Exact GUE-zero insertion amplitudes and a coefficient-space obstruction

Definitions and mathematical statement

The source asks for natural physical examples of random functions whose real zeros have GUE statistics. It does not specify whether this means an exact finite-dimensional Gaussian unitary ensemble eigenvalue law, a bulk scaling limit, an edge scaling limit, or a universality prediction.

In the supplied finite-$N$ model, $N$ measured fermion positions $X_1,\ldots,X_N$ in a harmonic-trap ground state have joint density proportional to \[ e^{-\alpha^2\sum_jX_j^2}\prod_{i<j}(X_i-X_j)^2, \qquad X_j\in{\mathbb{R}},\quad\alpha>0. \] This fixes a GUE scaling convention. The report describes the standard Slater-determinant insertion ratio, after division by its nonzero Gaussian envelope and monic normalization, as the random polynomial \[ P(z)=\prod_{j=1}^N(z-X_j). \] Its zeros therefore have that exact finite-$N$ law. The precise Hamiltonian, spin/noninteraction assumptions, orbital normalization and insertion amplitude are not specified in the record; the physical claim refers to the reported harmonic-trap Slater model, not arbitrary interacting fermion systems.

For $N\ge2$, write $P(z)=z^N+a_1z^{N-1}+a_2z^{N-2}+\cdots+a_N$, and let $\mathcal H_N\subset{\mathbb{R}}^N$ be the region of monic real-rooted polynomials. The candidate coefficient-space theorem gives density with respect to real coefficient Lebesgue measure \[ C_{N,\alpha}\,\mathbf1_{\mathcal H_N}(a) e^{-\alpha^2(a_1^2-2a_2)}\sqrt{\operatorname{Disc}(P)}, \] where $\operatorname{Disc}(P)=\prod_{i<j}(X_i-X_j)^2$ on that region and $C_{N,\alpha}$ is a normalizing constant. The multiplicity boundary has measure zero; ordering of roots is absorbed in the normalization. The report concludes that no full-rank jointly Gaussian real coefficient vector can realize exact GUE zeros for $N\ge2$. This obstruction concerns the coefficient law, not the Gaussian matrix-entry representation of GUE.

For the two limiting interpretations, the report names rigorous Sine$_2$ and Airy$_2$ entire-function/operator models, corresponding respectively to bulk and edge statistics. It supplies no model definitions or primary citations here. It contrasts these and the exact finite-$N$ insertion example with chaotic spectral determinants, which it describes in terms of GUE-type universality rather than an exact finite ensemble law. The status and hypotheses of particular physical universality assertions are not given. Thus the three meanings of GUE zeros remain distinct, and "natural physical" is not converted into an unspecified universal theorem.

Short English statement

The supplied report offers a harmonic-trap fermion insertion polynomial with exact finite-GUE real zeros, plus a non-Gaussian coefficient-density characterization. It separately mentions bulk and edge limiting models and distinguishes them from physical universality predictions. The source leaves which meaning of GUE statistics is intended unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $20000634$, source code catalogue entry; AIM analysis corpus, aim-analysis-notes.json, notes[165]. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM Random analytic functions, Question 9 attributed to Ashkan Nikeghbali: https://aimath.org/WWN/randomzeros/arcc1.pdf. The extraction appends a control character, page marker and funding footnote naming NSF DMS-0211458, CAREER DMS-0349309 and AIM; these are source artifacts, not mathematical hypotheses. The verification text is truncated after "The canonical JSON record contains".
• The report names harmonic-trap Slater amplitudes, Sine$_2$, Airy$_2$ and chaotic spectral determinants without complete model definitions or bibliographic references. Original remarks and literature fields are empty.
• Registry dated August 14, 2026: statement reconstructed_unverified, status partially_solved, classification PARTIAL-PROGRESS; attempt 1 labelled partial result, rigor proved, novelty confidence low. The physical example and coefficient-space result are machine-generated research aids, not peer reviewed, requiring independent verification before citation.