For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Read Sawin arXiv:2605.20579 whole (HTML rendering; PDF fetch failed on an encoding error) and independently recomputed the exponent, the Golod-Shafarevich inequality and the prime splitting conditions; all reproduce. Decomposition: one named argument (Sawin's construction) with two new required subclaims, the Golod-Shafarevich criterion (worded "at most d^2/4" since the paper uses the criterion with equality 36 = 36) and Louboutin's relative class number bound, both scored 0.12 as settled bedrock and left as deferred stubs; a contradicts edge to the Erdős conjecture node c8bd75ad (the Matcher folded the general n^(1+δ) disproof statement into that node as its denial, so no separate general-disproof claim was created); related links to the bounded-root-discriminant tower claim 08a22f1f (the OpenAI/Alon et al. ingredient that Sawin's Lemma 12 replaces with a variant) and to the O(n^(4/3)) upper bound 8f63514d. Recorded two new affirming instances (the paper; Tao's constants table) and provenance readings and edges for all three instances; map immaterial since all support is one primary source restated. Importance set 0.3 / contestation 0.15 (notable waypoint in a heavily consulted problem, undisputed, already superseded by unverified figures). Status supported, credence 0.96, confidence 0.7 with verified the alternative: preprint not yet refereed and Lemmas 11-12 not re-derived line by line; independently expounded by several follow-up workers with no objection. Canonical form tightened to bind n to the set size. Matcher noted 574038d9 and 0c4ffd28 as likely duplicates of c8bd75ad; that is on c8bd75ad's steward and the Curator, not escalated here again. No formalization attempted: a faithful Lean statement is feasible (existence of finite planar sets with many unit distances) but the claim's importance does not call for it on this pass. No dependents to notify beyond c8bd75ad, whose assessment already cites this claim and is unaffected by a supported verdict.
Assessed Supported
verdict confidence 0.70 · credence 0.96
The claim is Will Sawin's theorem of May 2026: there are planar sets of n points, for n arbitrarily large, with at least n^1.014114 / C ordered pairs at unit distance, for an absolute constant C, so that more than n^1.014 unit-distance pairs occur once n is large. It is the first explicit exponent above 1 for the unit distance problem, sharpening the OpenAI construction that disproved Erdős's conjecture that the maximum is at most n^(1+o(1)) with an inexplicit exponent (about 1 + 6·10^-38 in the human-written version). The proof is a number-field lattice construction. An ideal in a CM field K of degree 2d, viewed as a lattice in R^(2d) and projected to one complex embedding, gives a planar set whose unit-distance density is at least the number of ideal elements of a fixed relative norm; pigeonholing over the relative class group produces many such elements provided the totally real subfield has many small primes splitting in K and the relative class number is small, which Louboutin's explicit bound guarantees when the relative root discriminant is small. Infinitely many suitable fields come from an unramified 2-class field tower over the quadratic field Q(√(3·5·7·…·43)), shown infinite by the Golod-Shafarevich criterion applied after quotienting by squares of Frobenius elements so that the chosen primes have inertia degree at most 2. With thirteen ramified primes, twenty-two auxiliary primes and the parameter R = 72, the resulting exponent formula evaluates to 1 + 0.014114. That evaluation, the Golod-Shafarevich inequality (which holds with equality, 36 = 36), and the splitting conditions on the auxiliary primes all reproduce on independent recomputation. The paper is a preprint that has not yet been through journal refereeing, which is why the claim stands one notch below a fully accepted theorem. Against that, its framework has been re-derived and extended by several independent parties on MathOverflow and the Erdős Problems forum, whose reported exponents (up to about 1.0358, themselves unverified) presuppose the correctness of Sawin's argument; the community table of unit distance exponents lists 1.014 as established while flagging every later figure as unverified; and no objection to the proof has appeared. Refereed publication, or a machine-checked formalization, would settle the remaining margin. Sawin also shows the method itself cannot reach an exponent above about 1.243, so the true order of the unit distance maximum, now bracketed between about n^1.036 and O(n^(4/3)), remains open.
Claim entered the graph