Geometric spectrum conjecture for the finite free coupling matrix

Let $n\geq 3$ , let $h\in\mathbb{R}^n$ be the normalized Hermite root vector, and define

E_n:=\left.\frac{\partial\,\Omega_{\boxplus_n}(\cdot,h)}{\partial\alpha}\right|_{\alpha=h}\in\mathbb{R}^{n\times n}.

Let $W=\mathbf{1}^\perp=\{u\in\mathbb{R}^n:u^\top\mathbf{1}=0\}$ be the mean-zero subspace. Geometric spectrum conjecture. The singular values of $E_n$ restricted to $W$ are

\operatorname{spec}(E_n|_W)=\{2^{-k/2}:k=1,\ldots,n-1\},

independently of $n$ , and the singular value on $\operatorname{span}(\mathbf{1})$ is $1$ . This conjecture would identify the exact linearized spectral structure at the Hermite diagonal and, conditionally, yield uniform local stability and finite free central-limit convergence rates; the stated evidence is computational, so the claim remains open.

References

Primary source

Baran Hashemi, “Spectral Structure in Finite Free Information Inequalities and p-Stam Phase Transitions”, arXiv:2604.11922 (2026).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unverified posted argument claims a complete proof, but the published work records only numerical evidence and the conjecture has not been independently confirmed.

Baran Hashemi’s April 2026 preprint formulates the conjecture that the linearized finite-free coupling map at the Hermite configuration has geometric singular values $\{2^{-k/2}:k=1,\ldots,n-1\}$ on the mean-zero directions, with translation singular value $1$ . It presents the resulting stability and convergence consequences as conditional.

April 2026 numerical evidence

High-precision computations through $n=90$ match the proposed spectrum; apparent symmetry of $E_n$ is also numerical, not proved. The preprint explicitly leaves the conjecture open.

Posted attempt

A reader-written argument claims a complete proof using Hermite-polynomial perturbations, Gaussian quadrature, and an orthogonal diagonalization of $E_n$ . The attempt has not been independently verified.

Current status (as of August 2026): The conjecture has a complete-proof claim but no independent verification; the primary preprint establishes only numerical evidence, so the mathematical problem remains open.

Sources

ProofThis solution needs a summarySee full solutionHide full solution

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.