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 PSL₂ (Z) 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 ATuⱼ, uₖ=-2C₉₊q' (0) T^-3/2+O₉, ₊, ₐ (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 >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: 1RSR (Aₓ (ₑ) ;w) 0, (R) = (R) ^, >. Consequently, for every >0 one can choose a fixed compact Gevrey-linear profile for which the normalized variance remains positive on a certified O (R) path but collapses at T= (R) ^1+. 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`