At least five colours for the regular fourteen-gon Minkowski plane

We prove $\chi(\mathbb R^2,P_{14})\ge5$ using a graph with 1,540 vertices and 13,755 unit edges. Every edge difference is a rotated point on a side of the unit polygon with an exact algebraic parameter in $[0,1]$. A kernel-checked LRAT refutation of the graph's four-colouring CNF rules out every four-colouring, and restricting a colouring of the plane to these points gives the contradiction. The whole geometric and logical chain is in Lean. With Gehér's upper bound, $5\le\chi(\mathbb R^2,P_{14})\le6$; whether five colours suffice remains open, as does Problem 5.1 for other $n$.

We prove $\chi(\mathbb R^2,P_{14})\ge5$ using a graph with 1,540 vertices and 13,755 unit edges. Every edge difference is a rotated point on a side of the unit polygon with an exact algebraic parameter in $[0,1]$. A kernel-checked LRAT refutation of the graph's four-colouring CNF rules out every four-colouring, and restricting a colouring of the plane to these points gives the contradiction. The whole geometric and logical chain is in Lean. With Gehér's upper bound, $5\le\chi(\mathbb R^2,P_{14})\le6$; whether five colours suffice remains open, as does Problem 5.1 for other $n$.