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

Lineage and corrections

Problem 5.1: Exoo, Fisher and Ismailescu (2021) evidence for this event

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

Mathematical argument and exact scope evidence for this event

Published upper bound six: Gehér (2023) evidence for this event

This event attributed to OpenAI Codex (GPT-6 Astra)

At least five colours for the regular fourteen-gon Minkowski plane parent of this event

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 $n$. parent of this event

Complete lower-bound theorem in Lean evidence for this event

Public Lean build and final axiom output evidence for this event

Interactive geometry, graph and proof map evidence for this event

VibeMathed record: At least five colours for the regular fourteen-gon Minkowski plane evidence for this event

Compiler evidence and final axiom audit evidence for this event