The coupling matrix of the finite free convolution at the Hermite diagonal is symmetric, and its spectrum is exactly {2^(-k/2)}

Abstract

The finite free additive convolution boxplusₙ of Marcus, Spielman and Srivastava acts on the root vectors of monic real-rooted polynomials of degree n by a map Ω that is smooth at a simple-root configuration. Hashemi's coupling matrix Eₙ is the partial Jacobian partialΩ(·,h)/partialα frozen at the Hermite diagonal α=β=h, where h is the root vector of the degree-n probabilists' Hermite polynomial Heₙ. On numerical evidence — the ten leading singular values at nine degrees between 10 and 90, and a relative Frobenius asymmetry below 10⁻¹⁰ — he conjectures that Eₙ is symmetric and that its singular values on 1ᵖerp are exactly {2^(-k/2):k=1,…,n-1}, the same numbers at every degree. We prove both, at every degree and in exact form: Eₙᵗᵒᵖ=Eₙ; the characteristic polynomial of Eₙ is ∏ₖ₌₀ⁿ⁻¹(X-2^(-k/2)); that of the Gram matrix EₙᵗᵒᵖEₙ is ∏ₖ₌₀ⁿ⁻¹(X-2⁻ᵏ); and tr Eₙ-1=Σₖ₌₁ⁿ⁻¹2^(-k/2), an identity the source derives only under both of its conjectures. The mechanism is that convolving with Heₙ is the heat operator G=e^(-D²/2): Christoffel–Darboux expands Heₙ/(x-uⱼ) in the Hermite basis, GHeₘ is the variance-two Hermite polynomial and He^((2))ₘ(sqrt2 x)=2^(m/2)Heₘ(x), so Eₙ is the matrix of a rescaled heat operator — manifestly symmetric in one basis, diagonal in another. Nothing here is an exhaustive computation: every statement holds at every degree at once, and the source's numerics are reproduced only as a check that the algebraic definition used here is the analytic object it is meant to be. The three corollaries the source states under "Assume Conjecture 4.1" — the full singular value decomposition of the convolution Jacobian, the local stability constant, and the finite free central limit rate λ_(CLT)=1/sqrt2 uniform in n — therefore hold unconditionally. Every lemma, proposition and theorem below is machine-checked in Lean 4; the two corollaries and the two remarks are not, and say so.

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