SIGNIFICANCE

Checked 2026-08-23 · source current · v1

Claim from Evan Beller

The submission claims a proof of the first assertion of Erdős Problem #132 for n = 8: every eight-point planar set has two distinct distances occurring at most eight times.

Quoted from the source · b4ller · 2026-08-23 · View source

Problem link

ErdősProblems.com #132 · open the original problem page

Quoted from the source · Significance editor · 2026-08-23 · View source

Author involvement

Evan Beller reviewed the rendered draft and confirmed the scope, description, and two proposed checks on 2026-08-24.

Record note · b4ller · 2026-08-24

Summary · for readers

In one sentence
A public proof claim says the n = 8 case of the first assertion of Problem #132 is proved. It is not a claim to resolve the general problem.

Reader summary · Significance · 2026-08-23

Start here · for reviewers

Main deduction

Check the (9,9,9,1) multiplicity reduction, the two seven-point deletions, and the overlap-rigidity contradiction.

The proof PDF; formalization directory · The claim identifies the profile reduction and overlap-rigidity lemma as the central route. · Quoted from the source · b4ller · 2026-08-23 · View source

Risk points

  • Check that the cited classification of seven-point three-distance sets applies exactly to each deletion.

    Proof PDF, deletion and classification argument · Significance’s interpretation · Significance editor · 2026-08-23

  • Check the overlap-rigidity lemma and the contradiction from the two distinct diameter endpoints.

    Proof PDF, overlap-rigidity lemma · Quoted from the source · b4ller · 2026-08-23 · View source

Useful background

  • The Hopf–Pannwitz diameter bound, basic distance multiplicity counting, and the cited seven-point classification.

    Proof PDF references and introduction · Significance’s interpretation · Significance editor · 2026-08-23

Needs checking

  • Reconstruct the finite n = 8 argument and compare the Lean declarations with the informal statement. — The public claim reports formal verification conditional on cited literature inputs; no independent check is recorded.

    Proof PDF and formalization directory · Quoted from the source · b4ller · 2026-08-23 · View source

  • Suggest another focused check

Review activity

Independent reruns
0
Math assessments
0
Written reviews
0
Open checks
2

What Significance checked: Significance inspected the proof-claim page, downloaded the linked proof PDF, and recorded the linked formalization repository and its public commit.

Still open: Significance has not assessed the geometric deduction, the cited classification results, the Lean correspondence, or the finite computation.

Copy for sharing

Copy and paste this summary.

A public proof claim says the n = 8 case of the first assertion of Problem
#132 is proved. It is not a claim to resolve the general problem.

Checked: Significance inspected the proof-claim page, downloaded the linked
proof PDF, and recorded the linked formalization repository and its public
commit.

Not checked: Significance has not assessed the geometric deduction, the
cited classification results, the Lean correspondence, or the finite
computation.

As of 2026-08-23T19:11:00Z (freshness: current)
Full record: https://hjyuh.github.io/significance/2026-evanbeller-erdos-132/
Significance records evidence. It does not judge the mathematics.
Scope
This is only the n = 8 case of the first assertion; the general problem remains open. Quoted from the source · b4ller · 2026-08-23 · View source
Source
Erdős Problem #132, n = 8 proof writeup · github-EvanBeller-erdos-132-ffd0920dedff01f04f1c605754384a6a5207f662 · retrieved 2026-08-23T19:11:00Z
sha256 5d3faea03531…

Evidence — 2 entries

  1. Source inspection

    ev-source-inspection

    Public-source and version check only. This is not a mathematical review.

    The public proof-claim page and its linked PDF and formalization repository were inspected. This is source inspection, not a mathematical review.

    Significance’s interpretation · Significance editor · 2026-08-23 · View source

  2. Formal code published by the source

    ev-formalization-reported

    Published by the source. Any independent rerun appears as a separate entry.

    The claim page links a Lean formalization directory. No independent build receipt is recorded here.

    Open link @ ffd0920dedff… · Quoted from the source · b4ller · 2026-08-23 · View source

Plain-language explanation

This is a bounded n = 8 result, not a solution of the general distance problem. A reader can begin with the multiplicity profile and the two deletion arguments.

Explanation · Significance

Limits

  • This record does not issue a whole-paper correctness judgement.
  • Open invitations identify work not yet represented by a completed evidence entry.
  • No independent reproduction or mathematical assessment is represented unless an evidence entry above says otherwise.