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Persistent failure of Euclidean quartic SOS exactness

An explicit fixed rational cubic in 14 variables gives a quartically regularized polynomial whose exact minimum shift is nonnegative but not a sum of polynomial squares for every positive regularization parameter. A second explicit cubic has a unique global minimizer with positive-definite Hessian there and retains a strict SOS gap for every positive parameter.

Sources for this record

What was already known

The construction starts from a classical Choi–Lam quartic and addresses the stationary, non-positive-semidefinite-Hessian question in Cartis–Zhu, Remark 2.3. The quadratic and cubic data stay fixed while the regularization strength varies.

AI involvement AI-led

The disclosed Codex agents developed the construction, proof and rational witness. GPT-6 Pro supplied the unique nondegenerate-minimizer construction, which Codex checked and formalized. JD Jones is the human maintainer; a human-free research run is not documented.

Importance Notable

Provides explicit exact counterexamples and quantitative separation, including a nondegenerate unique-minimizer variant.

Limitations

The stationary origin has negative-definite Hessian; the positive-definite Hessian is at the unique minimizer in the companion construction. No optimal SOS value, minimal dimension, higher hierarchy result or worldwide priority is claimed. Separate agent-written Python checkers are not outside human review.

Technical verification Formal (Lean)

Palomar reports checking 35 selected contracts for the literal polynomials, minimum, ordinary SOS exclusion, quantitative gap and companion Hessian result, with Lean and independent kernel replay.

Formal checking

Palomar reports checking 35 selected contracts for the literal polynomials, minimum, ordinary SOS exclusion, quantitative gap and companion Hessian result, with Lean and independent kernel replay.

Checker
Palomar: Lean, NanoDa and con-ron
Scope
Literal fourteen-variable rational cubics; all σ,a,b>0; exact attained minimum, unrestricted polynomial SOS exclusion, quantitative gap and unique-minimizer positive Hessian; 35 selected contracts.
Basis
Reported by the linked checker
Check date
2026-10-18
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. Automated reviews and exact checks are described, but no named human explanation of the central construction is documented. Few is an evidence default.

Correction and publication history

2026-10-19 · publication

Added this sourced contribution with its stated scope, AI attribution, checking evidence and research limits.

Read this published revision