The Heawood family is not the complete list of excluded minors for Colin de Verdière invariant at most 5

We show that two intrinsically knotted graphs $G_1$ and $G_2$ with 23 edges, on 11 and 12 vertices, have Colin de Verdière invariant $\mu=6$ and no minor in the Heawood family, the 78 graphs obtained from $K_7$ and $K_{3,3,1,1}$ by $\Delta Y$- and $Y\Delta$-transformations. So the Heawood family is not the complete list of excluded minors for $\mu\le5$, as van der Holst proposed; Mattman and Pavelescu stated this proposal as a conjecture and derived from it that $G_1$ and $G_2$ have $\mu=5$. More precisely, either the common minor $H$ of $G_1$ and $G_2$, a 4-regular graph with 11 vertices and 22 edges, is an excluded minor for $\mu\le5$, or $G_1$ and $G_2$ both are. It follows that the following two conjectures of van der Holst cannot both hold: (a) a graph is 4-flat if and only if $\mu\le5$; (b) the Heawood family is the complete list of excluded minors for 4-flat graphs. The lower bound $\mu\ge6$ is proved by explicit matrices certified in interval and exact rational arithmetic, and the minor exclusion by exhaustive enumeration. This work has not yet been reviewed by experts. The proofs, the computer checks and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.