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Math-1987-A1 · revision r1 · on Math-1987

OpenAI claim summary for Math-1987

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.

For the main hyperbolic example any such Rips metric would give a geometric CAT(0) action: the finite generating set makes the Rips complex locally finite with finite quotient, and a compatible complete locally compact geodesic metric is proper. This contradicts the core no-geometric-CAT(0) conclusion. Within these assumptions, this is a claimed negative answer to the stated question.

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

Manuscript

Contents
  1. Questions in Geometric Group Theory — Q 1.5
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Questions in Geometric Group Theory — Q 1.5

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. A hyperbolic group with a finite classifying complex is claimed to admit no proper and cocompact isometric action with proper complete CAT$0$ target.

Mathematical statement of the claimed result

Result 1.
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

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 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