OpenAI claim summary for Math-245
mathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
Theorem 1.1 proves universal-cover contractibility for exactly the canonical Salvetti complex over arbitrary finite matrices, including disconnected diagrams and infinite labels. Within these assumptions, this is a claimed affirmative answer to the stated question.
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
K(pi,1) conjecture for Artin groups
OpenAI's claim
OpenAI presents a proof in the selected manuscript. Every finite-rank Artin Salvetti complex is claimed aspherical, allowing arbitrary Coxeter labels.Mathematical statement of the claimed result
Result 1.
Choose an arbitrary Coxeter matrix $M$ on a finite set $S$, allowing infinite labels. Let $W$ be its Coxeter group and $X(W,S)$ the Artin Salvetti complex, with a $|T|$-cell for each spherical subset $T\subseteq S$. The source claims \[ \widetilde{X(W,S)}\text{ contractible},\qquad \pi_n(X(W,S))=0\quad\text{for all }n\ge2. \] No connectedness condition is imposed on the Coxeter diagram.Outline of the argument
A framed category records spherical parabolic cosets and the Artin action. Weighted harmonic heights and two-point estimates organize convex layers; cohomological tests on those layers remove the higher homotopy obstruction and establish the classifying-space property.Formal verification
For family 254, the scope includes finite-rank Salvetti contractibility, the 116-generator Artin CAT$0$ counterexample, and arbitrary parabolic intersections with closure consequences. Each has its own Comparator statement. The listed Comparator files areArtinCAT0.lean, ArtinParabolicIntersections.lean, HarmonicArtin.lean. Named declarations include OAI.ArtinCAT0.main, OAI.HarmonicArtin.salvetti_cover_contractible. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.Status caveat
OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.Sources
- OpenAI, Harmonic heights and the Artin K($\pi$,1) conjecture, 2026-09-23; citation key
OAI:Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 254: Lean scope.
- Comparator ArtinCAT0.json.
- Comparator ArtinCAT0.lean.
- Comparator ArtinParabolicIntersections.json.
- Comparator ArtinParabolicIntersections.lean.
- Comparator HarmonicArtin.json.
- Comparator HarmonicArtin.lean.
- OpenAI catalogue entry.