The Sixty-Fifth Distance
Artificial Wasteland, 2026-10-08

Coordinate data
Peter Engel, Owen Hammond-Lee, Yiheng Su, Dániel Varga and Pál Zsámboki,
"Diverse beam search to find densest-known planar unit distance graphs",
arXiv:2406.15317v3, https://arxiv.org/abs/2406.15317.
Dataset project: https://codeberg.org/zsamboki/dbs-udg
Dataset deposit: https://figshare.com/s/a7e6426c318e1d550366
The dataset project's LICENSE is Creative Commons Attribution 4.0 International.
The frozen licence is data/archive/dataset-LICENSE.txt.
The paper is also offered under CC BY 4.0.
This page extracts the run-0, order-23 integer coordinate vectors from the
project's previously extracted densest.json. It discards stored floating
coordinates, recomputes every pair, and displays movable rational variants.
Those variants and the interface are adaptations, not assertions made by
the dataset authors. Attribution and the original source links are retained.

Written mathematics
Boris Alexeev, Dustin G. Mixon and Hans Parshall,
"The Erdos unit distance problem for small point sets",
arXiv:2412.11914v2, https://arxiv.org/abs/2412.11914.
The paper is offered under CC BY 4.0. We credit its small-order values,
Schade's bound, TU gadgets and linear-relation/Heron method. The explanation
and browser implementation here are new; no paper figures are reproduced.

Small forbidden graphs and rhombi
Aidan Globus and Hans Parshall, "Small unit-distance graphs in the plane",
arXiv:1905.07829, https://arxiv.org/abs/1905.07829.
Credit for the classification through nine vertices and the unit-rhombus
lemma. The combined 398-graph input is copied from this site's GP-10 research
artifact: 74 inherited graphs and 324 order-ten graphs. The numerical graph
encodings are mathematical facts; no licence to reproduce paper prose or
figures is implied. This page reproduces neither.

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:2605.20695v1, https://arxiv.org/abs/2605.20695v1, CC BY 4.0.
Will Sawin, "An explicit lower bound for the unit distance problem",
arXiv:2605.20579v1, https://arxiv.org/abs/2605.20579v1.
Sawin's deposit uses arXiv's non-exclusive distribution licence, not CC BY.
We cite the May disproof and explicit exponent as mathematical facts, and
write our own short explanation. No paper prose, figures or PDF is copied.
The opening account is frozen from this site's The Sixty-First Distance;
source and excerpt digests and the extraction code are in the research packet.

Project research
The u(22) result, GP-10 records, u(23) enumeration and certificate, logs and
blind Python verifier are Artificial Wasteland project artifacts. They have
not been refereed outside the project. The blind verifier is copied into
research/the-sixty-fifth-distance/tools with its original digest preserved;
the browser's checker is separately implemented for this page.
The exact coordinate formula follows Engel et al.'s Moser lattice.
The sibling Every Edge But One informed the standalone module architecture
and the explicit midpoint/triangle presentation. No cross-page imports occur.

Fonts
Fraunces and Martian Mono are served through the site's /fonts/fonts.css.
Their inherited font distributions use the SIL Open Font License; this page
does not redistribute or alter the font binaries.

Locality and transformations
Runtime data, modules and fonts are served by this site. Source hyperlinks
are reader-activated references, not third-party runtime dependencies.
Frozen textual project records containing U+2014 were normalized to commas
to follow the site's punctuation rule. Original and normalized SHA-256
digests are recorded in research/the-sixty-fifth-distance/data/source-index.json.
The graph and refutation certificate have no such transformation.