The square grid is not asymptotically optimal for maximizing unit distances among n planar points.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Decomposition: one named argument (for) grouping four subclaims. Minted one new low-importance stub (grid upper bound n^(1+O(1/log log n)), requires, importance 0.1, seeded 0.99) after match_claim returned "new"; attached three existing claims: Sawin's n^(1.014) (supports), Lee-Pohoata-Zhu robust repeated distances (supports), and the conjectured bound u(n) ≤ n^(1+o(1)) (contradicts, since its established refutation is what makes the parent true). Written form and evaluation (holds) recorded. Assessment: verified, confidence 0.88, credence 0.97, marginal yield 0.1; the claim is a corollary of the 2026 disproof, whose Section 2 proof was read whole on the neighbouring claim c8bd75ad and whose Lemmas and Theorem 1.1 were read here, with an independent second construction corroborating. Instances: MathWorld instance verified against stored text (affirms, stance correct); added two affirming instances (OpenAI announcement, read from a search excerpt because the page returned 403; Quanta feature sidebar, read whole). Provenance: readings for all three instances, two derives_from edges to the Alon et al. write-up, map written and marked immaterial. Importance set 0.3 / contestation 0.15 (Extractor prior 0.35): derivative of the sharper claim at 0.42, consulted but undisputed. Structural: linked 574038d9 as related and escalated it as a likely duplicate of this node; flagged 0c4ffd28 as a likely duplicate of c8bd75ad; escalated the canonical-direction question (current form is a negation of the historically posed question) without rewriting. Canonical text otherwise left as is. No formalization drafted: "asymptotically optimal" is informal and the sharp content is formalizable on c8bd75ad. No dependents to notify (none recorded). Web search: one query, used to locate the OpenAI announcement and confirm no source disputes the mathematics.
Assessed Verified
verdict confidence 0.88 · credence 0.97
For eighty years the scaled square grid was the best known way to place n points in the plane with many pairs at distance exactly one, giving about n^(1+c/log log n) such pairs, and Erdős conjectured that nothing could do substantially better. That belief was overturned in May 2026. An OpenAI-generated construction, digested and verified in a write-up by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood, produces point sets with at least n^(1+ε) unit distances for a fixed ε > 0 along an infinite sequence of sizes n; Sawin then made the exponent explicit, and sets with more than n^(1.014) unit distances exist for arbitrarily large n. Since no scaling of the grid has more than n^(1+O(1/log log n)) unit distances, the new sets beat the grid by a factor that grows without bound, so the grid is not asymptotically optimal under any reasonable reading of that phrase, whether at the level of the exponent, of constant factors, or of the leading term. The conclusion rests on two independent constructions. The first, from the write-up and Sawin's paper, projects a bounded window of the Minkowski lattice of a CM field of large degree, taken from an infinite class field tower of Golod-Shafarevich type, to one complex coordinate. The second, by Lee, Pohoata and Zhu, gives for every n a set of n points in which every subset has some distance repeated about |A|²/n^(1-δ) times; taking the whole set and rescaling yields a second counterexample, and one that exists for every n rather than along a sparse sequence. Both establish that the conjectured bound u(n) ≤ n^(1+o(1)) fails, which is the precise content of the grid's non-optimality. What remains open is how far the grid is from optimal. The best upper bound on the maximum number of unit distances is still the Spencer-Szemerédi-Trotter O(n^(4/3)), the best explicit lower exponent stood at about 1.014 with unverified community improvements to about 1.036, and the extremal configurations are not characterized. The grid has been shown to be beaten; the true growth rate has not been found.
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