Primal–Dual Mellin Neutralization for Arithmetic Quantum Observables: Incomplete-Eisenstein Matrix Elements, Exact Harmonic Variance, and Noncommuting Limits

Abstract

This paper develops a theorem-level application of primal–dual Mellin neutralization to arithmetic quantum observables on the modular surface $$\operatorname{PSL}_2(\mathbb Z)\backslash\mathbb H.$$ Starting from compactly supported logarithmic profiles, we construct a common family of mean-zero incomplete-Eisenstein observables $A_T$. A finite-order differential neutralizer removes the actual diagonal Rankin–Selberg pole at $s=1$ exactly while preserving compact support. The remaining centered multiplier contains the universal vanishing factor supplied by $1/\zeta(2s)$ at $s=1/2$. For each fixed spherical Hecke–Maass cusp form $u_j$, the resulting diagonal matrix element satisfies $$\langle A_Tu_j,u_j\rangle = \mathfrak B_j q'(0)T^{-3/2} + O_{q,M,\varepsilon}\left( (1+t_j)^\varepsilon T^{-M-3/2} \right),$$ for center-flat profiles of arbitrary fixed order $M$. The coefficient $\mathfrak B_j$ is explicit and is governed by the symmetric-square values $$\frac{L(1/2,\operatorname{sym}^2\pi_j)}{L(1,\operatorname{sym}^2\pi_j)}.$$ Using short-interval mean-Lindelöf estimates for symmetric-square $L$-functions, the paper derives natural and harmonic spectral-family bounds at the arithmetic quantum-variance scale. In particular, for suitable smooth spectral weights, $$\mathcal V_h^{\mathrm{har}}(R,T;q) \ll_{h,q,M,\varepsilon} R^{1+\varepsilon}T^{-3}$$ in the admissible mesoscopic range. The paper then inserts the neutralized observable into the weight-zero incomplete-Eisenstein quantum-variance formula of Sarnak–Zhao–Zhao. The physical-space variance transform is diagonalized by the Mellin transform, yielding $$Q_{\mathrm{SZ}}(A_T,A_T) = \frac{1}{2\pi} \int_{\mathbb R} \mathcal G(u/T) \left\vert{}\widehat q(u)\right\vert{}^2\, du,$$ with an explicit arithmetic-archimedean multiplier $\mathcal G$. Consequently, $$Q_{\mathrm{SZ}}(A_T,A_T) = c_{\mathrm{IE}}\vert{}q\vert{}_2^2 + O_q(T^{-2}),$$ where $$c_{\mathrm{IE}} = \frac{\zeta(1/2)^2\Gamma(1/4)^4}{576\pi} > 0.$$ This gives a full even asymptotic expansion in powers of $T^{-2}$. The principal conceptual result is an exact noncommutativity of the high-energy and broad-window limits. Every fixed Maass matrix element vanishes at the rate $T^{-3/2}$, but the high-energy harmonic quantum variance retains the positive collective limit $$c_{\mathrm{IE}}\vert{}q\vert{}_2^2.$$ Thus fixed-state decay and ensemble-level arithmetic quantum variance exhibit genuinely different limiting behavior. Version v0.5r1 completes two proof-hardening steps required by earlier reviews: A spectral-uniform holomorphic-disk factorization, including explicit cancellation between the archimedean spectral power and the fixed-strip symmetric-square convexity power; A complete $L^2$ Mellin–Plancherel justification for the incomplete-Eisenstein variance transform. The paper does not claim a new QUE rate, a spectral-form-factor theorem, eigenvalue pair correlation, or an asymptotic evaluation of the remaining reciprocal-adjoint symmetric-square moment. Those problems are identified as separate arithmetic inputs for subsequent work. This paper applies and extends the framework developed in: Lee Byoungwoo, Primal–Dual Mellin Neutralization: Exact Residue Cancellation, Sharp $T^{-3/2}$ Tail Suppression, and Finite-Rank Window Design, Version v0.24r1, Zenodo, DOI: 10.5281/zenodo.21809522.