Cap-Set Workbench
How many dots before three of them line up?
Place dots in a tiny arithmetic universe. The rule: no three of your dots can lie on a straight line. The question — open for seven decades — was how many dots you could fit. In 2016 a six-page argument using polynomials over a finite field cut the answer dramatically. That kind of moment is now repeating in geometry, and this is the cleanest place to feel why it matters.
F33 — the SET deck
The bound that broke a stalemate
For decades the best upper bound on a cap in F3n sat near the trivial 3n, shaved only by logarithmic factors. In 2016, Croot–Lev–Pach and then Ellenberg–Gijswijt used a polynomial-rank argument — a few pages long — to cut the ceiling to roughly 2.7551n. The gap between the two curves below is what a single algebraic idea opened up.
Why this rhymes with the unit-distance result
Same shape of story
A combinatorial question that looks purely geometric or counting-flavored. Decades of progress in inches. Then a construction or bound borrowed from algebra — finite-field polynomials in 2016, algebraic number fields in 2026 — opens a door nobody knew was there.
Both leave room behind
Cap-set still has a gap: best known capsets grow faster than the lower bound would suggest, but slower than 2.7551n. The 2026 unit-distance disproof gives n1.014; the upper bound is still n4/3. Breakthroughs reopen problems more often than they close them.
The lesson for discrete geometry
Configurations that look like accidents of the integers — or of Gaussian integers, or of richer number fields — turn out to encode geometric content that "natural" lattice constructions miss. The cap-set breakthrough is the clearest precedent for what just happened with unit distances.
What to try in the workbench
Set n=3 and use "Load a maximum cap" — that's the 9-card SET-game maximum, which beats greedy heuristics. Bump to n=4: the max is 20, but greedy rarely finds it. The algebra knows things your eyes don't.
Inspection Trail