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Formula Audit: 0.10 Fluid Dynamics and Chaos

Scope

This registry covers the current primary benchmark gate and the formulas that directly support it: the embedded simplified Navier-Stokes comparator, the UET master-equation update, the UET 2D fluid mobility bridge, the stress-test stability check, and the speedup metric. Many exploratory fluid scripts exist, but they are not promoted here unless a verifier artifact ties them to data, thresholds, and units.

Formula Registry

formula_id relation code surface variables and units constant_origin proof_status verification_role failure_mode next_hardening_step
FD-NS-DIFFUSION u <- u + dt * nu * Laplacian(u) and same for v Proof_Turbulence_Benchmarks.SimplifiedNSSolver.step u, v dimensionless velocity arrays in embedded benchmark; dt solver time units; nu dimensionless benchmark diffusivity; dx, dy unit-square spacing benchmark_anchor; embedded simplified comparator checked local comparator formula primary speed baseline Comparator is not a full CFD solver; speedup can overstate practical CFD advantage if treated as external validation. Add external CFD validation cases and compare against a documented solver/version.
FD-NS-POISSON-JACOBI p_ij <- 0.25*(p_E+p_W+p_N+p_S) repeated 20 times Proof_Turbulence_Benchmarks.SimplifiedNSSolver.step p dimensionless pressure-like array benchmark_anchor checked local comparator formula primary speed baseline cost driver Fixed 20 Jacobi sweeps are an implementation choice and can bias speed comparison. Record solver tolerance/sweep sensitivity and competitor solver variants.
FD-UET-MASTER-STEP C,I <- UETMasterEquation.step(C, dt, dx, I) Proof_Turbulence_Benchmarks.time_uet_once; docs/core/uet_master_equation.py C, I dimensionless fields; dt solver time units; dx=1/grid_size topic_derived_relation from core UET master equation checked local implementation relation candidate solver under primary benchmark If master-equation internals change, speed/stability result may change without topic docs noticing. Hash core master-equation file in artifact and add formula-level regression tests.
FD-UET-FLUID-LAPLACIAN Laplacian(C) = d2C/dx2 + d2C/dy2 by central differences Engine_UET_2D.compute_laplacian C dimensionless in benchmark; dx, dy grid spacing standard numerical identity identity / checked local implementation engine formula registry, not primary artifact gate Unit-square benchmark may hide physical-unit behavior. Add physical-unit benchmark with declared density, viscosity, Reynolds number, and boundary conditions.
FD-UET-MOBILITY u = -M * grad_x(C), v = -M * grad_y(C) Engine_UET_2D.step u, v velocity-like arrays; M mobility scale; grad(C) per length heuristic_bridge via FLUID_MOBILITY_BRIDGE / mu / rho when physical properties are used heuristic bridge engine diagnostic and future physical benchmark path Mobility bridge can be mistaken for a validated physical constitutive law. Source-lock physical fluids and validate velocity/pressure fields against external cases.
FD-PHYSICAL-KAPPA kappa = min(mu/rho, stability_limit); stability_limit = 0.4/(dt*(1/dx^2+1/dy^2)) Engine_UET_2D._derive_uet_parameters mu Pa s; rho kg m^-3; mu/rho m^2 s^-1; stability_limit numerical diffusion bound source_locked_physics_relation for kinematic viscosity plus numerical stability cap checked local implementation relation future physical-unit gate Stability cap can silently move from physical viscosity to numerical constraint. Artifact should record whether cap is active for each physical run.
FD-STABILITY-GATE stable = all(isfinite(C_stress)) after stress update Proof_Turbulence_Benchmarks.run_benchmarks Boolean gate over dimensionless stress field; stress seed 1e6 benchmark amplitude benchmark_anchor checked local benchmark gate primary stability gate Finite output does not prove boundedness or smoothness for arbitrary initial data. Add norm growth metrics and repeated stress amplitudes; separate theorem work from benchmark work.
FD-SPEEDUP speedup = median(t_NS_trials) / median(t_UET_trials) Proof_Turbulence_Benchmarks.run_benchmarks runtimes in seconds; speedup dimensionless topic_derived_metric metric definition primary PASS/FAIL metric; threshold > 2.0 Environment jitter or comparator choice can shift the pass result. Record hardware/runtime metadata and repeated runs across grid sizes.

Current Artifact Link

  • Primary command: python docs/topics/0.10_Fluid_Dynamics_Chaos/Code/02_Proof/Proof_Turbulence_Benchmarks.py
  • Artifact: Result/artifacts/fluid_benchmark_validation.json
  • Current gate: speedup greater than 2.0 and finite stress-test output.
  • Claim boundary: internal implementation benchmark only.

Current Formula Boundary

  • The benchmark gate can support speed/stability wording for the embedded comparator and configuration.
  • It cannot support external CFD accuracy, arbitrary-turbulence generalization, or Millennium-problem closure.
  • Paper-facing claims require external validation datasets, physical-unit Reynolds-number cases, and theorem-target assumptions separated from implementation benchmarks.