GPT-5.4 Pro: counterexample to GCI for non-centered measures

GPT-5.4 Pro Finds Counterexample to Hypothesis on Non-Centered Gaussian Measures

The GPT-5.4 Pro model generated a counterexample to the hypothesis extending the Gaussian Correlation Inequality (GCI) to non-centered Gaussian measures. The solution, formatted as a three-page article, was accepted by the solveall.org platform on March 12, 2026. The counterexample disproves the assumption that the inequality holds when shifting the mean of the Gaussian distribution.

Gaussian Correlation Inequality: History and Essence

GCI is formulated for centered Gaussian distributions: the probability of the intersection of two symmetric convex sets is at least the product of their marginal probabilities. The two-dimensional case was proved by Loren Pitt in 1977. A complete proof for arbitrary dimensions was presented by Thomas Royen in 2014—simple and elegant, but initially underrated by the community.

In 2025, Shehei Nakamura and Hiroshi Tsuji generalized the result to sets with a common Gaussian barycenter. The question of arbitrary non-centered Gaussian measures, where the mean does not coincide with the origin, remained open.

Counterexample from GPT-5.4 Pro

The model constructed a two-dimensional counterexample: two nearly parallel symmetric strips, with the Gaussian mean shifted such that the intersection of the strips has a lower probability than the product of the individual ones. The construction leverages geometric intuition—the shift in the mean leads to a rapid decay in the intersection probability in the limit.

The counterexample generalizes to dimensions n ≥ 2. In the one-dimensional case, the model proved that the inequality holds: symmetric convex intervals on the line are nested within each other.

The solution is formatted in LaTeX as a full article with definitions, lemmas, proofs via polar coordinates and limiting arguments, as well as a bibliography.

The solveall.org Platform and Context

Solveall.org is a curated collection of open problems in probability theory and related fields, created by Edgar Dobriban from the Wharton School of Business. The platform serves as a benchmark for AI reasoning. The GCI problem had been unsolved since it was added.

The solution was submitted by user Liam Price, confirming generation by GPT-5.4 Pro and verification by other models.

Previous Successes of GPT-5

The GPT-5 series demonstrates progress in solving mathematical problems:

• January 2026: GPT-5.2 Pro solved Erdős problem #728 on factorial divisibility.
• March 2026: GPT-5.4 tackled the FrontierMath problem by Bartosz Naskrencki (20 years in development).

The GCI counterexample stands out for its creative approach: not computations, but a geometric construction that traditionally requires human intuition. The problem was a "low-hanging fruit"—under-explored after 2025, solvable with elementary methods.

Key Takeaways

• GPT-5.4 Pro disproved the GCI hypothesis for non-centered measures via a two-dimensional counterexample with symmetric strips.
• The construction generalizes to n ≥ 2; in 1D, the inequality holds.
• The solution uses polar coordinates and limits—basic techniques.
• Solveall.org validates as an AI benchmark in mathematics.
• Highlights the shift from computation to geometric thinking in AI.

This case illustrates how modern models tackle problems requiring insight rather than brute force. For AI developers, testing on such platforms is increasingly relevant.

— Editorial Team