Projective Entropy Deletion and Tilted Bulk Spectral Measures Below BBP

This record contains a preprint manuscript submitted to the Journal of Statistical Physics. The paper introduces projective entropy-deletion landscapes (PEDL), directional entropy-response functions on projective space obtained by rank-one compression and renormalization of a density matrix. The manuscript develops finite-dimensional PEDL calculus, including a Cauchy secular equation for compressed spectra, an exact entropy formula, Morse-Hessian coefficients at eigendirections, first-variation formulas, and a contour identity for Cauchy-root transport. The high-dimensional application studies rank-one spiked covariance models below the BBP transition. The relevant object is the spike-direction spectral measure in the sample eigenbasis, whose Stieltjes transform is the diagonal resolvent entry. Using standard anisotropic local laws, the manuscript derives a compactly subcritical tilted Marchenko–Pastur limit and the associated effective-rank-normalized entropy-deletion coefficient. Numerical simulations validate the tilted spectral-measure prediction, the normalized PEDL coefficient, and the interpretation that below BBP the spike direction leaves a distributed tilted bulk footprint rather than a persistent outlier-coordinate signal. This preprint is provided for scholarly access and citation. The manuscript is under journal review and may be updated in later versions.