Unit-distance disproof
OpenAI announced an AI-produced disproof of Erdos's unit-distance conjecture in May 2026. The result uses algebraic number theory where planar intuition expected a lattice-like story.
Sundog Geometry
Dots, halos, and hidden machinery.
This is the public shelf for geometry workbenches: small visual problems where the visible pattern is not the whole story. Cap sets are the live demo. The unit-distance disproof is the fresh external discovery. Sundog's role is the apparatus beside them: source credit, hidden structure, visible signature, and named boundaries.
Discovery credit
The May 2026 unit-distance result is external work announced by OpenAI and checked by mathematicians. Sundog did not prove it.
Cap-set credit
The cap-set demo points to Croot-Lev-Pach and Ellenberg-Gijswijt's 2016 polynomial-method breakthrough.
Sundog boundary
The page is a primer and apparatus exhibit, not a claim that Sundog has solved a discrete-geometry problem.
Live demo
Capset first: the normal-person version.
The cap-set problem is a good first stop because the rule is tactile: place dots, avoid three-term lines. The shock is that the winning idea was not more dot-gazing. It was a polynomial method over a finite field.
What the demo shows
On `/capset`, a visitor can click points in F_3^n, watch forbidden triples light up, load small maximum cap sets, and compare the trivial 3^n ceiling with the Ellenberg-Gijswijt bound curve.
The point is not to teach the full proof. It is to make the shape of the breakthrough feel concrete: the visible grid is easy, but the decisive structure lives elsewhere.
Hidden
Finite-field algebra and polynomial constraints.
Visible
Dots, forbidden triples, and a bound curve a visitor can see.
Action
Use the workbench as a primer before reading the unit-distance discovery.
Boundary
Primer only. No Sundog-original cap-set proof or evaluator claim.
The side-by-side apparatus
What Sundog is allowed to add.
A new discovery does not become Sundog evidence just because it rhymes with Sundog. The useful move is side-by-side: cite the discovery, explain the visible problem, then show the apparatus Sundog uses when it suspects a visible signature is downstream of hidden structure.
Evidence-tier overlay
Planned exhibit: split the announcement into proved, externally verified, observational, editorial, and aspirational claims. The output is a bounded reading, not a counter-proof or endorsement shortcut.
Three-gate note
Planned note: classify the long stalemate as residual, coverage, or detection failure, then ask outside mathematicians whether that classification is coherent or just Sundog vocabulary pasted onto history.
Claim boundary: this geometry page can say that cap-set, halo geometry, and unit-distance have a shared educational shape: visible pattern, hidden construction, boundary-aware reading. It cannot say Sundog proved cap-set, solved unit-distance, or validated a general geometry theorem.
Workbench shelf
What belongs here next.
The geometry page is a container, not a single demo. Capset is active now. The halo atlas and h-of-x are already public. The AI-math apparatus rails are staged until their notes and review paths earn a stronger public position.
Capset
Clickable F_3^n primer for finite-field lines, small maximum caps, and the polynomial-method bound.
Halo Atlas
Atmospheric-optics atlas for parhelia and halo-family vocabulary, separated from the broader geometry shelf.
h-of-x
A small inverse handle: recover eligible hidden sun altitude from photographed parhelion offset.
Unit-distance overlay
Future side-by-side reading of the OpenAI result with citation rail, evidence tiers, and explicit boundaries.
Substrate rhyme note
A careful note asking whether parhelion, cap-set, and unit-distance share structure or only a convenient slogan.
Isotrophy
Three-body symmetry descent remains a research sidecar. Its current lesson is process discipline: do not rescue a count after a cheap precheck collapses.
Sources and credit
Start with the originals.
OpenAI unit-distance article
Primary announcement for the May 2026 result and its context.
Open sourceUnit-distance proof
The proof document linked from the announcement.
Open PDFCompanion remarks
Mathematician commentary and context around the result.
Open PDFEllenberg-Gijswijt
Cap-set bound paper: large subsets of F_q^n with no three-term progressions.
Open arXivCroot-Lev-Pach
The polynomial-method precursor on progression-free sets in Z_4^n.
Open arXivSundog capset ledger
Internal claim-boundary staging for this geometry shelf.
Open ledger