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Math-2656-A1 · revision r1 · on Math-2656

OpenAI claim summary for Math-2656

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.

Closed-surface immersion in four-space: The paper claims a smooth isometric immersion of every closed smooth Riemannian surface into $\mathbb R^4$. Nonorientable surfaces are included, and no embedding conclusion is asserted. 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. Smooth isometric immersion of closed surfaces into four-space
  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

Smooth isometric immersion of closed surfaces into four-space

OpenAI's claim

Closed-surface immersion in four-space: The paper claims a smooth isometric immersion of every closed smooth Riemannian surface into $\mathbb R^4$. Nonorientable surfaces are included, and no embedding conclusion is asserted.

Mathematical statement of the claimed result

Mathematical claim 1.
For every closed smooth Riemannian surface $(M,g)$ there is a smooth immersion $F:M\to\mathbb R^4$ with \[F^*\left(\sum_{i=1}^4dx_i^2\right)=g.\] Closedness means compactness without boundary. The assertion is an isometric immersion, which allows self-intersections; it does not assert a globally injective isometric embedding.

Outline of the argument

Closed-surface immersion in four-space: A short initial immersion into a small three-sphere gives a global radial normal and a finite decomposition of the metric deficit into rank-one terms. Local primitive additions use velocity loops and exact corrections to realize those terms. The corrections preserve the geometric and boundary conditions required for the next addition, completing the finite smooth construction.

Formal verification

Family 333 describes Lean coverage for closed-surface smooth isometric immersion into four-space, including nonorientable surfaces.

Closed-surface immersion in four-space: The manuscript links SurfaceImmersion.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

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