{even}}}$, not in $\ker A$. The earlier statement $\ker A = \{0\}$ is false: at $n = 10^4$, $\dim \ker A = \dim C_{\mathrm{even}} - \operatorname{rank}(A) = 3030 - 2141 = 889$. What is true is that $\ker A$ is strictly smaller than $\ker A_\uparrow|_{C_{\mathrm{even}}}$. (iii) The values $c_*$ reported in the earlier revisions correspond to $A_\uparrow$, not to $A$; at $n = 10^4$, $\operatorname{rank}(A_\uparrow) = 1125$, $\operatorname{rank}(A) = 2141$, so $c_* = 0.185$ while the full density is $c = 0.352$. (iv) The kernel of $H_n$ therefore has density $1 - 2c \approx 0.296$, not $1 - 2c_* \approx 0.630$. New exact results. (v) We prove that the maximal elements of $[1, n]$ are exactly $m = P_k m'$ with $P_k = p_1 \cdots p_k$ the $k$-th primorial, $m'$ squarefree coprime to $P_k$, and $m' \in (n/(p_{k+1} P_k), \, n/(p_k P_k)]$. This yields the exact formula for $M_{\mathcal{M}}(n)$. (vi) We prove that the correct decomposition of the Mertens invariant is $M(n) = M_{\mathcal{M}}(n) + \dim R_n^\uparrow - \dim R_n^{\uparrow, T}$, not the identity previously conjectured. Exact kernel decomposition (new in the sixth revision). (vii) We introduce the exact kernel $c_R^{\mathrm{exact}}(\rho, n)$ of the residual $\mathcal{R}(n) := M(n) - M_{\mathcal{M}}(n)$, prove that it converges to the asymptotic kernel $c_R^{\mathrm{asym}}(\rho)$ of the fifth revision as $n \to \infty$, and establish numerically that the convergence is power-law, $|c_R^{\mathrm{exact}}(\rho_k, n) - c_R^{\mathrm{asym}}(\rho_k)| \sim n^{-\alpha_k}$, over the first $175$ zeros of $\zeta$. The fitted exponents have sample mean $0.4918$ and sample standard deviation $0.0506$, consistent with the fifth-revision values $(0.4888, \, 0.0498)$ within the standard error. Because the $175$ fits share the same grid in $n$ ($80$ log-spaced points in $[10^2, 10^8]$), the naive standard error of the mean $0.0506/\sqrt{175} \approx 0.0038$ underestimates the true uncertainty: a block bootstrap with $B = 8$ contiguous blocks gives $\mathrm{SE}_{\mathrm{boot}} = 0.0078$, so the deviation of the sample mean from $1/2$ is approximately $1.0$ bootstrap standard error and the hypothesis $\alpha = 1/2$ cannot be rejected. A logarithmic-correction hypothesis $\Delta_k(n) \propto n^{-1/2} (\log n)^{-\beta_k}$ is rejected by the data: the fitted $\beta_k$ have sample mean $-0.081$ (wrong sign), the predicted deficit has the wrong sign, and the $t$-statistics $\beta_k / \mathrm{SE}(\beta_k)$ are distributed symmetrically around zero. Sensitivity to the truncation parameter $K \in \{10, 20, 30, 50\}$ in (24) is null to machine precision, so the truncation is not a source of systematic error. Correlations with $\log \gamma_k$ and with $k$ are $-0.011$ and $-0.067$ respectively. Rigorous recovery. (viii) We prove truncation bounds for the prime-power sum and the archimedean integral in the explicit formula, both of order $O(e^{-c^2/4})$ in the truncation parameter. (ix) We prove a stability theorem for ESPRIT: the recovery error scales as $\varepsilon \, e^{\sigma^2 \gamma_M^2}$, with the exponential amplification being intrinsic to the inverse problem. (x) We prove a bound on the Vandermonde conditioning $C_{M, \delta, K}$ 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 $\tilde{H}_n = (I - P_n) H_n (I - P_n)$ conjectured in the fourth revision cannot converge to the zeros of $\zeta$. (I) For $s > 1/2$, $\|H_n\|_{\mathrm{op}} \leq 2 P(s)$, 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 $p^j$, $j \geq 2$. 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 $\gamma_\rho$ is logarithmic; unlike (I) and (II), obstruction (III) is not proved rigorously here. We also correct an error of the fourth revision: the operators $\gamma_p = T_p + T_p^*$ do not commute in the truncated complex. No proof of the Riemann Hypothesis is claimed.