Almost-sure global well-posedness for high-regularity random data in energy-supercritical defocusing NLS (Deng-Nahmod-Yue Open Problem 3), refuted on the twelve-torus

Consider the defocusing nonlinear Schrodinger equation $i\partial_t u+\Delta u=|u|^{p-1}u$ on the torus $\mathbb T^d$ with odd $p$ , in the energy-supercritical range $s_{cr}=d/2-2/(p-1)>1$ , where the conserved energy does not control the regularity needed for global classical solutions. Deng, Nahmod and Yue (2020) observed that if classical solutions with random initial data of high regularity, such as Gaussian Fourier series $\sum_n g_n\langle n\rangle^{-\alpha}e^{in\cdot x}$ with $\alpha$ large, were almost surely global, then blowup for these equations would be non-generic and unstable, as the known whole-space blowup examples are. Their Open Problem 3 asks: in the energy-supercritical case, does almost-sure global well-posedness hold for random initial data of high regularity?

Result: Disproved(see note)
Status: Variant only
AI contribution: AI-discovered
Method: Argument
Field: Dispersive PDE; nonlinear Schrodinger equation; random data; blowup
Posed by: Yu Deng, Andrea R. Nahmod and Haitian Yue, Random tensors, propagation of randomness, and nonlinear dispersive equations (arXiv:2006.09285v1, 2020), Section 9.2.3, Open Problem 3
Year posed: 2020
Years open: 6y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 33 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims: there are an odd $p$ (which can be taken arbitrarily large), an integer $k>8$ and a nonempty open set $\mathcal U\subset H^k(\mathbb T^{12})$ such that every solution of the defocusing NLS with data in $\mathcal U$ blows up in finite time $T$ with $(T-t)^{1/(p-1)}|u(t,x_*)|\to c_0>0$ , a type-I self-similar singularity with logarithmic phase. Since Gaussian measures charge open sets, for every $\alpha>k+6$ the Gaussian Fourier datum blows up with positive probability, so almost-sure global existence for high-regularity random data fails for this $(d,p)$ . It does NOT give probability-one blowup, a bound uniform in $\alpha$ , an explicit $p$ , or anything for low dimensions such as $d\le 11$ ; the question for other pairs $(d,p)$ stays open.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. The proof is computer assisted: the release ships an exact rational-arithmetic verifier (Python standard library only) for the finite comparisons behind the self-similar profile and its spectral stability; the README says these checks do not verify the analytic arguments linking them to the blowup theorem. The main theorem and its Gaussian corollary also have a Lean comparator challenge in the release.

Verification

No independent mathematician has checked this yet. Theorem 1.1 and Corollary 1.2 were read against Open Problem 3. The Lean challenge lean/ComparatorChallenges/DefocusingNLS.json (solution module OAI.MathematicalPhysics.DefocusingNLS.MainTheorems, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/371.md. Its statement was read here: stable_blowup gives, for every lower bound, an odd $p\ge3$ above it, an integer $k>8$ and a nonempty open set of Fourier-side $H^k(\mathbb T^{12})$ data whose classical solutions blow up at the self-similar rate $(T-t)^{-1/(p-1)}$ at a point and do not extend continuously past $T$ ; gaussian_blowup and gaussian_not_almost_sure_global give positive Gaussian probability of that set for every $\alpha>k+6$ . This is the headline. The Sobolev product is a challenge definition whose summability proof the solution must supply. Not rebuilt here.

Almost-sure global well-posedness for high-regularity random data in energy-supercritical defocusing NLS (Deng-Nahmod-Yue Open Problem 3), refuted on the twelve-torus

What was actually shown

What the AI did

Verification

Sources

Discussion