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Benefit Ledger · provisional · Mathematics

Long, using Sol Pro and Fable 5, posts low-degree Gaussian Moments counterexamples

On 20 July 2026 Christopher D. Long posted explicit polynomials showing the Gaussian Moments conjecture fails in every dimension n≥3: a cubic in four variables found by GPT-5.6 Sol Pro, then a quartic in three variables found by Claude Fable 5. This is not a Jacobian-conjecture score and not a Clay problem.

20 Jul 2026Tier 2 NotableMethodology 0.1

Current score

+0.30

3 base · Notable (tier 2 of 5, 3 pts)
× 0.7500 attribution · Critical contribution
× 0.3500 evidence · Technical disclosure or preprint
× 0.5500 realization · Experimentally validated
× 0.7000 durability
Event-level product before credit split: 0.30

Explicit low-degree counterexamples to a 2017 conjecture are notable (tier 2). Dual named-model discovery (0.75). Evidence is an arXiv note with computer-checkable polynomials (0.35). Realization is a public construction (0.55).

What happened

Derksen, van den Essen, and Zhao (2017) proposed the Gaussian Moments conjecture as a route toward the Jacobian conjecture. After public claims of a Jacobian counterexample, Long asked models for small GMC failures. Sol Pro produced a six-term cubic in four Gaussians; Fable 5 then produced a five-term quartic in three variables. Both satisfy E(P^m)=0 while E(QP^m)=m!≠0. The examples are explicit and checkable by computer algebra; the preprint is unrefereed. The two-variable case is left open here. We do not import the Jacobian n=3 claim as a resolved event.

Model attribution

+0.15

Claude

Found the three-variable quartic counterexample after being shown the four-variable example.

The paper credits Claude Fable 5 with the three-variable construction and independent algebraic checks.

Attribution 0.7500 · Credit share 50% · Anthropic

+0.15

GPT-5.6

Produced the four-variable cubic counterexample after the initial prompt, according to Long.

The paper says the four-variable construction was produced by ChatGPT 5.6 Sol Pro without human intervention after the initial prompt.

Attribution 0.7500 · Credit share 50% · OpenAI

Claims

  • Long posted explicit 3- and 4-variable counterexamples showing the Gaussian Moments conjecture fails for every n≥3, crediting Sol Pro and Fable 5.

    outcome · supported

  • This event records a verified solution of the Jacobian conjecture.

    significance · disputed

Sources

primary sources

Revision history

  • 13 Sep 2026 · 0.00 → 0.30

    Imported events under methodology 0.1.