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

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/ζ(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 + Oq,M,ε( (1+tⱼ)^ε 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,sym²πⱼ)/L(1,sym²πⱼ). 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, Vₕʰᵃʳ(R,T;q) h,q,M,ε R1+εT⁻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 QSZ(AT,AT) = 1/2π ∫R G(u/T) q(u)²\, du, with an explicit arithmetic-archimedean multiplier G. Consequently, QSZ(AT,AT) = cIEq₂² + Oq(T⁻²), where cIE = ζ(1/2)²Γ(1/4)⁴/576π > 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 cIEq₂². 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.