This paper develops a theorem-level application of primal–dual Mellin neutralization to arithmetic quantum observables on the modular surface PSL₂ (Z) H. Starting from compactly supported logarithmic profiles, we construct a common family of mean-zero incomplete-Eisenstein observables AT. 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/ (2s) at s=1/2. For each fixed spherical Hecke–Maass cusp form uⱼ, the resulting diagonal matrix element satisfies ATuⱼ, uⱼ = Bⱼ q' (0) T^-3/2 + Oₐ, ₌, ( (1+tⱼ) ^ T^-M-3/2), for center-flat profiles of arbitrary fixed order M. The coefficient Bⱼ is explicit and is governed by the symmetric-square values L (1/2, sym²ⱼ) L (1, sym²ⱼ). 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, Vₕ^har (R, T;q) ₇, ₐ, ₌, R^1+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 Qₒₙ (AT, AT) = 12 ₑ G (u/T) q (u) ²\, du, with an explicit arithmetic-archimedean multiplier G. Consequently, Qₒₙ (AT, AT) = c₈₄q₂² + Oq (T^-2), where c₈₄ = (1/2) ² (1/4) ⁴576 > 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 c₈₄q₂². 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 L² 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.
Byoungwoo Lee (2026) studied this question.