Hinrichs–Vybíral positive-definite kernel lower bound

For every continuous, nonnegative, positive definite function f on Euclidean space with f(0) = 1, does the difference kernel matrix [f(x_j - x_k)] always dominate the matrix (1/n) * 1 1^T in the Loewner order?

Sep 18, 2026

Seven-dimensional trigonometric polynomial counterexample

Model / AI: ChatGPT 6, Claude Opus 5

self-reported (counterexample)

Jan Vybíral presents an explicit counterexample in R^7 using eight evaluation points and a nonnegative trigonometric polynomial where the quadratic form evaluates to 22/5 against a conjectured minimum of 9/2, yielding a gap of -1/10. The author notes the counterexample was found using ChatGPT 6 in a single prompt query by a colleague, with Claude Opus 5 used during manuscript preparation.

Source read 2026-10-08. Read the arXiv v1 abstract, explicit 8-point coordinates, Gram matrix evaluation, and model acknowledgement. The trigonometric evaluations were spot-checked against the manuscript values.

Paper — Paper