Primal–Dual Mellin Neutralization in Arithmetic Quantum Chaos: Fixed-State Decay, Persistent Quantum Variance, and Noncommuting Limits

This paper studies how the scale of an observable interacts with the high-energy limit in arithmetic quantum chaos on the modular surface. We construct a family of mean-zero, weight-zero incomplete-Eisenstein observables using a primal–dual Mellin differential neutralizer. The neutralizer exactly cancels the relevant diagonal Rankin–Selberg pole and exposes a central Mellin zero. As a result, the matrix element of the observable against each fixed Hecke–Maaß eigenstate decays at the sharp scale T-3/2. Uniform Taylor estimates and symmetric-square L-value moment bounds extend this fixed-state result to natural and harmonic spectral families. In particular, the normalized variance collapses along polynomially broadening mesoscopic paths. For every fixed broadening parameter T, the observables also lie in the Sarnak–Zhao harmonic quantum-variance framework. We derive an exact Mellin-space variance transform and prove QSZ(AT,AT) = cIEq₂² + Oq(T⁻²), where cIE = ζ(1/2)²Γ(1/4)⁴/576π > 0. This produces a striking contrast. Each fixed eigenstate becomes asymptotically insensitive to the broadened observable, but the high-energy ensemble retains a positive collective variance. Consequently, the high-energy limit and the broad-window limit do not commute: limT→∞limR→∞ Wₕ(R,T;q) ≠ limR→∞limT→∞ Wₕ(R,T;q). The paper further proves explicit support and profile-seminorm growth estimates, a mean-zero ($m=0$) Euler–Maclaurin profile bound, and polynomial mesoscopic collapse for admissible rapidly decreasing spectral weights. A profile-uniform finite-R trace ledger is formulated as a separate conditional interface. Under this assumption, positive harmonic variance persists for an explicit logarithmically broadening range. This conditional statement is kept strictly separate from the unconditional fixed-state, fixed-T, noncommuting-limit, and polynomial-collapse results. Version v0.8r2 is a proof-correction and source-reconciliation release. It aligns the inverse Mellin formula with the convention f(s) = ∫₀^∞ f(ξ)ξ⁻ˢdξ/ξ, completes the associated Fourier–Plancherel justification, and preserves the mathematical statements, variance transform, and broad-window constant.