The Stable Coordinate Conjecture in ambient dimension five
A polynomial is a coordinate if it is part of a polynomial coordinate system, that is, the image of a variable under an automorphism. The Stable Coordinate Conjecture, recorded by Makar-Limanov, van Rossum, Shpilrain and Yu (2004, p. 342), asks whether stabilization can create coordinates: if becomes a coordinate of after adjoining one variable, must already be a coordinate of ?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Polynomial automorphisms, coordinates
- Posed by
- L. Makar-Limanov, P. van Rossum, V. Shpilrain and J.-T. Yu, The stable equivalence and cancellation problems, Comment. Math. Helv. 79 (2004), p. 342
- Year posed
- 2004
- Years open
- 22y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 22 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Corollary 1.2: the explicit is a coordinate of (by the Dutta-Lahiri theorem, since is a residual coordinate over ) but not a coordinate of , because otherwise , contradicting Theorem 1.1. So the conjecture fails over in ambient dimension five after one stabilization. Lower ambient dimensions are not addressed. A four-variable counterexample, the first possible dimension, was given in a later OpenAI math release manuscript (5 October 2026), see stable-coordinate-conjecture-four-variables.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, the introduction and Corollary 1.2 with its proof, read against the conjecture as the manuscript cites it. The deduction is two lines given Theorem 1.1 and the published Dutta-Lahiri theorem. Not Lean-checked as stated: the release's Lean scope note says the stable-coordinate consequence is not formalized; only the non-polynomiality and stabilization of P/(H) are (Comparator challenge ComplexCancellation), and the step that H itself is a coordinate of P[w] is not in the formal statement.