Singularity Observatory
Explore the concentrating geometry and an executable outer-flow component from OpenAI’s Navier–Stokes construction. See why increasing velocity and shrinking length scales challenge numerical measurement.
Core geometry
Boundary X = 1, |η| ≤ 0.8, using q = τ/(1−η²). This selected region illustrates the coordinate map; it is not a certified core boundary of the completed construction. Coordinates and unit-normalized coefficients use ν = 1.
Growth and concentration
Power laws with coefficients set to one; these are neither sampled velocity norms nor a total-energy computation. Both axes are logarithmic. Moving the time control selects the vertical marker.
Numerical lab: where precision defeats refinement
The heat exterior satisfies an azimuthal diffusion equation. Here its time derivative comes from autodiff, while radial derivatives use a separate three-point finite-difference stencil. Refinement first reduces spatial error; roundoff can then dominate, especially in FP32.
| Stencil Δr | FP32 |R| | FP64 |R| |
|---|
This experiment checks one exterior component at fixed positive radius. It does not evolve the full PDE toward blowup, and finite-difference convergence does not verify the theorem. The exterior alone is singular at the axis at every time.
What the flagship is working toward
A reproducible answer to: How close to the constructed singular time can a numerical solver follow the full known solution, and what causes it to fail? A faithful inner profile, residual corrections, compact localization, and error-controlled truncation must be implemented before this app can answer that question for the complete construction.
Available now: differentiable simulation, exact smooth benchmarks, similarity coordinates, viscosity rescaling, and heat-exterior quadrature. Not Yet Tested: complete blowup reproduction, independent theorem verification, solver comparisons near the singularity, or Blender integration.
Paper: equations (3.2), (4.1), Lemma A.6 · Pinned Lean source