oai-math-revised beta

Browse › Partial differential equations › 376 Universal computation in forced Navier–Stokes flows

Prefix Instructions and Incompressible Flows

OpenAI · September 27, 2026

Lean

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

Finite prefix instructions may change area and erase information. We give explicit history processors and smooth incompressible motions that retain that information and realize every instruction on its full domain. A first application assigns each machine and finite input a smooth mean-zero force on the flat unit three-torus, at any fixed positive computable viscosity. The solution starts from rest; a fixed particle enters a fixed open strip exactly when the machine halts. The force repeats with period one after an initial loading interval. We then prove alternative realizations using full boxes, normal compensation, invariant planes, and a spatial clock. The constructions specify their initialization, comparison class, derivative bounds, and continuous-time observation. Exact formulas and effective cutoffs provide finite descriptions of the fields and all their derivatives.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.

Comments and discussion

Digests and expositions

News and blogs

  • Riley Scott Jacob (@RilScoJa), post on X (8 Oct 2026). Noting the released proofs give "no worked example", reports building one with Claude Opus 5.5: a smooth 3D flow from rest carrying a fluid particle through a shift-and-add multiplier computing 13 × 11 = 143. [family-level] [AI-generated per source]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Prefix-Instructions-and-Incompressible-Flows-September-27-2026,
  author = {{OpenAI}},
  title = {{Prefix Instructions and Incompressible Flows}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf}{OAI:Prefix-Instructions-and-Incompressible-Flows-September-27-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/376.md. The Lean statements have not been checked by people against the paper's claims.

Family 376: Universal computation in forced Navier–Stokes flows

Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and input into a finite program for the force, so a designated particle reaches a fixed region exactly when the machine halts. The viscosity is fixed, positive and computable.

Other papers in this family: