Neutralizing the Eisenstein Continuum (v1.8): A First-Principles Tool for High-Precision Arithmetic Quantum Chaos

# Summary (v1.8) **Title:** Neutralizing the Eisenstein Continuum (v1.8): A First-Principles Tool for High-Precision Arithmetic Quantum Chaos **Author:** Lee Byoungwoo **Date:** 2026-01-02 ## OverviewThis note develops a **first-principles “neutralization” engine** for arithmetic quantum chaos (AQC): an operator-level preprocessing step that removes the dominant **Eisenstein (continuous-spectrum) contamination** from smoothed spectral observables. The core mechanism is a theorem-level cancellation of the ubiquitous square-root drift \(cX^{1/2}\) by **orthogonalizing admissible windows** against the rank-one continuous direction \(v(t)=t^{1/2}\). :contentReference[oaicite:0]{index=0} The pipeline is organized as:1. **Structural neutralization** in a Hilbert–Plancherel model \(H_{\mathrm{mock}}\). :contentReference[oaicite:1]{index=1} 2. **Automorphic projection + exact Mellin–L identity** with audited local/archimedean normalizations. :contentReference[oaicite:2]{index=2} 3. **Transduction:** transferring \(t\)-aspect subconvex (or averaged) inputs for \(L(1/2+it,\pi)\) into improved neutralized remainder exponents. :contentReference[oaicite:3]{index=3} ## Closed results### (1) Neutralization removes the \(cX^{1/2}\) drift at theorem levelFor admissible windows \(\psi_T\), define the orthogonalized window\[f_T \;=\; \psi_T \;-\;\frac{\langle \psi_T, v\rangle}{\langle v,v\rangle}\,v,\qquad v(t)=t^{1/2}.\]Then the leading continuum main term cancels **exactly**, leaving a controlled tail. :contentReference[oaicite:4]{index=4} ### (2) Calibrated tail bound, uniform over the parameter boxWith calibrated effective scale \(T\) (dyadic: \(T\sim T\); log–Gaussian: \(T\sim \lambda^{1/2}\)), the neutralized continuous contribution satisfies\[\mathrm{Tail}(X;T)\;=\;O_\varepsilon\!\bigl(T^{-3/2+\varepsilon}\bigr)\;+\;O_\varepsilon\!\bigl(X^{-1-2\eta}\bigr),\quad T\in\mathcal{B},\]and the \(-3/2+\varepsilon\) exponent is explained by a **Mellin zero at \(s=1/2\)** plus **log-scale stationary phase**. :contentReference[oaicite:5]{index=5} ### (3) Exact Mellin–L identity via an automorphic projectionA bounded Hecke-equivariant projection is recorded in the form\[P_{\mathrm{aut}} \;=\; \mathrm{Sh}\circ\xi_{1/2}\circ K,\]and the normalization is audited so that the exact identity holds:\[\int_0^\infty \Theta_\pi(t)\,t^{s-1}\,dt\;=\;\Gamma(s)\,L(2s,\pi).\](See the paper’s audit checklist A1–A8.) :contentReference[oaicite:6]{index=6} ### (4) Transduction: subconvexity \(\Rightarrow\) neutralized remainder exponentIf\[L\!\left(\tfrac12+it,\pi\right)\;\ll_{\pi,\varepsilon}\;(1+|t|)^{\,1/4-\delta_{\mathrm{in}}+\varepsilon},\]then the neutralized remainder obeys\[R_\pi(X)\;\ll_{\pi,\mathcal{B},\varepsilon}\;X^{\,1/4-\delta_{\mathrm{in}}+\varepsilon}.\]Under averaged/hybrid inputs, a baseline proof-level efficiency \(c\ge 2/3\) is recorded. :contentReference[oaicite:7]{index=7} ## What is new in v1.8- A fully **three-step “engine” presentation** (problem \(\to\) solution \(\to\) prediction), with a uniform calibration statement across dyadic and log–Gaussian windows. :contentReference[oaicite:8]{index=8} - Explicit **finite-rank (multi-cusp) neutralization** as a reader-facing box: orthogonal projection against the cusp span \(\mathrm{span}\{v_{a_k}\}_{k=1}^m\) cancels the same \(X^{1/2}\) drift in the multi-cusp setting (basis-invariant). :contentReference[oaicite:9]{index=9} - A **Maass-targeted interface** promoted from “future work” to a **conditional mini-theorem box** (assumptions \(\Rightarrow\) outputs), to address completeness expectations in AQC audiences. :contentReference[oaicite:10]{index=10} - Two end-to-end worked examples (testbed + cross-domain): - \(X_0(11)\) neutralized spectral masses consistent with the transduction slope. :contentReference[oaicite:11]{index=11} - 2D CFT / AdS\(_3\) modular-bootstrap mapping yielding improved neutralized Cardy–Rademacher remainders. :contentReference[oaicite:12]{index=12} - A numerical protocol (Huber regression + bootstrap CIs) and a simulation study quantifying bias removal under hidden \(cX^{1/2}\) contamination. :contentReference[oaicite:13]{index=13} ## Scope & non-toy status- **Unconditional core:** the neutralization (main-term removal) is structural and does not rely on subconvexity; quantitative tail estimates use standard scattering hypotheses (SD1)–(SD3). :contentReference[oaicite:14]{index=14} - **Target sector in worked examples:** holomorphic/\(\theta\) sector with \(P_{\mathrm{aut}}\) fully audited; the **Maass-targeted** pipeline is recorded as a conditional interface (mini-theorem box), not merely an outlook bullet. :contentReference[oaicite:15]{index=15} - **What is not claimed:** this note does not claim a new unconditional Weyl remainder bound; the goal is a reproducible removal of continuum leakage so discrete-spectrum statements can be tested and sharpened under known or conjectural inputs. :contentReference[oaicite:16]{index=16} ## Program closure and targetsThe engine is designed to be reusable across observables once three inputs are verified:1. admissible window family + calibration,2. scattering/control hypotheses for contour shifts,3. an automorphic interface giving an exact Mellin–L identity. Immediate targets for strengthening:- Implement and audit a **fully Maass-targeted encoder** (beyond the conditional interface box).- Push **window optimization** beyond the baseline efficiency \(c\ge 2/3\) by proving sharper Mellin concentration with controlled derivative-loss bookkeeping. :contentReference[oaicite:17]{index=17} ## Reproducibility (what to release / how to rerun)The paper specifies a reproducibility manifest including:- window definitions (dyadic / log–Gaussian) and calibration rule,- normalization audit checklist (A1–A8),- scripts to compute discrete remainders and to fit local log–log slopes (Huber + bootstrap),- diagnostic artifacts (orthogonality check, tail-budget overlays, negative controls),- worked-example figure artifacts (e.g., \(X_0(11)\), CFT remainder) and the bias-simulation plot/table. :contentReference[oaicite:18]{index=18} ## KeywordsArithmetic quantum chaos; Eisenstein series; continuous spectrum; scattering; neutralization; orthogonal projection; admissible windows; stationary phase; Mellin transform; automorphic forms; subconvexity; spectral form factor; Weyl remainder; modular bootstrap; Cardy–Rademacher.