The sharp ordinary-double-point volume gap at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

A
D
V
E
R
T

I
S
E
M
E
N
T

The sharp ordinary-double-point volume gap

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

thinking... about 3 hours remaining

If this game doesn't work on your computer, we can't help you. No Lean version yet. Some unformalized games could have issues!

expertly designed by an internal OpenAI model

Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	shapes made of equations	Levels:	2
Category:	Algebraic and complex geometry	Lean version:	not yet
Rate this game! 4.7 out of 5 (4,392 votes)

>>> How to Play <<<

The ordinary-double-point volume gap. Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most $2(n-1)^n$. Equality holds precisely for an analytic ordinary double point.

>>> Level Select <<<

LEVEL 1 The ordinary-double-point gap in every dimension

released 2026-09-24 | 8 theorems · 30 lemmas · 44 proofs · 24,998 words | PLAY LEVEL 1 » (pdf)

We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most $2(n-1)^n$, with equality precisely at an analytic ordinary double point.

LEVEL 2 The normalized-volume gap in dimension four

released 2026-09-24 | 1 theorem · 5 lemmas · 11 proofs · 8,071 words | PLAY LEVEL 2 » (pdf)

We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.

More Algebraic and complex geometry Games!
The equivariant cohomological Hikita conjecture for quivers	Counterexamples to Shafarevich holomorphic convexity	Complex counterexamples to cancellation and affine fibrations	A characteristic-zero counterexample to Lipman–Zariski
A stable-coordinate counterexample in four variables	A counterexample to Griffiths' positivity conjecture	Kobayashi's canonical-ampleness conjecture	Tangent-bundle splittings and universal covers

COOLMAITH GAMES SPONSORS: (not real)
Big Al's USED GPUs	Epsilon & Delta LLP attorneys at limit	QED Bail Bonds	Hilbert's Hotel always a vacancy!

Cool Links: openai/math Lean Mathlib arXiv the real Coolmath Games

You are visitor number 000719

Thanks for visiting CoolmAIth Games
© 1997-2026 CoolmAIth Games. A parody. Not affiliated with Coolmath.com LLC or OpenAI.
Every proof here was written by an internal OpenAI model and lives at github.com/openai/math.
Best viewed in Netscape Navigator 4.0 at 800x600