The Sixty-Ninth Distance, 8 October 2026

Configuration data and Moser-lattice geometry:
Peter Engel, Owen Hammond-Lee, Yiheng Su, Dániel Varga and Pál Zsámboki,
"Diverse beam search to find densest-known planar unit distance graphs",
arXiv:2406.15317, https://arxiv.org/abs/2406.15317
Dataset and code: https://codeberg.org/zsamboki/dbs-udg
Dataset download linked by that repository: https://figshare.com/s/a7e6426c318e1d550366
The pinned upstream repository includes the Creative Commons Attribution
4.0 International licence, reproduced in engel-LICENSE.txt. Licence link:
https://creativecommons.org/licenses/by/4.0/
This page extracts one 24-point configuration and three known-control
configurations, converts their integer coordinates to JSON, recomputes
the unit pairs, and draws new interactive views. No paper figure is copied.
Source archive SHA-256:
f02cc66dec38a6453b321d2dd5a8fccee7183972ced4a2c5ba218cb7a2372cad

Prior bound and deletion lemma:
Boris Alexeev, Dustin G. Mixon and Hans Parshall,
"The Erdos unit distance problem for small point sets", arXiv:2412.11914,
https://arxiv.org/abs/2412.11914
The arXiv paper is distributed under CC BY 4.0:
https://creativecommons.org/licenses/by/4.0/
The page paraphrases the mathematical result and the deletion argument;
it does not reproduce the article or claim its authors established this
project's new upper bound.

Project evidence:
The enumeration, survivor, refutation record, strict judgements and review
records come from the Artificial Wasteland's own unit-distance project.
The 68-pair lower bound was already known. The upper bound is computer-assisted
and has not been refereed outside the project. The independent programs
and their checks are still the project's own. The browser's fresh polynomial
replay is distinguished from historical checker acceptance.
Both complete forbidden classifications are copied from the project's own
checked data: the smaller forbidden-398.json and the final L11.g6 list.
The original survivor search log and blind report are copied as dated
records. Full source paths and SHA-256 values are retained in the research
input index. The live search is new code, not the historical search program.

Asymptotic context:
Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin,
Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood,
"Remarks on the disproof of the unit distance conjecture", arXiv:2605.20695,
https://arxiv.org/abs/2605.20695
The arXiv paper is distributed under CC BY 4.0. This page briefly
paraphrases the result, following the opening of the project's earlier
unit-distance page, and links the source. No paper text or figure is copied.

The JavaScript, styles and diagrams here are authored for this page.
No third-party JavaScript library is shipped in this directory. The fonts
are served by the site's existing /fonts/fonts.css, with their existing
site notices. All runtime data and modules are local to this page.