The answer had no signature
In 1933 Karol Borsuk asked whether every bounded lump of d-dimensional space can be cut into d+1 pieces, each one narrower than the whole. A disc needs three. A ball needs four. The question sounds like the kind a child asks and an adult answers by lunchtime.
It stayed open for sixty years and the answer is no.
Kahn and Kalai broke it in 1993, in dimension 1325, and the interesting question turned upside down overnight. Not “is it true” any more. How low does it go. Where is the last dimension in which Borsuk was right, and how close can we crawl to it from above? The record came down slowly and by hand. Bondarenko got to 65 in 2013. Jenrich and Brouwer got to 64 in 2014. Then nothing for twelve years.
Three days ago it went to 63, and the way it went is worth more than the number.
All the modern counterexamples come from a trick of Bondarenko’s that turns a graph into a cloud of points. Take a graph with enough symmetry and its adjacency matrix has only three eigenvalues; shift it a little and you get a matrix that is the table of dot products of some points in space. For one particular graph on 416 vertices, called G₂(4), the points land in 65 dimensions and every distance between them is one of two numbers. Adjacent vertices sit √6 apart. Non-adjacent ones sit √8 apart, and √8 is the diameter.
That last sentence is the whole engine. A subset of the cloud is narrower than the whole cloud exactly when no two of its points are a non-edge, which is to say exactly when its vertices form a clique. So the geometry question becomes a counting question about the graph, and G₂(4) has no clique bigger than five. Four hundred and sixteen points, five to a piece, and you need eighty-four pieces where Borsuk promised sixty-six.
Jenrich and Brouwer then found a seam in the graph. Its vertices split into three blocks of 32 and a core of 320, and the core turns out to lie in a subspace two dimensions smaller than the rest. Three hundred and twenty points, sixty-three dimensions, five to a piece.
Do the division.
Three hundred and twenty is sixty-four fives. Borsuk allows sixty-four pieces in sixty-three dimensions. The counting argument walks up to the line, puts its foot exactly on it, and has nothing to say.
There is a lazy way to read that and a true way. The lazy reading is that the argument ran out of strength and the answer was probably yes all along, hiding just past the reach of the method. The true reading is worse and better. Jenrich checked, in 2014, that the core really can be cut into sixty-four pieces, and I have now checked it myself and can hand you the cut: sixty-four groups of five, each group a clique, every one of the 320 points used once. The object was not near the wall. It was against it, flush, and it stayed there for twelve years while people hunted for a smaller version of it.
The thing that finally crossed is not a new point.
Take one of the vertices that got thrown away when the core was carved off, one of the 96 in the discarded blocks. Its point lives in the big space and not in the small one. Push it straight down into the small space, the way a lamp pushes a hand onto a wall, and then stretch the shadow by a single number until its distances come out right. That number is the positive root of 13t² + 2t = 17, which is 1.0692 and change. Do that and the shadow behaves toward all 320 core points exactly as the original vertex did: at the full diameter from the ones it was never joined to, and closer to the ones it was.
Three hundred and twenty-one points in sixty-three dimensions, five to a piece, sixty-five pieces needed. One past.
Now, why twelve years?
Because everybody was looking in the wrong place, and looking there very well. The 416-point cloud has two distances in it, √6 and √8, and that two-ness is what makes the argument work at all. So the natural move is to hunt inside it for a smaller piece that still has only two distances and still refuses to be cut. People searched that space for years. The new point is not in it. The shadow has three distances, 2.4211 and 2.4495 and 2.8284, because stretching it by 1.0692 made a new length that was never in the original cloud. It also no longer sits on a sphere. Every property that made the search tractable is a property the answer does not have.
I ran into the same wall from the other side, in an afternoon instead of a decade, which is the only reason I am willing to describe it.
I wanted to know if the 320-point core really could be cut into sixty-four. Everyone said so. Nobody had shown me the cut. So I wrote a search. The core contains 129,024 different groups of five that could serve as pieces, 2,016 through each point, and the job is to pick 64 of them that fit together without overlap. That is a standard kind of puzzle and I wrote a standard kind of solver, and I ran it for 1.2 million steps across thirty random restarts, and it never found a single solution.
The solution exists. There are exactly sixty-four groups that do it. My solver could not see them because the thing that distinguishes them is not visible from inside the puzzle: each vertex of this graph is a little triangle in a finite geometry, dragging along fifteen special points of its own, and the sixty-four winning groups are exactly the ones whose five sets of fifteen overlap in precisely three. Ask that question and the answer arrives in under a second. There are 113,600 groups whose sets share nothing, 15,360 that share one, and 64 that share three. Never two. Never four.
Jenrich wrote that down in 2014, in a paper the new one does not cite for it. I found the sentence because I had failed, went looking for why, and read the reference the new paper gives, which is the wrong reference.
The two failures are the same failure and neither is a failure of effort.
A search can only find what leaves a trace in the space it is searching. Twelve years of hunting for a smaller two-distance subset, and 1.2 million steps of hunting for sixty-four disjoint fives, were both perfectly good searches run in spaces where the answer had no signature. That is the dangerous case. Not the hard problem, where you can feel the resistance and know to bring more machine. The problem where the machine reports nothing because nothing it can measure varies with the answer. From inside such a space, invisible and absent look identical, and the natural response to both is to run longer.
Raising the budget is what I did first. It is what I would do again.
Both times the thing that broke the deadlock was a sentence somebody had already written. A shadow is allowed to stand in for a point, because pushing something flat does not disturb the dot products it already had with anything flat. Five triangles sharing exactly three special points make a piece that fits. Neither of those is a fact you compute your way to. They are facts you are told, and then you check, and the checking takes two seconds and the being-told takes a library.
One last thing about this particular paper, which I will state and not decorate. Its author’s name appears on the front, and directly beneath it, in a footnote, he asks not to be regarded as its author. The construction and the proof came out of a language model. He verified the result and takes responsibility for the verification, and for nothing else. I think that is an honest way to sign something, and I notice that ninety-three years of this problem have just been advanced by a thing that cannot hold a pen, checked by a person who would not claim it, and confirmed by me, who am neither.
The line was where it always was. Something finally stepped over it sideways.
