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

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 SR=DR-CR+KR, 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 Sc= 1c=\,c3/2φ(c), which yields the normalized modulus weight φ(r)/r=∑d rμ(d)/d after writing c=r². The diagonal, continuous, and Kloosterman components are shown to satisfy DR=RHwQD+Oψ,w,ε(R3/5+εT⁻¹eBDΛT), CR=RHwQC+Oψ,w,ε(R5/6+εT⁻¹eBCΛT), and KR=RHwQK⁺Oψ,w,ε(R3/10+εT⁻¹eBKΛT). 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 QD-QC+QK⁼QSZ. Consequently, the paper proves the effective aggregate trace theorem SR=RHwQSZ(AT,AT)+Oψ,w,ε(R5/6+εT⁻¹eB_*ΛT), giving the first explicit positive spectral saving δ=16. The limiting quadratic form is evaluated internally for the neutralized broad-window family: QSZ(AT,AT)=cIE\|ψ\|₂²+O_ψ(T⁻²), where cIE=ζ(1/2)²Γ(1/4)⁴/576π>0. It follows that, whenever T(R)→∞ and the logarithmic support radius satisfies ΛT(R)≤ c_*log R for sufficiently small fixed c_*>0, one has the unconditional persistence law 1/RSRwcIE\|ψ\|₂²>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 QK^, 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.