What was already known
Litvak conjectured the regular-simplex covariance minimizes this quantity for every n≥3 and p>0. Earlier comparisons with identity covariance did not settle that claim.
AI involvement
AlphaEvolve found the cosine covariance in a search designed by Dmitriy Kunisky. Kunisky recognized its structure and supplied the analytic law and proof; GPT-5.5 Pro’s proposed proofs of a broader conjecture were incorrect.
AlphaEvolve supplied the decisive counterexample candidate within a human-selected optimization objective; Kunisky supplied the mathematical analysis and proof.
Importance
Gives a rigorous four-variable counterexample to the universal conjecture.
Limitations
The proposed optimality of cosine matrices for all n and p and the related volumetric-zone claim remain conjectures. Kunisky proved this specific comparison after the search found the candidate.
Technical verification
Kunisky's Theorem 2.2 gives a rigorous human proof of the exact n=4,p=2 inequality for the AI-found matrix, replacing the initial numerical search with explicit analytic bounds.
Formal checking
analytic calculation and rigorous bounded comparison in Theorem 2.2
- Checker
- Dmitriy Kunisky (author)
- Scope
- At n=4,p=2, the AlphaEvolve-found cosine covariance has smaller expected squared Gaussian minimum than the regular-simplex covariance
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Kunisky explains the rank-two regular-polygon representation, derives the candidate's law and expectation, and isolates the exact inequality that refutes the conjecture.
Correction and publication history
Gives a rigorous four-variable counterexample to the universal conjecture.
Read this published revision