Spectral Capacity and Edge–Bulk Laws for Berezin Quantization

We study the distribution of stable information across the spectrum of the Berezin transform associated with coherent-state quantization. An abstract trace–purity identity shows that an informationally complete POVM on an $N$-dimensional Hilbert space may have algebraic rank $N^2$ while carrying only $O(N)$ eigenvalues above any fixed positive threshold. For Berezin–Toeplitz quantization on a compact polarized Kähler manifold of complex dimension $n$, this square-root loss is sharp: for every fixed $s>0$, $$\\frac{1}{N_k}\\,\\mathord{\\#}\\{\\mu_{k,j}\\ge e^{-s}\\}\\longrightarrow\\frac{s^n}{n!}.$$ The proof comes from the fixed-power trace law $$\\frac{\\operatorname{Tr}(B_k^m)}{N_k}\\longrightarrowm^{-n},$$ and yields a universal edge measure. At exponentially deep scale we factor the positive Berezin spectrum as the Gram spectrum of restriction from $X\\times\\bar X$ to its maximally totally-real diagonal. Finski's logarithmic transfer theory then gives a macroscopic depth distribution governed by a Mabuchi transfer function $\\Phi_\\Delta$. Using attached analytic discs and rooftop contact, we prove $$\\Phi_\\Delta(y)\\asymp d(y,\\Delta)^2,\\qquad\\nu_Y\\{\\Phi_\\Delta\\le a\\}\\asymp a^n.$$ Thus the same exponent governs the universal fixed-depth edge and the zero-depth boundary of the global spectral bulk. Finally, using the classical exact projective spectrum, we derive on $\\mathbb{CP}^n$ the full mesoscopic law $$\\mathord{\\#}\\{\\mu_{k,j}\\ge e^{-s_k}\\}\\sim\\frac{k^n s_k^n}{(n!)^2}$$ for every $1\\ll s_k\\ll k$. No growing-window heat approximation or unproved quantitative transfer rate is used in the main theorem chain.