Two-Scale Quantum Variance for Mellin-Neutralized Incomplete Eisenstein Observables: Logarithmic Persistence, Near-Logarithmic Collapse, and Noncommuting Limits

# Two-Scale Quantum Variance for Mellin-Neutralized Incomplete Eisenstein Observables **Version v1.0 — public preprint** This paper studies a two-parameter quantum-variance problem for a moving family of mean-zero incomplete Eisenstein observables on the modular surface $$\mathrm{PSL}_2(\mathbb Z)\backslash\mathbb H.$$ A compact Mellin neutralization removes the diagonal Rankin-Selberg pole at $s=1$. After this cancellation, the centered reciprocal Euler factor forces every fixed cuspidal matrix element to disappear at the explicit scale $$\langle A_Tu_j,u_k\rangle=-2C_{jk}q'(0)T^{-3/2}+O_{j,k,q}(T^{-5/2}).$$ The high-energy ensemble behaves differently. For the same observable family, the Sarnak-Zhao variance form has a strictly positive limiting plateau, and an effective finite-height Kuznetsov analysis proves persistence along an explicit logarithmic support corridor. For compact Gevrey profiles of order $\sigma>1$ that are exactly linear near the origin, a spectral-uniform Gevrey refinement together with a sparse-square spectral large-sieve estimate yields a complementary simultaneous zero regime: $$\frac{1}{R}S_R(A_{T(R)};w)\longrightarrow0,\qquadT(R)=(\log R)^\gamma,\quad \gamma>\sigma.$$ Consequently, for every $\delta>0$ one can choose a fixed compact Gevrey-linear profile for which the normalized variance remains positive on a certified $O(\log R)$ path but collapses at $$T=(\log R)^{1+\delta}.$$ Thus one moving observable family exhibits: - explicit fixed-state disappearance;- persistent positive high-energy quantum variance;- a certified positive simultaneous logarithmic regime;- a certified near-logarithmic simultaneous collapse regime;- two genuine iterated limits that both exist but do not commute. The paper also isolates the analytic ingredients relevant to possible extensions to other arithmetic spectral settings: Mellin neutralization, an effective high-energy variance formula, and spectral-uniform control strong enough to enter a simultaneous limit. **Claim boundary.** The paper does **not** prove monotonicity in the moving scale, a unique critical scale, or a universal crossover law. The intrinsic two-parameter crossover remains open. ## Related public technical records - Finite-$R$ Poisson transfer / effective aggregate variance: DOI `10.5281/zenodo.21953833`- Earlier integrated Mellin-neutralization preprint: DOI `10.5281/zenodo.21954332`- Gevrey corridor technical source: DOI `10.5281/zenodo.21896694`- Fixed-shift/source-ancestry technical source: DOI `10.5281/zenodo.21976051`