Number-field lattice construction with explicit parameters (Sawin 2026)
The argument
Take a totally real field F of degree d with a CM extension K; an ideal of K, viewed as a lattice in R^(2d) and projected to one complex embedding, yields a planar set with unit-distance density at least the number of ideal elements of a fixed relative norm, and pigeonholing over the relative class group gives at least ∏(k(p)+1)^(d/(e f)) / (2^d h^-(K)) such elements. Because the relative class number is bounded explicitly by Louboutin's inequality, and because the Golod-Shafarevich criterion yields infinitely many totally real unramified 2-extensions of Q(√(3·5·...·43)) with inertia degree at most 2 at the chosen primes, the exponent formula evaluates, with T the thirteen odd primes up to 43, twenty-two auxiliary primes and R = 72, to 1 + 0.014114, so point sets with more than n^(1.014) unit distances exist for arbitrarily large n.
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 →A finitely presented pro-p group with d generators and at most d^2/4 defining relations is infinite.
- 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 →The relative class number of a CM field is bounded explicitly in terms of its relative root discriminant and degree (Louboutin's bound).
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: given the two named results, the lattice-projection, pigeonhole and class-number steps are elementary and the final exponent is a computation that reproduces exactly. Both premises are settled theorems that nobody disputes, the Golod-Shafarevich criterion supplying the infinite family of fields and Louboutin's relative class number bound supplying the size of the exponent gain. The residual risk lies not in the premises but in the paper's own Galois-group counting (Lemmas 11 and 12), which is unrefereed and was not re-derived line by line here.