Mock Theta Neutralization v2.0d: Full-Stack Neutralization Theorem with Calibrated \(T\)-Exponents, \(P_{\mathrm{aut}}\) Pipeline, and a Worked Application on \(X_0(11)\)

# Mock-Theta Neutralization (v2.0) ## Overview This release (v2.0) refines and extends the neutralization engine introduced in earlier versions, whosegoal is to remove the recurring continuous-spectrum main term of size \(c X^{1/2}\) from mock-thetamass integrals and to push the leading behavior into the discrete spectrum. The core perspective is Hilbert--Plancherel: in the ambient space \(H_{\mathrm{mock}}\) the continuousdirection responsible for the drift is isolated explicitly (rank-one in the one-cusp model), andadmissible windows are orthogonalized against that direction. The resulting neutralized quantitiesobey a uniform quantitative tail bound and are compatible with an automorphic projection\(P_{\mathrm{aut}}\) that yields a Mellin--\(L\) interface and a transduction principle. ## Core results ### Structural neutralization A canonical continuous direction \(v(t)\) (rank-one in the basic model) is identified in\(H_{\mathrm{mock}}\). For any admissible window \(f_T\), the orthogonalized window\(f_T^\perp := f_T - \langle f_T,v\rangle v\) eliminates the \(X^{1/2}\) drift term at the theorem level. ### Quantitative tail After neutralization, the tail satisfies\[\mathrm{Tail}(X;T)= O_{\varepsilon}\!\bigl(T^{-3/2+\varepsilon}\bigr) + O_{\varepsilon}\!\bigl(X^{-1-2\eta}\bigr),\]uniformly over calibrated window families \(T \in \PB\). The \(T^{-3/2+\varepsilon}\) saving comes froma Mellin zero at \(s=1/2\) together with a log-scale stationary-phase refinement.The same exponent applies to dyadic bumps and log-Gaussian windows (only implicit constants vary within \(\PB\)). ### Automorphic projection and Mellin--\(L\) interface A bounded, Hecke-equivariant automorphic projection is packaged as\[P_{\mathrm{aut}} = \mathrm{Sh} \circ \xi_{1/2} \circ \mathcal{K},\](plus-space \(\to\) Bruinier--Funke \(\xi_{1/2}\) \(\to\) Shimura/Niwa--Shintani), yielding an exactMellin--\(L\) identity of the form\[\int_0^\infty \Theta_\pi(t)\, t^{s-1} dt = \Gamma(s)\, L(2s,\pi)\]for \(\Re s\) large, with meromorphic continuation. ### Transduction (inputs \(\Rightarrow\) remainder exponents) Neutralization plus the Mellin--\(L\) interface yields a transduction theorem: any \(t\)-aspect input\[L(1/2+it,\pi) \ll t^{1/4-\delta_{\mathrm{in}}+\varepsilon}\]transfers to a neutralized remainder bound\[R_\pi(X) \ll_\varepsilon X^{1/4-\delta_{\mathrm{in}}+\varepsilon},\]and in averaged/hybrid settings to\[R_\pi(X) \ll X^{1/4-c\,\delta'_{\mathrm{in}}+\varepsilon},\quad c \ge 2/3,\]with \(c\) depending only on the window model. ## What is new in v2.0 ### (1) Multi-cusp / finite-rank neutralization made explicit The rank-one picture is upgraded to a finite-rank projection framework for cofinite groups withfinitely many cusps. The finite set of cusp directions \(\{v_{\mathfrak a}\}\) is identified and removedsimultaneously by projecting away from their span. This clarifies that the mechanism is not arank-one accident. ### (2) Scattering-matrix residue formula for cusp directions The cusp coefficients are expressed explicitly in terms of the scattering matrix \(\Phi_{jk}(s)\) andMellin data \(\mathcal{A}_j(s)\) attached to the constant terms, in the residue form\[\kappa_{\mathfrak a_k}= \mathrm{Res}_{s=1/2}\Bigl(\sum_{j=1}^m \mathcal{A}_j(s)\, \Phi_{jk}(s)\Bigr),\]which makes the dependence on the cofinite group and cusp geometry completely transparent. ### (3) Worked model: \(X_0(11)\) as a finite-rank snapshot A dedicated worked-model appendix fixes the cusp set for \(\Gamma_0(11)\), records widths and scaling,and provides a checklist (cusps \(\to\) \(\Phi\) \(\to\) Mellin constant terms \(\to\) residues \(\to\)orthogonalization verification) showing how the finite-rank neutralization is executed in practice. ### (4) Maass-targeted variant promoted to a mini-theorem box The previously "future work" Maass-targeted interface is upgraded to a one-page mini-theorem box:assumptions (MT1)--(MT4), the conditional statement (boundedness of a Maass-targeted projection anda Mellin--\(L\) identity for nonholomorphic observables), and the exact external inputs required.This addresses completeness concerns and clarifies the shortest path to a full Maass closure. ## Scope and applications The engine is designed to translate physics-facing observables into \(L\)-value problems by removingcontinuous-spectrum contamination at the theorem level and then transferring analytic inputs viatransduction. Target domains include: - arithmetic quantum chaos (neutralized spectral masses, heat traces, SFF windows),- BPS state counts / black-hole microstates,- 2D CFT / AdS\(_3\) modular bootstrap (neutralized Cardy--Rademacher remainders),- scattering / heat-kernel / resonance remainders on arithmetic surfaces. ## Classification MSC (2020): Primary 11F72, 11F30; Secondary 11M06, 46E35