The Stable Coordinate Conjecture in ambient dimension five

Corollary 1.2: the explicit $H\in P=\mathbb C[p,s,u,F,J]$ is a coordinate of $P[w]$ (by the Dutta-Lahiri theorem, since $H$ is a residual coordinate over $\mathbb C[p]$) but not a coordinate of $P$, because otherwise $P/(H)\cong\mathbb C^{[4]}$, contradicting Theorem 1.1. So the conjecture fails over $\mathbb C$ 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.

Corollary 1.2: the explicit $H\in P=\mathbb C[p,s,u,F,J]$ is a coordinate of $P[w]$ (by the Dutta-Lahiri theorem, since $H$ is a residual coordinate over $\mathbb C[p]$) but not a coordinate of $P$, because otherwise $P/(H)\cong\mathbb C^{[4]}$, contradicting Theorem 1.1. So the conjecture fails over $\mathbb C$ 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.