OpenAI claim summary for Math-2465
mathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
The theorem settles precisely that bundle clause for all finite-rank Artin groups; it does not classify the various Helly/systolic/injective spaces or lattices in the other questions. This entry records partial support; complete coverage of the question is not established by this catalogue.
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
Artin Groups in Helly and Injective Geometry
OpenAI's claim
OpenAI presents a mathematical result relevant to this question in the selected manuscript. Every intersection, including an infinite intersection, of parabolic subgroups in a finite-rank Artin group is claimed parabolic.Mathematical statement of the claimed result
Result 1.
Let $A_M$ be the Artin group of a finite Coxeter matrix on $S$; $A_X$ denotes its standard parabolic subgroup for $X\subseteq S$. For arbitrary $X,Y\subseteq S$ and $g,h\in A_M$, the claim is \[ \exists Z\subseteq S,\ k\in A_M:\qquad gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}. \] Every family of parabolics has the same intersection as some subfamily of size at most $|S|$, so arbitrary intersections are parabolic and every subset has a least parabolic closure. The algorithmic corollary takes $M,X$ and words $u,v$ and decides \[ v\in uA_Xu^{-1}; \] choosing $X=\varnothing$ yields a uniform word-problem algorithm. If the diagram is irreducible and has an infinite label, the additional corollary gives acylindrical hyperbolicity and, for every proper parabolic $P$, a $g$ with $P\cap gPg^{-1}=\{1\}$.Outline of the argument
Relative categorical layers track positive divisors and caps. Two-point uniqueness and a relative frame control null layers, while harmonic heights produce connected minima and bounded positive-height representatives; these give the parabolic carrier of the intersection.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, Parabolic Intersections in Artin Groups, 2026-09-23; citation key
OAI:Parabolic-intersections-in-Artin-groups-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: cor:membership; Statement source: thm:main; Statement source: cor:arbitrary-intersections; Statement source: cor:nonclique-dynamics. - 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.