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Claim. OpenAI, The Euclidean plane is not five-colorable, OpenAI Math Release preprint, 23 September 2026, linked above at the pinned revision and carded in the library (card). A proper $k$-coloring of the plane is a map $c:\mathbb R^2\to\{1,\dots,k\}$ with $c(x)\ne c(y)$ whenever $\|x-y\|=1$, with no restriction on the color classes, and the chromatic number $\chi(\mathbb R^2)$ asked for by Problem 508 is the least $k$ for which one exists. The paper's Theorem 1.1 states that the plane has no proper five-coloring, even when arbitrary color classes are allowed, so that

$$6\le\chi(\mathbb R^2)\le7.$$

The upper bound is the classical seven-coloring by the hexagons of a triangular lattice, which the paper writes out with every boundary point assigned to an incident hexagon. The lower bound has two parts, both proved in ZFC. Theorem 1.3, the transfer theorem, states that for every positive integer $k$ a proper $k$-coloring exists exactly when a weak measurable $k$-coloring exists: a Lebesgue measurable coloring whose same-color unit pairs form a null set of the product of the plane with the unit circle (Definition 1.2). Its forward direction restricts a proper coloring to the plane of real algebraic numbers, averages its translates and rotations into an invariant probability law on labelings, and applies a Haar rigidity theorem for rotation-invariant measures on the character group of the algebraic plane (Theorem 2.3) to discard the part of each label that correlates with no continuous character, which keeps the zero unit-pair correlations and yields measurable fields on the plane. Theorem 1.4 states that no weak measurable five-coloring exists; its proof constructs the connected interfaces between color regions that a planar-map argument would assume and ends at a placement the Moser spindle forbids. The paper contrasts this with the measurable problem, where Falconer's bound of five for measurable classes does not by itself say anything about arbitrary colorings. Corollary 1.5 gives a positive lower bound on the fraction of monochromatic unit pairs in every five-coloring; its proof obtains from Theorem 1.1 and the de Bruijn–Erdős compactness theorem a finite unit-distance graph that is not five-colorable, which is existential and not exhibited. The paper also records, in a footnote, that a public manuscript of Reed dated 14 May 2026 announces $\chi(\mathbb R^2)=7$ by a circle-density argument, and that the manuscript's proposed strict bound (angular measure less than $\pi/3$) on a unit-independent subset of the unit circle fails for the half-open arc $\{e^{it}:0\le t<\pi/3\}$, which has measure $\pi/3$ and contains no unit pair, while its displayed formal theorem assumes that bound as a hypothesis; that announcement is the rejected claim on [[problems/discrete_geometry/E0508/claims/2026_05_13_reed|Reed's claim page]].

Covers. Bounds only, not the value. No coloring of the plane with at most five colors, with arbitrary color classes, avoids two same-colored points at distance $1$, so $\chi(\mathbb R^2)\ge6$, which raises the lower bound from de Grey's five (claim page). The hexagonal seven-coloring gives Isbell's classical bound $\chi(\mathbb R^2)\le7$. The two bounds are separate theorems on two models of the plane related by an isometry. Whether $\chi$ is six or seven is open.

Depends on. No page of this wiki.

Formal verification. The release's Lean development, the lean/ folder at the pinned revision linked above, states the two bounds as separate theorems on two models of the plane. OAI.EuclideanFiveColor.no_proper_five_coloring, in OAI/Geometry/PlaneColoring/Five.lean, proves ¬ ∃ c : ℂ → Fin 5, ProperColoring 5 c, where ProperColoring k c, defined in OAI/Geometry/PlaneColoring/Coloring.lean, is ∀ x y : ℂ, ‖x - y‖ = 1 → c x ≠ c y. The complex numbers with Mathlib's norm are the Euclidean plane, the coloring is an arbitrary function with no measurability hypothesis, the five labels need not all be used, so every coloring with at most five colors is excluded, and $x=y$ is excluded because $\|0\|=0$. The proof obtains a Borel weak coloring from the proper one (proper_to_borel_weak, the transfer) and refutes it (no_measurable_weak_five_coloring). OAI.Problem160.properColoring_seven, in OAI/Geometry/PlaneColoring/Seven.lean, proves ProperColoring 7 on Plane := EuclideanSpace ℝ (Fin 2), where ProperColoring k is ∃ c : Plane → Fin k, ∀ x y : Plane, dist x y = 1 → c x ≠ c y, a total coloring that assigns every boundary point; the namespace Problem160 is the release's internal label and does not refer to another catalog problem. The isometry between the two models is not in Lean. The corpus's verification built both declarations and checked their axioms: each uses only propext, Classical.choice and Quot.sound. The comparator challenges lean/ComparatorChallenges/EuclideanFiveColor.lean and lean/ComparatorChallenges/PlaneColoring.lean pin the two statements, each together with its ProperColoring definition and the second with the Plane abbreviation, and the pinned fingerprints were identical at the build. Neither theorem defines $\chi(\mathbb R^2)$ or determines it.

Acceptance. The evidence is formalized: the kernel-checked proofs whose statements the corpus audited against the problem and built as described above. The acceptance is of the Lean statements so audited; the prose proof of the manuscript is unreviewed. Not reviewed and not refereed: no outside review of the manuscript is recorded, the release is a preprint with no journal record, and the site's page labels the problem OPEN, was last edited on 22 January 2026 with the bounds $5\le\chi\le7$ in its remarks, and its proof-claims thread listed no claim on 6 October 2026. The release's own README says that its manuscripts were produced by an internal OpenAI model and are at different stages of verification. The claim is partial because the question asks for the value of $\chi(\mathbb R^2)$, and the theorems leave six and seven both possible.