The Well harness
PolymathicAI's The Well is a 15 TB collection of physics simulations across fluid dynamics, plasma, astrophysics, and reaction–diffusion systems. BPR's scaling predictions are checked against it automatically.
Read this first. Most of these checks test spectral exponents that standard physics also predicts (Kolmogorov −5/3, enstrophy −3, and so on). They show that BPR's scaling relations are consistent with known physics, not that BPR is right about particles. They are labelled consistent for that reason. Two are BPR-specific and labelled conjectural.
| ID | Dataset | Prediction | Predicted | Observed | σ | Status |
|---|---|---|---|---|---|---|
| PW2.1 | acoustic_scattering_inclusions | Mode entropy H/Hmax > 0.5 | > 0.50 | 0.696 | pass | consistent |
| PW3.1 | rayleigh_benard | Nu ~ Raβ (Class C) | 0.307 | 0.287 | 0.82 | consistent |
| PW4.1 | active_matter | Kuramoto transition direction | — | correct | pass | consistent |
| PW5.1 | MHD_64 | E(k) ~ k−5/3 | −1.667 | −2.180 | 0.45 | consistent |
| PW6.1 | brusselator | Turing wavelength λT = 2π/kc | 0.393 | 0.333 | 1.51 | consistent |
| PW7.1 | turbulent_radiative_layer_2D | 2D enstrophy cascade α = −3 | −3.000 | −3.006 | 0.01 | consistent |
| PW8.1 | turbulence_gravity_cooling | Stratified Fr < 1 cascade | −3.0 | −3.15 | 0.30 | consistent |
| PW9.1 | rayleigh_taylor_instability | Stratified buoyancy α = −3 | −3.000 | −3.661 | 1.32 | consistent |
| PW10.1 | supernova_explosion_64 | Post-shock Kolmogorov | −1.667 | −2.1 | 2.11 | consistent |
| PW11.1 | turbulent_radiative_layer_3D | Cooling-steepened spectrum | −3.5 | −3.96 | 0.91 | consistent |
| PW12.1 | acoustic_scattering_maze | Mode entropy > 0.5 | > 0.50 | 0.870 | pass | consistent |
| PW13.1 | shear_flow | 2D enstrophy k−3 | −3.0 | −3.9 | 1.85 | consistent |
| PW14.1 | planetswe | Geostrophic k−3 | −3.0 | −3.5 | 1.05 | consistent |
| PW15.1 | helmholtz_staircase | Mode spacing CV < 0.3 | < 0.30 | 0.100 | 0.50 | consistent |
| PW16.1 | viscoelastic_instability | α = −(3 + Wi1/3) | −6.68 | −6.82 | 0.27 | conjectural BPR-specific |
| PW17.1 | euler_multi_quadrants_openBC | Burgers shock α = −2 | −2.000 | −2.973 | 1.95 | conjectural |
| PW18.1 | convective_envelope_rsg | Compressible stellar α = −4 | −4.000 | −4.023 | 0.05 | consistent |
| PW19.1 | post_neutron_star_merger | Compressible MHD α = −3 | −3.000 | −3.085 | 0.17 | consistent |
| PW1.1 | gray_scott | Exempt. Gray–Scott spots are self-replicating structures, not Turing instabilities: det J = F(F+k) > 0 at the trivial state, so the bifurcation condition is never met. | exempt | |||
18 validators pass, 1 exempt. Several observed exponents differ from the predicted ones by well over 10 percent while the quoted σ is small; the σ reflects the fit uncertainty of the observed exponent, not a claim of precision agreement. Source: experiments/the_well_harness.py.
What was learned from it
- Turing wavelength corrected. The original validator targeted Gray–Scott, which is not Turing-unstable. Switching to the Brusselator gives a genuine test, at 1.5σ.
- Class B vs Class C. The wrong universality class was initially applied to Rayleigh–Bénard convection. Corrected.
- Radiative cooling steepens spectra from k−5/3 towards k−3 to k−4, consistent with the simulations.
- One new formula, α = −(3 + Wi1/3) for viscoelastic turbulence, agrees at 0.27σ and is not in the existing literature. It is a conjecture.