Neutralizing Shadow Terms in BPS/Microstate Counts: A Projection Framework and L-Value Transduction (v2.6)

## OverviewThis paper develops a projection-and-neutralization framework for BPS / black-hole microstate counting kernels in which a dominant continuous-spectrum “shadow” contribution obstructs Mellin-type extraction of the genuinely automorphic/discrete term. The method is operational: we define a canonical orthogonalization that removes a universal continuum mode at the *observable level*, and then connect the remaining discrete component to automorphic \(L\)-values via a Mellin–\(L\) identity after an explicit automorphic projection \(P_{\mathrm{Aut}}\). The v2.6 update strengthens the “assumption-free” core by moving the key continuum decomposition from a standing assumption into lemma-level derivations, and by adding a self-contained appendix proof sketch explaining (i) which automorphic kernel is subjected to Selberg spectral decomposition and (ii) why the continuum contribution enforces an \(X^{1/2}\) barrier. ## One-line summaryGiven a window \(\psi_T\) and the continuum mode \(v(t)=t^{1/2}\), define the neutralized window\[f_T \;=\; \psi_T \;-\;\frac{\langle \psi_T,v_T\rangle}{\langle v_T,v_T\rangle}\,v_T,\qquad\langle f,g\rangle_{t^{-1}dt}=\int_{\mathrm{supp}(\psi_T)} f(t)\overline{g(t)}\,\frac{dt}{t},\]so that \(\langle f_T,v_T\rangle_{t^{-1}dt}=0\). Applying the observable \(D_X[\cdot]\) to \(\psi_T\) vs. \(f_T\) removes the continuum main term and isolates the discrete automorphic remainder up to a controlled tail budget. ## Closed results (v2.6)- **Structural neutralization theorem (protocol-level):** the orthogonalization \(f_T\) kills the continuum obstruction term in the windowed observable \(D_X[\cdot]\), leaving a discrete term plus a tail budget.- **Automorphic projection \(P_{\mathrm{Aut}}\):** for holomorphic/theta targets (plus-space), a concrete pipeline is given (Kohnen plus slice \(\to\) Bruinier–Funke \(\xi_{1/2}\) \(\to\) Shimura/Niwa–Shintani) to align with a target representation \(\pi\).- **Mellin–\(L\) bridge:** for the aligned kernel \(\Theta_{\pi,T}(t)\), for \(\Re s>1\), \[ \int_{0}^{\infty}\Theta_{\pi,T}(t)\,t^{s-1}\,dt \;=\; \Gamma(s)\,L(2s,\pi), \] with an explicit normalization audit and a compact cheatsheet table.- **Transduction principle:** analytic input bounds for \(L(\tfrac12+\mathrm{i}t,\pi)\) (pointwise or mean/hybrid) transfer to the post-neutralization output exponent extracted from \(\log D_X\) vs. \(\log X\), up to the structural tail budget. ## What is new in v2.6- **(SC3) promoted from assumption to lemmas:** the “observable decomposition” is now derived as lemma-level statements from the definition of \(D_X[\cdot]\) and the standard spectral splitting of automorphic kernels.- **Self-contained Appendix proof sketch (1–1.5 pages):** - specifies the automorphic kernel \(K_{\pi,T}(z)\) on \(Y_0(N)=\Gamma_0(N)\backslash\mathbb H\) to which Selberg spectral decomposition is applied, - shows how the Eisenstein constant term produces a rank-1 continuum contribution at the observable level, - explains the origin of the continuum scaling \(\mathsf{C}_X \asymp X^{1/2}\) via Mellin scaling at \(s=\tfrac12\).- **Camera-ready layout:** the main statements are organized as three theorems plus an “acceptability matrix” (who needs what), with a strict separation between structural claims and input-dependent gains. ## Scope & non-toy status- This paper is **methodological**: it provides a canonical neutralization operator and a normalization-audited Mellin–\(L\) bridge for windowed observables arising from microstate-count kernels.- It does **not** claim a full derivation of all BPS degeneracy formulas; rather, it isolates and removes a universal continuous-spectrum obstruction and provides a reproducible extraction engine for the discrete automorphic component.- Immediate scope: holomorphic/theta targets (plus-space). Extensions to Maass targets and multi-cusp normalizations beyond finite rank are deferred.