Geometric spectrum conjecture for the finite free coupling matrix

The conjecture holds for every degree, and the coupling matrix is in fact real symmetric with an explicit orthogonal diagonalization.

Let $H_m$ be the monic probabilists' Hermite polynomial,

$$H_m(x)=e^{-\partial_x^2/2}x^m, \qquad H_m'(x)=mH_{m-1}(x),$$

and let $h_1<\cdots<h_n$ be the roots of $H_n$. Common rescaling of both root vectors leaves the Jacobian unchanged, so this is equivalent to the normalized Hermite convention in the question.

For degree-$n$ polynomials, the defining finite free convolution coefficient formula gives

$$f\boxplus_n H_n = \sum_{j\ge0}\frac{(-1)^j}{2^j j!}f^{(2j)} = e^{-\partial_x^2/2}f.$$

Therefore

$$H_n\boxplus_n H_n = e^{-\partial_x^2}x^n = 2^{n/2}H_n(x/\sqrt2),$$

whose roots are $\sqrt2\,h_i$.

For $0\le m<n$, perturb the first polynomial by

$$f_\varepsilon(x)=H_n(x)+\varepsilon H_m(x).$$

Its input root-velocity vector is

$$v_i^{(m)} = -\frac{H_m(h_i)}{H_n'(h_i)}.$$

The corresponding output polynomial is

$$f_\varepsilon\boxplus_n H_n = 2^{n/2}H_n(x/\sqrt2) + \varepsilon\,2^{m/2}H_m(x/\sqrt2).$$

Differentiating its roots at $\varepsilon=0$ gives

$$E_n v^{(m)} = 2^{-(n-1-m)/2}v^{(m)}.$$

To identify singular values rather than merely eigenvalues, apply $n$-point Gaussian quadrature. At the roots $h_i$, its weights are

$$w_i=\frac{n!}{H_n'(h_i)^2},$$

and the rule is exact through degree $2n-1$. Since $\deg(H_mH_\ell)\le2n-2$,

$$\sum_{i=1}^{n} \frac{H_m(h_i)H_\ell(h_i)}{H_n'(h_i)^2} = \frac1{n!} \int_{\mathbb R}H_m(x)H_\ell(x) \frac{e^{-x^2/2}}{\sqrt{2\pi}}\,dx = \frac{m!}{n!}\mathbf1_{m=\ell}.$$

Thus the $v^{(m)}$ form an orthogonal basis. Set

$$Q_{i,m} = \sqrt{\frac{n!}{m!}}\, \frac{H_m(h_i)}{H_n'(h_i)}, \qquad 0\le m<n.$$

Then $Q^{\mathsf T}Q=I$, and

$$\boxed{ E_n = Q\,\operatorname{diag} \left( 2^{-(n-1)/2}, 2^{-(n-2)/2}, \ldots, 2^{-1/2}, 1 \right)Q^{\mathsf T}. }$$

In particular, $E_n$ is symmetric positive definite, proving also the separate symmetry assertion of Remark 4.2. Because $H_n'=nH_{n-1}$, the $m=n-1$ eigenvector is constant, with eigenvalue $1$. All other eigenvectors belong to $W=\mathbf1^\perp$. Consequently

$$\boxed{ \operatorname{spec}_{\mathrm{sing}}(E_n|_W) = \{2^{-k/2}:k=1,\ldots,n-1\}. }$$

This proves Conjecture 4.1 and removes the corresponding conditional assumptions from its stated spectral and local-stability consequences.

Source: B. Hashemi, “Spectral Structure in Finite Free Information Inequalities and $p$-Stam Phase Transitions,” arXiv:2604.11922v2, equations (3)–(4), Conjecture 4.1, and Remark 4.2.