Primal–Dual Mellin Neutralization for Arithmetic Quantum Observables: Incomplete-Eisenstein Matrix Elements, Unconditional Logarithmic Persistence, and Noncommuting Limits

Abstract

This work develops a theorem-level arithmetic-quantum application of primal–dual Mellin neutralization on the modular surface. The basic observables are incomplete Eisenstein series built from compact logarithmic windows, and the analysis combines exact Rankin–Selberg unfolding, residue neutralization, harmonic quantum variance, and an effective finite-$R$ Kuznetsov reduction. For fixed Hecke–Maaß cusp forms, exact unfolding produces a Rankin–Selberg Mellin multiplier whose only possible pole in the working half-strip is the diagonal pole at $s=1$. A constant-coefficient differential operator acting on the compact logarithmic profile removes this actual pole divisor exactly. After neutralization, the arithmetic factor $1/\zeta(2s)$ has a simple zero at the centered point $s=1/2$. Under the $L^2$-normalized dilation $$q_T(x) = T^{-1/2}q(x/T),$$ this central arithmetic zero yields the fixed-pair asymptotic $$\langle A_T u_j, u_k \rangle = O(T^{-3/2}),$$ with an explicit leading coefficient whenever the first centered term is nonzero. In the spherical diagonal channel, the same neutralizer applies simultaneously to the full Maaß spectrum and produces a mean-zero incomplete-Eisenstein observable $A_T$. The resulting diagonal coefficients are expressed through central symmetric-square $L$-values. Using the short-interval mean-Lindelöf theorem of Khan–Young together with the Hoffstein–Lockhart lower bound, the corresponding natural and harmonic coefficient moments are placed at the expected $R^{1+\varepsilon}$ variance scale. For every fixed $T$, the observable $A_T$ belongs to the Sarnak–Zhao test class. The weight-zero incomplete-Eisenstein quantum-variance formula is Mellin-diagonalized to obtain $$Q_{\mathrm{SZ}}(A_T,A_T) = \frac{1}{2\pi} \int_{\mathbb R} G(u/T) \vert{}\widehat q(u)\vert{}^2 \, du,$$ and hence $$Q_{\mathrm{SZ}}(A_T,A_T) = c_{\mathrm{IE}} \vert{}q\vert{}_2^2 + O_q(T^{-2}),$$ where $$c_{\mathrm{IE}} = \frac{\zeta(1/2)^2\Gamma(1/4)^4}{576\pi} > 0.$$ Thus the high-energy and broad-window limits do not commute: fixed matrix elements decay like $T^{-3/2}$, whereas the fixed-observable harmonic quantum variance approaches a positive constant. The principal new feature of Version v0.9r1 is the completion of the previously conditional finite-$R$ effective trace interface. The exact same-sign Kuznetsov decomposition is analyzed through its diagonal, continuous-spectrum, and Kloosterman components. The proof includes the finite Fourier transform of the square-index Kloosterman phase, the square-modulus zero-frequency collapse, a uniform Whittaker–Fresnel reduction, mean-zero divisor packets, a pre-Poisson physical-$q$ split, a core bound of size $R^{39/40+\varepsilon}$, a large-modulus bound of size $R^{9/10+\varepsilon}$, a three-pole continuous-spectrum analysis, and an independent square-zero-mode truncation. The resulting effective theorem is $$\Lambda_T \le \frac{\log R}{400} \quad \Longrightarrow \quad \left\vert{} R^{-1}S_R(T;w) - H_w Q_{\mathrm{SZ}}(A_T,A_T) \right\vert{} \ll_{q,w} R^{-1/100},$$ where $\Lambda_T = L_q T$ is the logarithmic support radius and $w$ is a same-sign Kuznetsov-admissible spectral cutoff. Consequently, if $T = T(R) \to \infty$ while $$\Lambda_T \le \frac{\log R}{400},$$ then $$R^{-1}S_R(T(R);w) \longrightarrow H_w c_{\mathrm{IE}} \vert{}q\vert{}_2^2 > 0.$$ This gives an unconditional logarithmic-persistence theorem for the moving observable family. At the opposite end, high-order center-flat profiles satisfy unconditional polynomial-scale collapse. For $T = R^\theta$ and sufficiently large fixed center-flat order $M$ with $M\theta \ge 1/2$, the normalized harmonic variance tends to zero. The paper therefore establishes the rigorous partial two-parameter phase diagram: $$\text{logarithmic persistence} \quad \longrightarrow \quad \text{unresolved crossover corridor} \quad \longrightarrow \quad \text{polynomial collapse}.$$ No intrinsic critical scale, universal crossover function, or physical interpretation of the constants $1/400$, $39/40$, or $1/100$ is claimed. These are proof constants arising from the present effective estimates. Determining the actual logarithmic-to-polynomial crossover law remains a separate two-parameter arithmetic-quantum problem. Version v0.9r1 is the final integrated publication-audit freeze of this manuscript. Relative to public Version v0.8r2, the principal advance is the replacement of the conditional logarithmic-persistence interface by a proof-complete unconditional finite-$R$ theorem, together with a fully reconciled continuous-spectrum and square-zero-mode analysis.