Finite-R Poisson Transfer for Mellin-Neutralized Incomplete Eisenstein Observables I: Effective Aggregate Sarnak–Zhao Identification and Logarithmic Persistence

Abstract

This paper provides the technical proof carrier for the finite-$R$ reconstruction of harmonic quantum variance for a Mellin-neutralized moving family of incomplete Eisenstein observables on the modular surface. Starting from the exact parity-complete Kuznetsov decomposition $$S_R=D_R-C_R+K_R,$$ the paper develops a uniform moving-profile analysis of all three components. The main technical ingredients include finite-order Whittaker-to-physical-kernel transfer, unrestricted Eisenstein-depth localization, two-dimensional Poisson summation, exact dual arithmetic support, global three-variable nonstationarity, suppression of all nonzero Poisson aliases, mesoscopic low/high spectral assembly, exact Whittaker-defect closure, and demotion of all positive-order Whittaker zero modes. A central arithmetic result is the square-supported Poisson zero-mode identity $$\mathfrak S_c=\mathbf 1_{c=\square}\,c^{3/2}\varphi(c),$$ which yields the normalized modulus weight $$\frac{\varphi(r)}{r}=\sum_{d\mid r}\frac{\mu(d)}{d}$$ after writing $c=r^2$. The diagonal, continuous, and Kloosterman components are shown to satisfy $$D_R=RH_wQ_D+O_{\psi,w,\varepsilon}\left(R^{3/5+\varepsilon}T^{-1}e^{B_D\Lambda_T}\right),$$ $$C_R=RH_wQ_C+O_{\psi,w,\varepsilon}\left(R^{5/6+\varepsilon}T^{-1}e^{B_C\Lambda_T}\right),$$ and $$K_R=RH_wQ_K^\square+O_{\psi,w,\varepsilon}\left(R^{3/10+\varepsilon}T^{-1}e^{B_K\Lambda_T}\right).$$ After freezing one numerical profile and comparing the resulting scale-free component limits with the fixed-observable Sarnak–Zhao theorem, the main functionals are identified by $$Q_D-Q_C+Q_K^\square=Q_{\rm SZ}.$$ Consequently, the paper proves the effective aggregate trace theorem $$S_R=RH_wQ_{\rm SZ}(A_T,A_T)+O_{\psi,w,\varepsilon}\left(R^{5/6+\varepsilon}T^{-1}e^{B_*\Lambda_T}\right),$$ giving the first explicit positive spectral saving $$\delta=\frac16.$$ The limiting quadratic form is evaluated internally for the neutralized broad-window family: $$Q_{\rm SZ}(A_T,A_T)=c_{\rm IE}\|\psi\|_2^2+O_\psi(T^{-2}),$$ where $$c_{\rm IE}=\frac{\zeta(1/2)^2\Gamma(1/4)^4}{576\pi}>0.$$ It follows that, whenever $T(R)\to\infty$ and the logarithmic support radius satisfies $$\Lambda_{T(R)}\le c_*\log R$$ for sufficiently small fixed $c_*>0$, one has the unconditional persistence law $$\frac{1}{R}S_R\longrightarrowH_wc_{\rm IE}\|\psi\|_2^2>0.$$ The paper is designed as a public quantitative proof and audit carrier. It records the detailed Whittaker, Poisson, arithmetic, spectral, and continuous-spectrum estimates underlying the integrated parent theorem while maintaining an acyclic proof provenance. The exact finite-$R$ trace decomposition and moving Mellin ledger are restated and proved within the present manuscript, so no inaccessible internal theorem core is required as a proof input. The following are deliberately not claimed: a direct closed second-theta/Mellin formula for $Q_K^\square$, a unique intrinsic crossover scale, monotonicity in the moving width, a full two-parameter crossover law, or superlogarithmic reconstruction. These remain separate structural or crossover problems. Technical Proof Carrier Version v1.0.