Mock Theta Neutralization v1.8: Full-Stack Neutralization Theorem with Calibrated \(T\)-Exponents, \(P_{\mathrm{aut}}\) Pipeline, and a Worked Application on \(X_0(11)\)

# Mock-Theta Neutralization v1.8 ## OverviewVersion v1.8 is a stabilization and reference-completion update to v1.4.The framework structurally removes the recurring continuous-spectrum main term from mock-thetamass integrals and pushes the leading behavior into the discrete spectrum. The targeted drift term is\[c\,X^{1/2}.\] ## Core results ### Neutralization (structural + quantitative)In the Hilbert--Plancherel setup, we isolate the rank-one continuous direction\[v(t) = t^{1/2}.\]After orthogonalization against \(v\), the neutralized tail satisfies\[\mathrm{Tail}(X;T) = O_{\varepsilon}\!\left(T^{-3/2+\varepsilon}\right) + O_{\varepsilon}\!\left(X^{-1-2\eta}\right),\]and the same interpretation applies to dyadic and smoothed window variants. ### Automorphic projection and Mellin--L interfaceA bounded, Hecke-equivariant automorphic projection is packaged as\[P_{\mathrm{aut}} = \mathrm{Sh} \circ \xi_{1/2} \circ K.\]This supports the Mellin--\(L\) interface used to convert neutralized spectral quantities into \(L\)-value problems. ### Transduction (inputs imply remainder exponents)The transduction mechanism converts \(t\)-aspect subconvex savings into improved remainder exponents.In averaged or hybrid settings, the efficiency constant satisfies\[c \ge 2/3.\] ## What is new in v1.8- Build and citation stability: compilation and reference-hygiene fixes (operator/environment definitions, ASCII-safe fallbacks, and duplicate-label cleanup) so the document compiles cleanly and references remain stable.- Reference completion: bibliography expanded to cover plus-space, Shimura/Niwa/Shintani lifts, and standard spectral/scattering tools required by \(P_{\mathrm{aut}}\) and contour-shift steps.- Constants bookkeeping finalized: the "where constants come from" dependency table is stabilized, clarifying which quantitative bounds depend on the parameter box PB, the level/nebentype \((N,\chi)\), the window class, and scattering assumptions (SD1)--(SD3), and which parts are parameter-free.- Presentation fixes: minor layout repairs (e.g., wide-table handling) to prevent margin spillover. Note: v1.8 does not introduce new mathematical claims beyond v1.4. It is intended as a citeable,stable baseline for a later v2.0 expansion (end-to-end worked closures and sharper effectivity targets). ## Scope and applicationsThe framework bridges mock-theta kernels to automorphic \(L\)-functions and is designed to translatephysics-facing observables into \(L\)-value problems, including:- arithmetic quantum chaos,- BPS state counts / black-hole microstates,- 2D CFT / \(\mathrm{AdS}_3\) modular bootstrap,- scattering / heat-kernel / resonance problems. ## ClassificationMSC (2020): Primary 11F72, 11F30; Secondary 11M06, 46E35