How many unit distances can points in a plane have?

The paper claims that for arbitrarily large n there is a set of n points in the plane with more than n^1.014 pairs at distance exactly 1; Theorem 1 states the exponent as 1.014114/C for an absolute constant C. This improves on an OpenAI result in which Lijie Chen, "using an internal OpenAI model," showed a lower bound n^(1+δ) with δ "not made explicit," and Mark Sellke and Mehtaab Sawhney checked correctness. A simplified human version of that argument obtained δ ≈ 6 × 10^-38. The paper's exponent is explicit and, in its words, "not exorbitantly small," differing from the best upper bound O(n^4/3) (Spencer, Szemerédi and Trotter) "by a factor of less than 24."

The mechanism is a lattice with many short vectors whose projections to the plane have length 1. The paper takes the lattice from the ring of integers of a CM field K over a totally real subfield F. The squared length of a projected lattice vector lies in F, which has smaller degree than K, so many projections can share a length. The argument then chooses an ideal in which many elements share a norm (Lemmas 4 and 5), uses primes that split in K (Lemma 7), specializes to Galois K (Lemma 8), bounds the relative class number (Lemma 9), and uses a Golod-Shafarevich criterion to produce infinitely many suitable fields (Lemma 12, with group calculations in Lemma 11). The author says the paper makes "explicit and sharpen[s] every step" of the OpenAI argument, and that this "rarely makes the arguments much more complex."

The sharpest contrast is with Can identical outputs hide broken internal representations?. That note concerns outputs that match while the internals are broken, visible only under weight perturbation. Here the correct result comes with its steps written out lemma by lemma, and the gap between correct and understood is closed by human exposition: the excerpt credits the simpler account to "a team of mathematicians," not to a model. The contrast with Do foundation models learn world models or task-specific shortcuts? is in the question asked. That note asks what a model has internalized behind correct predictions; this paper asks which number-theoretic construction yields the count. Can AI systems improve themselves through trial and error? marks another standard: that system validates changes empirically on benchmarks, while this result rests on a formal argument with stated lemmas.

The excerpt does not establish that this proof has been independently verified. No proof assistant, referee or expert check appears in it, and the only verification it names is Sellke and Sawhney's check of the OpenAI argument. It also does not show the explicit bound is easier to follow than the original: the paper says its length is "essentially the same as the original OpenAI writeup, though longer than the simplified version prepared by human authors." The excerpt supports an explicit exponent for this construction. It does not support a claim that this proof is understood in its own right, and as the excerpt itself reports, the human-readable route runs through the separate simplification. Open questions the excerpt leaves are the density of the n for which such sets exist and the limits of the exponents, both posed by Thomas Bloom.