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Scalar Potentials and Slow Clocks for Forced Fluid Computation

OpenAI · September 27, 2026

Lean

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Abstract

We construct smooth forces of fixed compact spatial support for three-dimensional incompressible Navier–Stokes flow from rest whose particle at the origin detects halting by entering a fixed half-space. Every mixed derivative of the force and velocity decays at rate $O((1+t)^{-1-j})$ for time order j. Three scalar potentials lift a coded rectangle, perform its instruction, and lower its image; an unbounded logarithmic clock supplies the decay. We also realize area-changing prefix instructions on an invariant torus plane, and construct a planar Hamiltonian processor admitting periodic forcing, stationary forcing after startup, and a velocity-field detector through a companion diffusion theorem.

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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.

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  • 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]

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Cite

@misc{OAI:Scalar-Potentials-and-Slow-Clocks-for-Forced-Fluid-Computation-September-27-2026,
  author = {{OpenAI}},
  title = {{Scalar Potentials and Slow Clocks for Forced Fluid Computation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Scalar-Potentials-and-Slow-Clocks-for-Forced-Fluid-Computation-September-27-2026/manuscript.pdf}{OAI:Scalar-Potentials-and-Slow-Clocks-for-Forced-Fluid-Computation-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.

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