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Math-1679-A1 · revision r1 · on Math-1679

OpenAI claim summary for Math-1679

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: partially solved. 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.

A smooth three-space immersion obstruction: A smooth positive-definite metric with no immersion on any neighborhood of its origin also has no embedding there, exactly negating the universal local smooth embedding assertion. Local immersion/embedding equivalence is explicitly part of our definition.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

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

Manuscript

Contents
  1. Geometry of Curves and Surfaces — Problem 1.9
  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

Geometry of Curves and Surfaces — Problem 1.9

OpenAI's claim

A smooth three-space immersion obstruction: A smooth positive surface metric is claimed to have no isometric immersion into $\mathbb R^3$ on any neighborhood of the origin. All its derivatives at the origin nevertheless agree with the Euclidean metric.

Mathematical statement of the claimed result

Mathematical claim 1.
There exists a positive-definite smooth Riemannian metric $g$ on $(-1,1)^2$ such that for every open neighborhood $U$ of $(0,0)$ there is no $C^\infty$ immersion $F:U\to\mathbb R^3$ satisfying \[F^*(dx_1^2+dx_2^2+dx_3^2)=g|_U.\] Consequently smooth local isometric embeddings also fail for this metric.

Outline of the argument

A smooth three-space immersion obstruction: The necessary Darboux equation for an immersion height is obstructed on a sequence of small square patches. A candidate height with an appropriate Hessian component would violate the patch estimate. The metric is arranged to produce such a height from any hypothetical local immersion, while the patch amplitudes vanish fast enough to preserve smooth flatness at the origin.

Formal verification

Family 334 describes Lean coverage for a smooth metric for which every neighborhood of the origin fails three-space isometric immersion.

A smooth three-space immersion obstruction: The manuscript links IsometricImmersion.lean (declaration OAI.SmoothLocal.Geometry.exists_local_metric_without_local_immersion).

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