This is a paper posted to arXiv in March 2026 (2603.24880, revised in May) by Yuta Inoue, Ken-ichi Kawarabayashi, Atsuyuki Miyashita, Bojan Mohar, Carsten Thomassen and Mikkel Thorup, titled “The Four Color Theorem with Linearly Many Reducible Configurations and Near-Linear Time Coloring.” The four color theorem says every map in the plane can be colored with four colors so that neighbouring regions differ. The first proof (1976) and the later simplified proof by Robertson, Sanders, Seymour and Thomas both rely on a computer checking a list of “reducible configurations”, and both work one at a time: find one reducible local structure, shrink it away, color the smaller graph, then put the piece back. Coloring a planar graph with n vertices that way takes on the order of n² steps. The new paper proves that every planar triangulation contains linearly many pairwise non-touching reducible configurations, so a constant fraction can be shrunk in each round, for O(n log n) steps in all. According to Quanta (10 September 2026, Gregory Barber), the unavoidable set they use has 8,202 configurations.

What is new is where they look. The classical discharging method assigns each vertex a combinatorial curvature: a vertex of degree below 6 has positive curvature, and a vertex of degree exactly 6 has zero curvature, the “flat” region of a triangular lattice. Earlier proofs looked at the positively curved places, the clusters of vertices with few neighbours, because a reducible configuration is guaranteed to be there and it can be pointed to one at a time. This paper says the flat regions also contain reducible configurations, far more common and far more widespread; once both kinds are counted, there are always many reducible configurations that do not touch, and shrinking one never disturbs the coloring of another. In the abstract’s words: every planar triangulation contains “linearly many pairwise non-touching reducible configurations”, and “all configurations are D-reducible.”

This is the third ray, the Beauty of Mathematics, in the face of the shape of a gap, with some of the tension of the generative face. What a judge cannot not acknowledge here is a structural turn: the same objects, but where the old proofs treated the one reducible place as something to be found, the new proof sees that once the zero-curvature regions nobody had weighed are counted as objects too, the place that can be reduced is not one place but everywhere, and the places do not interfere with one another. The reason is written into the non-touching of the configurations and sits in front of you; you do not have to believe what a computer reports. It is not merely “faster”: n² becoming n log n is a consequence, not the beauty. It is not another confirmation that the theorem is true, since it was already true. The beauty is that something already proved is seen again, and you can see why it can be taken apart this way. It is chisel and construct as one event: the chisel is the unquestioned premise “reduce one place at a time”, the construct is linearly many places in parallel.

Its remainder is not hidden. Quanta quotes Thomassen: “What I would like is a proof without the use of a computer. And I will never stop thinking about that.” This proof still relies on a computer to verify that the configurations are reducible. So “why four colors suffice” still has no reason that can be laid whole before a person; the new proof makes the gap more exact: what is missing is not more configurations but another way of seeing. That is a direction, not a defect, and nothing here ranks one proof above another.

It is worth looking now because the proof will only be presented at FOCS in November, and its standing has not settled. What can be seen is concrete: in a structure everyone thought had one way of being looked at, a flat place that had been taken for “nothing there” has something to say.

Han Qin (秦汉) · @hqinjarsy · SAE Judgment & Aesthetics · Ray 3