GPT-5.6-sol Finds Counterexamples to the Symmetric-Maximizer Conjecture for Lyapunov Operators
Daniel Kressner and Bart Vandereycken give an exact-arithmetic order-seven counterexample showing that a Lyapunov operator's Frobenius-induced norm need not be attained on a symmetric matrix, and extend it to every order at least seven while leaving order six open; OpenAI's gpt-5.6-sol designed the numerical search and follow-up searches that found the published sparse integer matrix.
In a preprint first uploaded on August 21, 2026, Daniel Kressner and Bart Vandereycken disprove the conjecture that the Frobenius-induced norm of every Lyapunov operator is attained on a symmetric matrix. They give a sparse integer matrix of order seven for which the norm restricted to skew-symmetric matrices is strictly larger than the symmetric restriction.
A rational separator and exact-arithmetic certificates prove the strict inequality without relying on floating-point calculations. Taking direct sums with zero blocks yields counterexamples in every order . Since the conjecture is known for orders at most five, order six is the only unresolved case, and the paper does not claim that seven is the smallest possible counterexample order.
The authors report that OpenAI's gpt-5.6-sol designed a numerical search that maximized the gap between the two restricted norms, initially finding an order-nine example with Adam. Follow-up prompting reduced the dimension and coefficients to the published order-seven matrix; the candidate and all proof inequalities were then checked in exact arithmetic.