oai-math-revised beta

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

Incompressible Box Transport and Finite Computation

OpenAI · October 6, 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

We realize finite positive diagonal affine maps of determinant one by effective smooth incompressible flows on neighborhoods of entire closed rational solid boxes. The source and target families are each disjoint, but may overlap each other. The construction uses localized curls, evacuation to storage and obstacle detours. A balanced three-stack recorder then assigns every machine and finite input a smooth Navier–Stokes force with one compact spatial support, periodic after a loading interval, at any fixed positive computable viscosity. The fluid starts at rest, and one fixed particle enters one fixed open cube exactly when the machine halts. Further constructions give fixed torus charts, periodicity from time zero, alternative history guards and bounded or slab observers. Onto slow clocks yield separate decaying forces. Each construction includes an all-time observation proof, effective derivative bounds and a stated pressure comparison class.

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

Formalization

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:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026,
  author = {{OpenAI}},
  title = {{Incompressible Box Transport and Finite Computation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/manuscript.pdf}{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026}},
  year = {2026}
}

Versions

This version supplies effective torus-projection and common-clock decay estimates, including the projected-force and pressure bounds; see the previous version.

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: