Neutralizing Shadow Terms in BPS/Microstate Counts: A Projection Framework and L-Value Transduction (v2.3)

# Neutralizing Shadow Terms in BPS/Microstate Counts: A Projection Framework and L-Value Transduction (v2.3) **Author:** Lee Byoungwoo **Version:** v2.3 (2026-01-04) **Files:** PDF manuscript + LaTeX source + figure assets ## OverviewThis note develops a *projection-and-neutralization* method to separate (and remove) the dominant **continuous-spectrum “shadow” mode** that contaminates Mellin-type analyses of BPS/microstate counting kernels (and related mock/half-integral-weight inputs). The goal is operational: given a windowed observable derived from the kernel, we construct a canonical orthogonalization that **kills the universal continuum mode** and exposes the genuinely automorphic/discrete contribution in a way that is compatible with an explicit Mellin–\(L\) bridge. The technical backbone is a two-layer map: 1. **Automorphic projection** \(P_{\mathrm{Aut}}\): a pipeline that aligns half-integral weight inputs with a target automorphic representation \(\pi\) (via plus-space slicing and classical lifts/operators).2. **Neutralization**: an orthogonal projection that removes the universal continuum mode \(v(t)=t^{1/2}\) from a chosen window \(\psi_T\), producing \(f_T\) with \(\langle f_T,v\rangle=0\). This yields a practical “DMZ alignment” viewpoint in the spirit of Duncan–Mertens–Zagier (2012), *Mock modular forms and black hole entropy*, while emphasizing *observable extraction* and *windowed* diagnostics. ## Core object (one-line)Let \(\psi_T\) be a window (dyadic or log–Gaussian) and \(v(t)=t^{1/2}\) the continuum mode. Define the neutralized window\[f_T \;=\; \psi_T \;-\;\frac{\langle \psi_T,v\rangle}{\langle v,v\rangle}\,v,\qquad\langle f,g\rangle_{t^{-1}dt}=\int_0^\infty f(t)\overline{g(t)}\,\frac{dt}{t}.\]We evaluate a windowed observable \(D_X[\cdot]\) (defined in the manuscript) on \(\psi_T\) and on \(f_T\), and compare the scaling of \(\log D_X\) versus \(\log X\). ## Closed results (v2.3)- **Structural neutralization lemma (assumption-free):** the orthogonalization \(f_T\) removes the universal \(t^{1/2}\) component in the relevant Mellin geometry, eliminating the leading continuous-spectrum contribution at the level of the windowed observable.- **Projection framework \(P_{\mathrm{Aut}}\):** an explicit alignment path from half-integral inputs to a target \(\pi\), organized as: - Kohnen plus-space slice, - Bruinier–Funke operator \(\xi_{1/2}\), - Shimura / Niwa–Shintani lifts to \(S_2(\Gamma_0(N))\).- **Mellin–\(L\) bridge (checkable identity):** for the aligned kernel \(\Theta_{\pi,T}(t)\), \[ \int_0^\infty \Theta_{\pi,T}(t)\,t^{s-1}\,dt \;=\; \Gamma(s)\,L(2s,\pi), \qquad (\Re s>1), \] with an explicit normalization discussion and a compact “cheatsheet” table.- **Empirical diagnostics (worked example):** a reproducible slope comparison (pre-neutralization / post-neutralization / negative control) showing: - pre-neutralization slopes concentrate near \(\beta\approx \tfrac12\), - post-neutralization slopes track a discrete exponent \(\beta_{\mathrm{disc}}\), - negative control restores \(\beta\approx \tfrac12\), supporting that the effect is driven by neutralization rather than overfitting. ## What is new in v2.3 (camera-ready update)- Reorganized front-matter into a clean *3-step reading flow*: 1) Status dashboard (structural vs input-dependent), 2) Quantity map + DMZ alignment summary, 3) Informal main results \(\to\) proofs.- Removed overfull boxes and improved layout robustness: - DMZ comparison header moved to the top line, details wrapped below, - tables converted to \(\texttt{tabularx}\) where appropriate, with safer line breaks.- “Notation snapshot” and “Pipeline (E1–E5)” rewritten as scan-friendly itemizations with display equations. ## Scope & non-toy status- This work is **methodological**: it provides a projection/neutralization mechanism and a Mellin–\(L\) compatibility audit for windowed observables derived from microstate-count kernels.- The paper does **not** claim a full derivation of all BPS degeneracy formulas from first principles; rather, it isolates a universal continuous-spectrum obstruction and provides a checkable extraction engine for the discrete automorphic component.- Immediate scope: holomorphic/theta targets (plus-space). Maass targets and multi-cusp normalizations beyond finite rank are deferred. ## Reproducibility- The release contains the LaTeX source and the figure assets used in the manuscript: - `bps_slope_pre_post.png` - `bps_negative_control.png`- Build (typical): - `pdflatex Neutralizing_BPS_v2.3.tex` (run twice), or `latexmk -pdf`.- All definitions are included in-text; no external macros are required beyond standard LaTeX/AMS packages. ## Suggested citationLee Byoungwoo (2026), *Neutralizing Shadow Terms in BPS/Microstate Counts: A Projection Framework and L-Value Transduction*, version v2.3, Zenodo. ## KeywordsBlack hole microstates; BPS state counting; mock modular forms; continuous spectrum; shadow term; Kohnen plus space; Bruinier–Funke operator; Shimura lift; Mellin transform; \(L\)-functions; neutralization; orthogonal projection; windowed observables.