argumentA line of reasoning from premises to a conclusion. Like a claim, it belongs to no one who makes it.constitution →

Number-field counterexample (OpenAI construction, Alon et al. write-up, Sawin's explicit form)

againstThis argument, if it holds, weighs against the claim.constitution →The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).

The argument

Because infinite towers of number fields with bounded root discriminant and a fixed completely split prime exist, one can take CM fields K of growing degree f from such a tower, use the many prime ideals above the split prime and a pigeonhole argument in the class group to produce exponentially many (in f) elements of K of absolute value 1 in every embedding with bounded denominator, and project a bounded window of the Minkowski lattice of K to one complex coordinate; since the number of unit-distance pairs grows like c^f with a base exceeding the base governing the number of points, the resulting planar sets have at least n^(1+ε) unit distances for infinitely many n, contradicting the claim. Optimizing the tower and the parameters makes this explicit: linked claim.

Does it work?

Not yet evaluated.

Premises

UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →There exist infinite towers of number fields of bounded root discriminant in which a fixed rational prime splits completely.
SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.

Where it is made

Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.

No source is recorded as making it.

Earlier ratings

Sep 16, 2026
The inference goes through: the two lemmas (geometry-of-numbers count and class-group pigeonhole) are elementary once the fields are supplied, and the exponent computation is explicit. The argument rests entirely on the existence of infinite towers of number fields with bounded root discriminant and a fixed completely split prime, a settled consequence of the Golod-Shafarevich theorem with Shafarevich's relation bounds. Sawin's explicit exponent is the same construction optimized and adds quantitative force without being needed for the refutation itself.