HEXAGONbeta

An Exponent of 1.04273 for the Unit Distance Problem

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.NT — Number Theory

Contributed by Eric Naslund ↗

SubmitterEric Naslund

Version 1 / Oct 02, 2026 / CC BY 4.0

Abstract

Let u(U)u(U) count the unordered pairs at distance one in a finite planar set UU. We construct finite sets UjU_j with ∣Uj∣→∞|U_j|\to\infty and u(Uj)/∣Uj∣1.04273→∞u(U_j)/|U_j|^{1.04273}\to\infty. The largest exponent previously claimed, 1.03581.0358, is in the author's unpublished manuscript. The method is the number-field construction of OpenAI and Sawin: unit distances come from elements of relative norm one in quadratic extensions, and the fields come from an infinite pro-22 class field tower. The new ingredients are quadratic extensions of mixed signature, with an exact average over their norm-one units, and a tower over the real quadratic field $\Q(\sqrt{241})$, in which 22, 33 and 55 split. Its Golod--Shafarevich function contains two copies of the local conditions at these primes but only one constant term, and the extra room lets the tower be ramified only above 22, 33 and 55; the root discriminant of its fields is about 286286. The relative zeta value is bounded through the zeta function of the degree-512512 field generated over $\Q(\sqrt{241})$ by the square roots of its {2,3,5}\{2,3,5\}-units, which is the product of the Dedekind zeta function of $\Q(\sqrt{241})$ and 255255 quadratic Hecke LL-functions. Finite facts and numerical inequalities are certified by exact computation and interval arithmetic. A significant portion of this work was verified in Lean, reducing the result with exponent 1.04271.0427 to an explicit zeta function inequality (Palomar registry, PALOMAR-2026-10-01-000018, version 1).

Provenance statement

This work was done with extensive AI use, with agents developing the mathematics and writing the paper. I supervised and prompted the AI agents that produced this work. While this has been edited to improve readability, I recommend the reader also make use of their AI model of choice to ask the paper questions. The construction over $\Q(\sqrt{241})$, the computations that certify it, the Lean formalization of the reduction of the proof of exponent $1.0427$ to one explicit zeta inequality, and this paper were produced with a combination of Claude Opus 5.5, ChatGPT Sol 5.6, ChatGPT Astra 6, working as agents in Claude Code and Codex. The construction adapts the author's previous manuscript which obtained 0.0358 which was also written with AI tools.
FormalizationsPalomar

Tools used

OpenAI
ChatGPTVersion Astra 6
OpenAI
ChatGPTVersion Sol 5.6
Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1Initial depositCurrentOct 01, 2026