Navier-Stokes: the published solution, then an exact flow: for mathematicians
Learn what OpenAI solved, inspect the fluid equations in the Workbench, and explore computation and checked reasoning in Osprey.
For mathematicians. Choose the other audience version or browse the complete series.
Lean is the development oracle for compatible proofs; Osprey checks them independently. Known unsound reference behaviour must be refused, and changing a discipline requires fresh proof validation.
What this demonstrates
On 8 September 2026, OpenAI published a solution with a Lean formalization: smooth forcing produces a finite-time singularity from initially smooth fluid at rest while energy remains finite. The films explain that result before exploring Osprey. The periodic two-dimensional Taylor-Green flow is a separate numerical example with measured residuals; the Workbench exercise checks a propositional implication under explicit premises. Separately, Osprey checks selected computability, work-counter and number-theory exports from openai/math, while the complete headline theorem remains to be ported. The recorded computation and residual checker are production assets in the separate TikTok repository; the source link below opens the Workbench lesson.
Read the guide and try it yourself. Osprey is actively evolving; the guide distinguishes the recorded demonstration from current corpus progress.
Actual source
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Use the Workbench · Current Mathlib port evidence · All twelve films
Transcript
- Navier-Stokes equations model fluid motion. In September twenty twenty six, OpenAI published an analytical solution of the three-dimensional existence and smoothness problem, formalized in Lean. Smooth forcing drives an initially resting fluid to a finite-time singularity while energy stays finite. We will explain that result, then inspect the equations in Osprey, evaluate a separate two-dimensional solution, and construct a checked logical argument.
- In the workbench lesson, u is velocity, p is pressure, nu is viscosity, and f is an external force. The first equation balances acceleration against pressure, viscosity, and forcing. Zero divergence expresses incompressibility. Our computed example uses unit density, a periodic domain, and zero external force.
- Osprey evaluates separated spatial profiles and exponential time factors. Python reconstructs the field from that output and measures finite-difference residuals. Both momentum equations and divergence are checked. Using a four-times-finer spatial grid reduces the momentum residual by about fifteen times, consistent with second-order spatial convergence.
- The velocity amplitude decays exponentially at twice the viscosity parameter, and kinetic energy decays at four times that parameter. The changing energy readout follows this prediction. These numerical checks validate the evaluation of this particular exact periodic solution; they are not a proof term checked by Osprey’s kernel.
- The published proof is a much larger object than this lesson checks. The lesson supplies a propositional exercise: assume smooth or blowup, and assume not blowup. Derive smooth by cases, using contradiction for the excluded branch. Osprey checks that argument through its kernel. These propositions are placeholders for claims; no definition of fluid regularity is imported by naming them.
- Osprey checks selected supporting lemmas from pinned OpenAI sources; the complete Navier-Stokes theorem remains to be ported. In the lesson, inspect the hypotheses and changing goal, then run the remaining-outcome exercise and examine the kernel's verdict.