How Much Arithmetic Can Zero Data Recover? Localized Explicit Formulas, Bandwidth Obstructions, and a Spectral Inverse Program for L-Data

Abstract

## Overview This preprint develops a deterministic spectral inverse framework for Booker's distributional $L$-data. Its central question is: > Which arithmetic coefficients can be recovered from specified observations of the zero distribution, and with what stability? The analysis begins with the corrected explicit formula, in which the known archimedean contribution is separated from the zero/pole distribution. This distinction makes it possible to compare full spectral information with compressed statistics such as pair correlation or nearest-neighbor spacing. A test of arithmetic bandwidth $R$ can sample the coefficient site $\log n$ nontrivially only when $$\log n < R.$$ Equivalently, the visible arithmetic range satisfies $$n < e^R.$$ Thus finite bandwidth imposes an exact arithmetic resolution barrier, while shrinking support eventually loses all nontrivial Euler information. ## Main results The paper establishes the following theorem package. ### 1. Bandwidth obstruction and arithmetic visibility A finite-band explicit-formula observation sees only coefficient sites lying strictly inside its support. The largest potentially visible integer is $$X_{\mathrm{vis}}(R)=\left\lceil e^R\right\rceil-1.$$ ### 2. Two-point non-identifiability Noncongruent finite spectra can have the same signed difference multiset. Consequently, pair-difference statistics are not injective on general spectral configurations and cannot by themselves serve as arithmetic certificates. ### 3. Full corrected spectral injectivity Once the archimedean term is known and removed, equality of the full corrected zero-distribution functional forces equality of all arithmetic coefficients. ### 4. Finite-band and finite-height reconstruction Localized packet tests recover arithmetic coefficients from truncated zero data, with explicit dependence on bandwidth, test smoothness, zero-counting growth, and spectral height. ### 5. Imperfect finite-zero-list stability Smooth spectral tapering yields deterministic error propagation for: - missing or spurious spectral points;- multiplicity mismatch;- zero-location uncertainty;- archimedean approximation;- quadrature error; and- auxiliary numerical error. The paper propagates externally supplied zero-list certificates. It does not itself certify completeness of a reported zero list. ### 6. Automorphic exact-data recovery range For a fixed cuspidal automorphic $L$-function, the standard $O(T\log T)$ zero-counting law gives uniform exact-data recovery throughout every range $$N(T)\le T^{1-\delta},$$ where $\delta>0$ is fixed and the test smoothness is chosen sufficiently high. ### 7. Joint arithmetic frames and coherent-noise recovery Explicit localized tests form a finite arithmetic sampling frame. Under coherent perturbations in the real dual of the unit $C^k$ test ball, bounded-overlap packet synthesis yields matching terminal and joint minimax risk $$\mathfrak R_{N,R,k}(B,\varepsilon)\asymp\min\left\{B,\varepsilon N^k\right\}.$$ Thus pointwise terminal recovery and joint $\ell^2$ recovery have the same optimal crowding exponent under coherent functional noise. ### 8. Positive and prime-supported structured classes The logarithmic crowding obstruction persists, along infinitely many bounded prime clusters, under nonnegativity and prime or prime-power support. In particular, the corresponding minimax risk satisfies $$\mathfrak R_{\mathbb P(P_\nu),R_\nu,k}^{+}(B,\varepsilon)\asymp\min\left\{B,\varepsilon P_\nu^k\right\}$$ along an infinite sequence of terminal primes $P_\nu$. ### 9. Formal local Euler-factor model For the formal degree-one local relation $$f_\alpha(p^m)=\frac{\log p}{p^{m/2}}\alpha^m,$$ and below the square threshold, recovery of the local Euler parameter has minimax scale $$\mathfrak R_{\nu,R_\nu,k}^{\mathrm{Eul}}(B,\varepsilon)\asymp\min\left\{B,\frac{\varepsilon P_\nu^{k+1/2}}{\log P_\nu}\right\}.$$ The additional factor $P_\nu^{1/2}/\log P_\nu$ is the normalization cost for recovering $\alpha_p$ from $$f(p)=\alpha_p\frac{\log p}{\sqrt p}.$$ ### 10. Multilayer Euler observations beyond the square threshold For arbitrary finite bandwidth, each Euler layer is divided into three types: - complete layers;- invisible layers; and- partially visible layers. Complete layers retain divided-difference cancellation. A partially visible top layer is suppressed by the boundary flatness of compactly supported $C^k$ tests. Consequently, the local linear inverse modulus remains $$\kappa_{\nu,R_\nu,k}^{\mathrm{Eul,lin}}(a)\asymp\frac{P_\nu^{k+1/2}}{\log P_\nu},$$ independently of how many Euler powers are visible and even when the top layer is cut partially by the observation bandwidth. ### 11. Square-layer neutrality and cubic crossover A complete square layer is locally minimax-neutral. Cubes and higher powers first affect the symmetric hard-pair modulus through a cubic remainder. For the distinguished hard direction $h=t w_\nu$, the symmetric observation difference satisfies $$\left\|\mathsf E_{\nu,R_\nu}(a\mathbf 1+t w_\nu)-\mathsf E_{\nu,R_\nu}(a\mathbf 1-t w_\nu)\right\|_*\lesssimt\frac{\log P_\nu}{P_\nu^{k+1/2}}+t^3\frac{\log P_\nu}{P_\nu^{3/2}}.$$ This yields the lower minimax scale $$\mathfrak R_{\nu,R_\nu,k}^{\mathrm{Eul,loc}}\gtrsim\min\left\{B,\frac{\varepsilon P_\nu^{k+1/2}}{\log P_\nu},\left(\frac{\varepsilon P_\nu^{3/2}}{\log P_\nu}\right)^{1/3}\right\}.$$ In the ultra-low-noise regime $$\varepsilon\lesssim(\log P_\nu)P_\nu^{-3k/2},$$ the first-layer rate remains sharp even when the full visible Euler tower is used: $$\mathfrak R_{\nu,R_\nu,k}^{\mathrm{Eul,loc}}\asymp\min\left\{B,\frac{\varepsilon P_\nu^{k+1/2}}{\log P_\nu}\right\}.$$ ## Conceptual conclusion The results distinguish several levels of spectral information: $$\text{pair statistics}\;<\;\text{finite-band observations}\;<\;\text{expanding-band observations}\;<\;\text{full corrected spectral data}.$$ Pair statistics are generally non-injective. Finite bandwidth resolves only a finite arithmetic range. Full corrected spectral data recover the complete coefficient distribution. Higher Euler powers add nonlinear information, but they do not automatically remove the logarithmic crowding obstruction. Complete square layers are minimax-neutral, partially visible layers are boundary-suppressed, and the first possible nonlinear improvement appears through cubic and higher-order curvature. ## Scope and claim boundary The results concern deterministic inverse problems for corrected explicit-formula observations and finite coefficient models. The paper does not: - construct distinct global positive $L$-data;- construct genuine automorphic hard pairs;- prove a new automorphic converse theorem;- prove that pair correlation determines arithmetic origin; or- use random-matrix statistics as a modularity certificate. Random-matrix statistics are treated only as possible priors for spectral regularity. ## Status of earlier claims This manuscript supersedes the author's earlier SAPZ-FLT and MUGS proof declarations. The claims that those earlier notes supplied an unconditional spectral-entropy proof of Fermat's Last Theorem or a proved equivalence between exact GUE statistics and modularity are withdrawn. None of those claims is used in the present paper. ## Deposit contents This Zenodo record contains the PDF preprint only. **Version:** v0.8r2 **Author:** Lee Byoungwoo **Publication date:** August 2, 2026