Effective Quantum Variance for Mellin-Neutralized Incomplete Eisenstein Observables: Exact Kuznetsov Reduction and Finite-Complexity Bounds

This paper develops an exact trace-formula framework for studying effective quantum variance of Mellin-neutralized incomplete Eisenstein observables on the modular surface. Starting from a compactly supported profile q, we consider the broadened and Mellin-neutralized family qT(x) = T-1/2q(x/T), QT^Δ(x) = (-12-∂ₓ)qT(x), VT^Δ(y) = y1/2QT^Δ(-log y), AT = EVT^Δ. The neutralizer enforces the exact mean-zero relation VT^Δ(1) = 0 and produces a Mellin packet with explicitly controlled exponential type. The first main result is an exact unfolding formula for incomplete-Eisenstein matrix elements. Using the Hecke-square identity, the matrix element is rewritten as ωⱼ(EV) = (π tⱼ)/L(1,sym²πⱼ) ∑d,q≥ 1 λⱼ(q²) HV(tⱼ; dq), where HV is an exact Whittaker-square transform. This removes the need for a preliminary Whittaker–Stirling approximation. For the full parity-complete real Hecke–Maaß spectrum, the harmonic finite-energy trace therefore admits the exact decomposition SR(V₁,V₂;w) = DR(V₁,V₂;w) + CR(V₁,V₂;w) + KR(V₁,V₂;w), with explicit diagonal, continuous-spectrum, and same-sign Kloosterman components. For fixed mean-zero profiles, this identity is reconciled with the Sarnak–Zhao quantum-variance form as R → ∞. To control profiles whose logarithmic support grows with T, the paper introduces the weighted Mellin-line norm V_ PA,B = _σ≤ B ∫R (1+τ)A V(12+σ+iτ) \, dτ. For the neutralized family, VT^Δ_ PA,B q,A,B T-1/2eBΛT, ΛT = Lq T. This profile ledger is combined with uniform Whittaker-transform estimates, an explicit contour-shift argument for the Bessel transform, the Weil bound for Kloosterman sums, and the global imaginary-order Bessel estimate J₂ᵢₜ(x) ≤ √(2π t) ≤ e^πt. The resulting unconditional finite-complexity bounds are DR R⁶ PT², CR _ε R5+ε PT², KR _ε R⁶ PT², where PT = VT^Δ_ P12,5/2 q T-1/2e(5/2)ΛT. Consequently, SR(T;w) q,w,ε R6+εT⁻¹e5ΛT. These estimates are deliberately non-sharp. Their purpose is to prove that the exact diagonal, continuous, and Kloosterman kernels can all be controlled by one fixed Mellin-line profile norm, without superexponential loss from the moving support. The remaining central problem is to extract the known Sarnak–Zhao main term effectively and prove an aggregate power-saving estimate of the form DR + CR + KR = R Hw QSZ(AT,AT) + O( R1-δ+ε NA(VT^Δ)² ) for some δ > 0. Such an estimate would yield unconditional positive quantum-variance persistence in an explicit logarithmically broadening regime. Version v0.3r3 is a proof-hardened reduction paper. It establishes the exact finite-R Kuznetsov architecture and the neutralized-family finite-complexity bounds, but it does not yet claim the aggregate power saving or the resulting unconditional logarithmic-persistence theorem.