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Math-43-A2 · revision r1 · on Math-43

OpenAI claim summary for Math-43

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.

Extension from curved surfaces: The extension estimate establishes the stated p>3 branch in ambient dimension three for spheres or positively curved surfaces. The question asks the sharp two-exponent restriction range in all dimensions or for general submanifolds, which is broader in dimensions, surfaces, and/or exponents.

Sphere extension and packet capacity: The extension estimate establishes the stated p>3 branch in ambient dimension three for spheres or positively curved surfaces. The question asks the sharp two-exponent restriction range in all dimensions or for general submanifolds, which is broader in dimensions, surfaces, and/or exponents.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

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

Manuscript

Contents
  1. Sharp Exponents in the Fourier Restriction Conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Sharp Exponents in the Fourier Restriction Conjecture

OpenAI's claim

Extension from curved surfaces: For a compact smooth surface in $\mathbb R^3$ with positive curvature, the paper claims diagonal Fourier extension boundedness for every exponent $p>3$; it includes smooth surface boundaries and draws a two-dimensional Schrödinger smoothing consequence from the paraboloid case.

Sphere extension and packet capacity: For bounded data on the two-sphere, the announced extension bound has every output exponent larger than three; the threshold three is also asserted to be the sharp endpoint of this open range.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $\Sigma\subset\mathbb R^3$ be a compact smooth surface with positive curvature, allowing smooth boundary, and use surface area $\sigma$. For $E_\Sigma f(x)=\int_\Sigma e^{2\pi ix\cdot\xi}f(\xi)\,d\sigma(\xi)$, the claim is \[\|E_\Sigma f\|_{L^p(\mathbb R^3)}\le C_{\Sigma,p}\|f\|_{L^p(\Sigma,\sigma)}\quad(3<p<\infty),\] for all complex-valued $f\in L^p(\Sigma)$. It concerns this diagonal range in three dimensions, with a surface-dependent constant.
Mathematical claim 2.
For surface area $\sigma$ on $S^2$ and $Eg(x)=\int_{S^2}e^{2\pi ix\cdot\omega}g(\omega)\,d\sigma(\omega)$, every bounded measurable complex $g$ satisfies \[\|Eg\|_{L^p(\mathbb R^3)}\le C_p\|g\|_{L^\infty(S^2,\sigma)}\quad(p>3).\] The constant depends only on $p$; the endpoint $p=3$ and other ambient dimensions are outside this theorem.

Outline of the argument

Extension from curved surfaces: Graph patches are extended to phases compatible with an elliptic-capacity propagation theorem. That theorem is combined with weighted Kakeya control to sum packet coefficients by their capacity. A joint estimate for widely separated balls removes the local spatial loss, and chart assembly transfers the estimate to the original surface.

Sphere extension and packet capacity: Joint tests of position and velocity use ellipse weights that penalize eccentricity. A geometric propagation estimate controls how this capacity evolves along lines, using overlap savings from differently directed tests. Exact packet analysis and synthesis retain the oscillatory information needed to convert the positive capacity estimate into Fourier extension bounds.

Formal verification

Extension from curved surfaces: The pinned catalogue lists no Lean formalization for this result.

Sphere extension and packet capacity: 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