The Foregger–Sinkhorn Tie-Point Conjecture

What the AI did

The paper's acknowledgement, in full: "The proof of Theorem 1 was carried out by GPT-5.6-sol and Claude Fable 5, under the guidance of the author. The author has reviewed the resulting proof arguments. Responsibility for the final text rests with the author."

Verification

Independently recomputed by this site on 21 August 2026 from the paper's own data (arXiv:2608.13025v1), at 60-digit precision: $\beta$ is the unique root of the stated cubic in the stated bracket, on which the cubic is monotone; the matrix is doubly stochastic and nonnegative; its 19-entry support is nearly decomposable, being fully indecomposable while the removal of any single support entry destroys that; the permanent agrees with the paper's closed form to 1e-61; the cofactor gap agrees with its closed form to 4e-61 and exceeds the claimed 2047/240100; and the prescribed zero is not a tie point, with exactly one witnessing support entry. The conjecture's hypothesis also holds: the face is four-dimensional, and a 6561-point grid plus 61 local descents found nothing on it with a smaller permanent. Still a days-old preprint with no independent review, hence a candidate.

Source

• PaperA counterexample to the Foregger-Sinkhorn tie-point conjecture