The stable coordinate conjecture in four variables

Theorem 1.1: with $Q=x_2^2-x_4^2+x_1x_3$ and $f=x_1-2Q(Q(x_2+x_4)+x_1x_4)$ , a degree-five polynomial in $\mathbb C[x_1,\dots,x_4]$ , some automorphism of $\mathbb C[x_1,\dots,x_4,w]$ sends $f$ to $x_1$ but none of $\mathbb C[x_1,\dots,x_4]$ does. Since the conjecture holds in at most three variables, this is the smallest possible dimension, and one added variable suffices. Corollary 1.2: every fibre of $f$ is $\mathbb A^3$ and none is rectifiable. Not shown: anything about dimensions above four, which the release's earlier five-variable example already covered for five; the result is one explicit polynomial, not a classification.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Oct 5, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The stable coordinate conjecture in four variables · Stable coordinate conjecture, four variables

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed
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Unreleased internal OpenAI model
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Named evidence

VibeMathed record: The stable coordinate conjecture in four variables

specialist registry · passed

No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and Corollary 1.2 of the TeX source, read against Shpilrain-Yu's Conjecture 3 as cited. Lean-checked on the release's Comparator challenge StableCoordinateFour (declaration OAI.StableCoordinate.principal_package, solution module OAI.AlgebraicGeometry.StableCoordinate.Principal), added in the 7 October 2026 update and listed in lean/formalization.yaml. Its statement was read here: for $Q=x_2^2-x_4^2+x_1x_3$ and $f=x_1-2Q(Q(x_2+x_4)+x_1x_4)$ (the paper's polynomial, indices shifted), $f$ has total degree 5, some automorphism of $\mathbb C[x_1,\dots,x_4,w]$ sends $f$ to $x_1$ , no automorphism of $\mathbb C[x_1,\dots,x_4]$ does, and every fibre $f=t$ has coordinate ring isomorphic to $\mathbb C[y_1,y_2,y_3]$ while no automorphism maps its ideal to $(x_1)$ . That is the headline with Corollary 1.2. Not rebuilt here. Nothing is claimed about dimensions above four; the result is one explicit polynomial, not a classification.

Source ↗

The stable coordinate conjecture in four variables

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

An explicit noncoordinate polynomial with affine three-space zero fibre (companion)

paper · pending

paper link supplied by the VibeMathed registry.

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Lean Comparator challenge StableCoordinateFour (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge StableCoordinateFour (OpenAI math release)

lean artifact · pending

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No linked compute attempts recorded.

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An explicit noncoordinate polynomial with affine three-space zero fibre (companion) evidence for this event

This event attributed to Unreleased internal OpenAI model

The stable coordinate conjecture in four variables parent of this event

Lean Comparator challenge StableCoordinateFour (OpenAI math release) evidence for this event

The stable coordinate conjecture in four variables evidence for this event

VibeMathed record: The stable coordinate conjecture in four variables evidence for this event

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