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Math-205-A3 · revision r1 · on Math-205

OpenAI claim summary for Math-205

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.

Wave local smoothing: The critical L^3 three-dimensional smoothing theorem and its interpolated corollaries settle the n=3 branch. The question covers every n>=3 (and the cone-decomposition bundle also asks additional mixed-norm and cone-multiplier assertions).

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. Local Smoothing for the Euclidean Wave Equation
  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

Local Smoothing for the Euclidean Wave Equation

OpenAI's claim

Wave local smoothing: OpenAI claims the critical $L^3$ spacetime estimate for the wave equation in three dimensions with any positive Sobolev loss. This is the Euclidean three-dimensional smoothing assertion at that exponent.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $Uf(x,t)=(2\pi)^{-3}\int e^{i(x\cdot\xi+t|\xi|)}\widehat f(\xi)\,d\xi$ and $J^\epsilon=(1-\Delta)^{\epsilon/2}$. The claimed three-dimensional critical local-smoothing estimate is \[\|Uf\|_{L^3(\mathbb R^3\times[1,2])}\le C_\epsilon\|J^\epsilon f\|_{L^3(\mathbb R^3)}\quad(\epsilon>0,\ f\in\mathcal S(\mathbb R^3)).\] Every positive Sobolev loss is allowed, but zero loss is not asserted.

Outline of the argument

Wave local smoothing: Null coordinates put light rays into a quadratic velocity model, and packet synthesis reduces the physical estimate to a cubic mass sum. Entropy records retain changing shear frames as well as direction and time. Planar Furstenberg and cone-envelope inputs constrain those records, giving a frequency-annulus estimate that can be summed with arbitrarily small loss.

Formal verification

Wave local smoothing: 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