The generator problem for finite factors 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 generator problem for finite factors

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, go here for help. (Lean version available!)

expertly designed by an internal OpenAI model

Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	infinite matrices	Levels:	1
Category:	Operator algebras	Lean version:	YES! ✔
Rate this game! 4.6 out of 5 (3,718 votes)

>>> How to Play <<<

The generator problem for finite factors. Proves that every type II₁ factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $P\subset M$ of such factors, the unitaries u with $M=W^*(P,u)$ form a dense G_δ subset in the trace 2-norm topology.

>>> Level Select <<<

LEVEL 1 Relative generation and the generator problem for finite factors

released 2026-09-23 | 4 theorems · 8 lemmas · 13 proofs · 6,924 words | PLAY LEVEL 1 » (pdf)

We prove that every type II₁ factor with separable predual is generated by one operator, equivalently by two self-adjoint operators. Combined with the established direct-integral reduction to this case, this gives an affirmative solution of the generator problem for von Neumann algebras with separable predual.

More Operator algebras Games!
A counterexample to Kirchberg's norm-ultrapower embedding problem	A counterexample to the hyperinvariant-subspace problem HOT!	A counterexample to Kaplansky's quasitrace conjecture	The Kadison–Ringrose cohomology conjecture
A ZFC counterexample to Naimark's problem	A counterexample to Voiculescu’s free-entropy equality conjecture	The Kirchberg–Rørdam character criterion	The Popa–Vaes quadratic strong-operator paving conjecture

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