Global classical solutions of the relativistic Vlasov–Maxwell system
From OpenAI Aftermath
Global classical solutions of the relativistic Vlasov–Maxwell system concerns the large-data Cauchy problem for the system that describes a collisionless plasma whose particles move under their self-consistent electromagnetic field: whether smooth initial data in three space and three momentum dimensions always lead to a smooth solution for all time. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math), listed as its result 362, claims a unique global classical solution for the one-species system, with no smallness or symmetry assumption, and the collection includes a Lean formalization of this statement. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Wollman established local existence and uniqueness of classical solutions. DiPerna and Lions developed global weak solutions for large data, and Rein gave a direct formulation for the relativistic one-species system; these weak solutions do not give the regularity and uniqueness of a classical theory.
- Glassey and Strauss identified bounded momentum support as a continuation condition for classical solutions and proved global existence for initially dilute plasmas. Glassey and Schaeffer proved global classical existence in dimensional reductions and for the spherically symmetric electrostatic reduction, and Rein proved persistence under small perturbations of suitable global solutions.
- Later proofs and refinements of the continuation criterion are due to Klainerman and Staffilani, Bouchut, Golse and Pallard, Luk and Strain, Pallard, and Sospedra-Alfonso and Illner. Small-data and small-particle results are due to Schaeffer, Bigorgne, X. Wang, and Wei and Yang; Bigorgne also constructed particular large global solutions from prescribed asymptotic data.
- X. Wang's two papers of July 2026 establish global existence for large cylindrically symmetric data in a different data class; the manuscript states that the two data classes should not be identified.
The original result
- Claim. Theorem 1.1 of Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system states that every smooth admissible datum, namely a smooth nonnegative compactly supported particle density and fields with bounded derivatives of every order and finite energy satisfying both Gauss constraints, generates a unique global classical solution, smooth on every finite time interval, with compact phase-space support there and with the divergence constraints preserved. The manuscript notes that the momentum support bound may grow with the time interval and that a nonzero total charge is allowed.
- Method, as described: the central estimate bounds the signed change of a particle's momentum, with the norm taken after time integration; it implies that the time needed to double the largest momentum is bounded below by a sequence with divergent sum, so momentum support stays bounded on finite intervals and the Glassey–Strauss continuation criterion applies. The estimate uses the retarded representation of the fields, an identity writing the force along a source trajectory as a total derivative plus less singular terms, a bound on the number of direction changes, an improved occupation estimate and a selection argument, closed by a bootstrap.
- Source (openai/math repository, commit
7ef5c0c): Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system, dated 23 September 2026. - Formalization, as described by the repository. The scope note states that the formalized result gives global existence and uniqueness for the three-dimensional one-species system for the stated data, with the solution smooth on every finite time interval, compact particle phase support there, uniqueness among classical solutions and no smallness, symmetry or neutrality assumption. The Comparator statement is
VlasovMaxwell.lean.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | Global smoothness for relativistic Vlasov–Maxwell, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the manuscript's claim of a unique global classical solution of the three-dimensional one-species relativistic Vlasov–Maxwell system for arbitrary smooth compactly supported data, covering phase space and characteristics, why the conservation laws are weaker than the continuation criterion, and the signed impulse and momentum-doubling argument. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit c76f1e4, 14 October 2026) |
Review (fidelity of the formal statement) | Rates this family "full" and records a sandboxed Comparator pass for VlasovMaxwell. Its note lists the formal statement's hypotheses and conclusions and records that the Vlasov, Maxwell and Gauss equations are written out by hand in the formalization. |
MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | cert-machine, openai/math lane K: run 38019899278 | Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept the VlasovMaxwell solution, in 1,366 s. |
Repository. |
| 2026-10-11 | Azmat, Vlasov Maxwell System: How OpenAI Tackled a Decades-Old Plasma Physics Problem, Binary Verse AI | Reading (exposition) | Walks through the signed momentum estimate and the momentum-doubling argument, and states that collisions, several species, boundaries and uniform-in-time bounds fall outside the theorem and that formal machine checking and broad scientific acceptance answer different questions. | Blog article. |
| 2026-10-12 | Y. Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics, arXiv:2610.17090v1, section 8.2.3 | Reading (survey of methods) | Lists the signed momentum-increment bootstrap, the signed force identity along source labels and the direction-change counting and selection as the manuscript's new tools, attributing the time-integrated increment idea to Cheverry and Ibrahim. Its caveats state that OpenAI's published reasoning summary says the cited continuation theorem alone does not give the required estimates while the paper presents them as proved, that Cheverry and Ibrahim are cited in the summary but not in the paper, that one exponent comparison has a small margin, and that uniqueness is asserted only among global classical solutions. | Preprint; the survey states that it did not compile or kernel-check the formalization. |
| 2026-10-13 | M. Colbrook and contributors, AIM — Open Applied Problems (GitHub), problem 073 | Catalogue | The collection records its problem 073, large-data global classical solutions of relativistic Vlasov–Maxwell with no smallness or symmetry assumption, as solved by the manuscript. | The status review is described as an independent AI theorem-to-target statement comparison, not an independent complete proof audit; it records that no local kernel log was inspected. The collection is not affiliated with the American Institute of Mathematics. |
| 2026-10-19 | AI Proved What?, episode 362 Plasmas never blow up?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of the claim that, for one species in three dimensions, every smooth nonnegative compactly supported particle density with finite-energy fields whose derivatives are bounded and which satisfy Gauss's laws gives a unique classical solution that stays smooth on every finite time interval, with no smallness or symmetry assumption. The episode states that long-time behaviour, several species and particle densities without compact support are not claimed, that X. Wang's results for cylindrically symmetric data allow different data so that neither result contains the other, and that according to OpenAI's published summary the problem statement was given to the model in its prompt. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 49 pages, dated September 23, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: the full theorem. The page states that the site did not run the formalization of about 90,000 lines, that the formalization appears to prove its own continuation step rather than import it and that its uniqueness statement is slightly narrower, and that it found no specialist comment and no reported error. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
