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The generalized Lax conjecture holds for mixed determinants

A mix of human-written and AI-generated textHuman understanding: some partsmath.OC — Optimization and Controlmath.AG — Algebraic Geometrymath.FA — Functional Analysismath.CO — Combinatorics

Contributed by Suvrit Sra ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

We prove that the mixed determinant η(⋅)\eta(\cdot) of Borcea and Brändén has a spectrahedral hyperbolicity cone. More precisely, for Hermitian matrices A1,…,AmA_1,\ldots,A_m of order dd, the hyperbolicity cone of η(A1,…,Am)\eta(A_1,\ldots,A_m) is cut out by ∑jΦj(Aj)⪰0\sum_j\Phi_j(A_j)\succeq0, where Φ1,…,Φm\Phi_1,\ldots,\Phi_m are explicit unital completely positive maps. We argue by induction on dd, adding one row and one column to every argument and expressing the logarithmic derivatives of η\eta as extremal eigenvalues. By compressing along polynomial boundary kernels, we reduce the order of the pencil ∑jΦj(Aj)\sum_j\Phi_j(A_j) to exp⁡(O(md3log⁡m))\exp(O(md^3\log m)); for one fixed positive semidefinite n×nn\times n matrix AA a separate pencil yields the order exp⁡(O(n2log⁡n))\exp(O(n^2\log n)). We also answer four problems of Borcea and Brändén negatively.

Provenance statement

The results in this paper were developed through extensive use of AI assistants. None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. Earlier attempts with OpenAI's GPT Astra (Pro) to extend Brändén's matrix–tree argument (Brändén, 2014) to the fixed-matrix polynomial η(z₁A, …, zₙA) did not produce a proof. The investigation then searched for a counterexample, adapting the norm obstruction of OpenAI (2026) to the cubic of Appendix D; restricted Clifford and exceptional-Jordan-algebra ansatzes and a sextic square-domination candidate were also explored. The obstruction proved valid for pure powers but not for cones, and the author's suggestion to regard the mixed determinant as an “overgrown elementary symmetric polynomial” turned the search back to the positive direction, where GPT Astra (Pro) found the construction of Theorem 1.1. The search contributed the decision to permit determinant multipliers, to use extremal eigenvalues rather than traces, and to preserve linear rank-one factorizations at every step of the induction. The counterexamples of Appendix C come from an earlier, separate investigation (Sra, 2026). Those for Problems 35 and 38 were found with GPT-5 Pro in January 2026 and those for Problems 36 and 37 with GPT-5.6 Pro in July 2026; Anthropic's Claude (Opus 5) isolated the Schur-positive witness for Problem 37 in August 2026, and the author formalized and audited all four. For this version, Claude (Opus 5.5) recomputed every expansion, minor, and inequality of Appendix C in exact arithmetic. The finite exact checks mentioned in Section 3.4 and Appendix D were run within the same AI-assisted workflow and were not rerun for this version. Anthropic's Claude (Opus 5.5) prepared the manuscript before the compression bounds were added, and Anthropic's Claude (Fable 5.1) revised the presentation of the whole manuscript in this version. OpenAI's Codex developed and integrated the bounds in Section 4, and the specialization bounds in Appendix B and in the supplement, including the cubic resolvent construction and its decomposable continuation, independent assistant audits, and exact checks of the polynomial-kernel identities, degree bounds, coefficient ranks, and signed Kraus decompositions. A subsequent dimension-optimization package supplied the additional scalar bounds and the sharp constant-compression example in Section 5.1 and in the supplement. Its exact checks of the cubic pencil and of the induction, and its finite-field rank, smoothness, and corank certificates were rerun during integration. Assistant checks are not external peer review or formal verification. The main text has not yet been vetted line by line by the author.

Tools used

OpenAI
GPT Astra (Pro)
OpenAI
GPT-5 Pro and GPT-5.6 Pro
OpenAI
Codex
Anthropic
Claude Opus 5, Opus 5.5 and Fable 5.1

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Version history

  1. v1 · 31 pagesSubmitted by Suvrit SraInitial depositCurrentNov 05, 2026