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Math-2838-A1 · revision r1 · on Math-2838

OpenAI claim summary for Math-2838

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.

Stable defocusing Schrödinger blowup: The equation is defocusing on $\mathbb T^{12}$, with a selected odd power and a high but finite Sobolev index. The claimed open set has a fixed spatial point whose rescaled amplitude tends to a positive limit, so the conclusion is classical finite-time blowup of the required type.

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. Stable amplitude blowup for a supercritical defocusing torus 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

Stable amplitude blowup for a supercritical defocusing torus equation

OpenAI's claim

Stable defocusing Schrödinger blowup: The paper claims finite-time self-similar blowup on the twelve-dimensional torus for suitable large odd powers of a defocusing nonlinear Schrödinger equation. Blowup occurs on an open high-Sobolev set and has positive probability for sufficiently regular Gaussian Fourier data.

Mathematical statement of the claimed result

Mathematical claim 1.
For the defocusing equation $i\partial_tu+\Delta u=|u|^{p-1}u$ on $\mathbb T^{12}=(\mathbb R/2\pi\mathbb Z)^{12}$, the claim gives some sufficiently large odd $p\ge3$, fixed integer $k>8$ and nonempty open $\mathcal U\subset H^k$ such that every $u_0\in\mathcal U$ has a unique local solution in $C([0,T);H^k)\cap C^1([0,T);H^{k-2})$ with finite $T>0$ and no continuous $H^k$ extension through $T$. More precisely some $x_*,c_0>0$ obey \[\lim_{t\uparrow T}(T-t)^{1/(p-1)}|u(t,x_*)|=c_0.\] The exponent and Sobolev order are chosen by the construction, not quantified over every supercritical power.

Outline of the argument

Stable defocusing Schrödinger blowup: A smooth profile with a pressurized core is matched to a free exterior, and exact finite arithmetic checks the relevant profile and spectral signs. Outgoing spectral control gives linear decay in the rescaled equation. Nonlinear stability on expanding tori produces an open set of blowup data, after which Gaussian support gives the positive-probability consequence.

Formal verification

Family 371 describes Lean coverage for stable blowup for arbitrarily large selected odd powers and the positive-probability Gaussian consequence.

Stable defocusing Schrödinger blowup: The manuscript links DefocusingNLS.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