A Walk From Free Probability to Matrix Discrepancy III: Higher Rank Kadison-Singer and Spectrally Thin Trees

Abstract

Let $A_1,\ldots,A_N$ be positive semidefinite matrices of rank at most $r$, with $\sum_i A_i=I$ and $\norm{A_i}\le\varepsilon$. We prove that one sign can be assigned to each original matrix with discrepancy $O(\sqrt{\varepsilon\log(2r)})$, independently of their dimension and number, which is known to be optimal upto constants. We give a separate existence proof and a deterministic algorithm with polynomial work in a real-arithmetic model with semidefinite-value and exact spectral primitives. Separate Lean formalizations verify the existence proof and the algorithm in this arithmetic model. The proof extends the variational approach to discrepancy developed in the companion papers, motivated by operator-valued free interpolation, Lehner's formula, and spectral Tsallis regularization. A concave matrix power interpolates between a trace source and a sandwich source. Its concavity controls the response of the optimizing density. The walk maintains independent source reserves and a second matrix recording reserve expenditure. This matrix certifies contraction of the unfinished input mass between epochs; within an epoch, preparation and a negative-curvature step control the spectral potential while the coefficients advance toward signs. As applications, one spanning tree can be chosen simultaneously $O(\varepsilon\log(2s))$-spectrally thin for $s$ positive weightings of a common graph whose edge leverages are at most $\varepsilon$. The diagonal specialization gives discrepancy $O(\sqrt{L\log(2k)})$ for matrices with row sums at most $L$ and at most $k$ nonzero entries per column, through a walk of fractional colorings.