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Math-2679-A1 · revision r1 · on Math-2679

OpenAI claim summary for Math-2679

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: 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.

Spectral width in dimension three: A complete three-manifold satisfying $-4\Delta+\mathrm{Scal}\ge\lambda>0$ is claimed to map continuously to a graph with full fiber diameter at most $500/\sqrt\lambda$. No global bounded-geometry condition is assumed. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

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

Manuscript

Contents
  1. Spectral scalar-curvature control of width in dimension three
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Spectral scalar-curvature control of width in dimension three

OpenAI's claim

Spectral width in dimension three: A complete three-manifold satisfying $-4\Delta+\mathrm{Scal}\ge\lambda>0$ is claimed to map continuously to a graph with full fiber diameter at most $500/\sqrt\lambda$. No global bounded-geometry condition is assumed.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $(M^3,g)$ be connected, complete, smooth and without boundary. If $\lambda>0$ and \[\int_M(4|\nabla\phi|^2+\operatorname{Scal}_g\phi^2)\,dV_g\ge\lambda\int_M\phi^2\,dV_g\quad(\phi\in C_c^\infty(M;\mathbb R)),\] the claim supplies a simplicial complex $K$ satisfying $\dim K\le1$ and a map $f:M\to K$ which is continuous with $\operatorname{diam}_g f^{-1}(y)\le500/\sqrt\lambda$ for every fiber. Compactness, orientability, spin and bounded geometry are not required.

Outline of the argument

Spectral width in dimension three: A positive supersolution is converted into a drift-weighted scalar inequality without changing the original distance. Weighted stable surfaces inherit a two-dimensional inequality, whose geodesic index estimate bounds their intrinsic diameter. Local minimization and smoothing in compact regions then organize these surfaces into a graph-valued decomposition with a uniform bound on each complete fiber.

Formal verification

Spectral width in dimension three: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources