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Math-1793-A1 · revision r1 · on Math-1793

OpenAI claim summary for Math-1793

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The recorded question allows a proper isometric action without a cocompactness assumption. The selected Artin and hyperbolic examples rule out proper cocompact actions, with their indicated finite-dimensional classifying spaces. They do not rule out the noncocompact actions permitted here. These are related obstructions to a stronger action requirement; the existence question on this page remains unanswered by the selected theorems.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Questions in Geometric Group Theory — Q 2.2
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Questions in Geometric Group Theory — Q 2.2

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: A specific Artin group on 116 generators is claimed to have no proper cocompact isometric action on a proper CAT$0$ space; A hyperbolic group with a finite classifying complex is claimed to admit no proper and cocompact isometric action with proper complete CAT$0$ target. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Manuscript 1.
For an Artin matrix $M$ on 116 generators with off-diagonal entries only in $\{2,3,\infty\}$, the constructed Artin group $A_M$ is claimed to have no action that is simultaneously isometric, proper, and cocompact on any nonempty proper CAT$0$ metric space. In quantified form, \[ \nexists X,\ A_M\curvearrowright X:\quad X\text{ proper CAT}(0),\quad\text{action proper and cocompact}. \] The metric space is required to be proper, and the exclusion is independent of dimension.
Manuscript 2.
The source constructs a finite, connected, aspherical simplicial complex $K$ of dimension two. In its universal cover, a length-$\ell$ combinatorial loop admits a triangular disk diagram with at most $C\ell$ faces, for one finite constant $C$. Nevertheless, \[ \pi_1K\text{ has no proper cocompact isometric action on a proper complete CAT}(0)\text{ space}. \] The linear filling estimate makes $\pi_1K$ word-hyperbolic. Within the homotopy class of $K$, no simplicial complex with finitely many simplices admits a geodesic length metric compatible with its topology and locally CAT$0$; the claim also excludes such a locally CAT$-1$ metric.

Outline of the argument

Stable translation lengths and centralizer characters impose quantitative braid constraints. Six asymptotic sectors and labeled triangular diagrams yield a local winding count incompatible with those constraints; a finite Artin matrix realizes the contradictory configuration.

Nested incidence tables produce finite relations with a linear disk-filling bound. Conditional sampling creates a limiting configuration of controlled energy, and its geometric obstruction contradicts any proposed CAT$0$ action; the finite aspherical model preserves the topological finiteness claim.

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 are ArtinCAT0.lean, ArtinParabolicIntersections.lean, HarmonicArtin.lean. Named declarations include OAI.ArtinCAT0.main, OAI.HarmonicArtin.salvetti_cover_contractible. For family 257, the documented construction gives a finite aspherical hyperbolic group model without a geometric proper-complete CAT$0$ action. The stronger dimension-two and all locally CAT$0$ model claims are excluded. The listed Comparator files are HyperbolicObstruction.lean. Named declarations include OAI.HyperbolicObstruction.main. 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