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Math-2678-A1 · revision r1 · on Math-2678

OpenAI claim summary for Math-2678

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.

Higher-dimensional spectral width: The spectral lower bound $-4\Delta+\mathrm{Scal}\ge1$ is claimed to imply uniform continuous $(n-2)$-width for every complete $n$-manifold with $n\ge4$. The bound uses the initial metric rather than an auxiliary conformal distance. 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 codimension-two width
  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 codimension-two width

OpenAI's claim

Higher-dimensional spectral width: The spectral lower bound $-4\Delta+\mathrm{Scal}\ge1$ is claimed to imply uniform continuous $(n-2)$-width for every complete $n$-manifold with $n\ge4$. The bound uses the initial metric rather than an auxiliary conformal distance.

Mathematical statement of the claimed result

Mathematical claim 1.
For $n\ge4$, a connected complete smooth boundaryless $(M^n,g)$ satisfying \[\int_M(4|\nabla\phi|^2+\operatorname{Scal}_g\phi^2)\,dV_g\ge\int_M\phi^2\,dV_g\quad(\phi\in C_c^\infty(M;\mathbb R))\] is claimed to have a continuous $f:M\to K$ with $\dim K\le n-2$ and all fiber diameters at most $C_n$, depending only on $n$. Thus its Urysohn $(n-2)$-width, the infimum of these uniform fiber-diameter bounds, is at most $C_n$.

Outline of the argument

Higher-dimensional spectral width: A positive supersolution produces a signed drift with a weighted curvature bound. An auxiliary graph metric and updated drift add a quadratic reserve even when the original drift is unbounded. The subsequent stretching and wall construction uses that reserve to preserve curvature while reducing intersection multiplicity, yielding fibers bounded in the original geometry.

Formal verification

Higher-dimensional spectral width: 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