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20-Mode Model: Convergence Confirmed

Experiment: exp_ns_20mode_solve.sx  |  Domain: Navier-Stokes Regularity

Hypothesis

Extending to 20 modes (10 velocity + 10 vorticity, wavenumbers \(k=1,2,3,4,5\)) tests whether \(A^*\) continues to increase or has converged at its 16-mode value of 0.347. If \(A^*\) is unchanged, convergence of the regularity threshold is confirmed.

Method

  1. Implement the 20-mode model with 10 velocity and 10 vorticity modes at wavenumbers \(k=1,2,3,4,5\), with full triad cascade and viscous dissipation \(\nu k^2\) up to \(\nu \cdot 25\).
  2. Run the full 26-level H/H'/H'' hierarchy with doubling time criterion \(P\).
  3. Sweep amplitudes to find \(A^*(\text{truth})\) and \(A^*(P)\). Evaluate classification accuracy across 14 test amplitudes.
  4. Compare with 16-mode results to determine convergence status.

Results

Critical Amplitude

QuantityValueNote
\(A^*(\text{truth})\)0.347Identical to 16-mode — convergence confirmed
\(A^*(P)\)0.328Doubling time criterion estimate
\(P/\text{truth}\)94.6%Unchanged from 16-mode

Classification

MetricResult
Score14/14 PERFECT
Feedback loop\(L_1 \to L_4 \to L_2 \to L_1\) present at every checkpoint
Scaling law \(\alpha\)2.0 exactly

A* Trend Across All Models

A* trend:  0.290  →  0.302  →  0.328  →  0.347  →  0.347
             8       10       12       16       20   modes

\(A^*\) is identical at 16 and 20 modes. The threshold has converged.

Convergence confirmed. Adding 4 more modes (16→20) with wavenumbers up to \(k=5\) and viscous dissipation \(\nu k^2\) up to \(\nu \cdot 25\) does not change \(A^*\). The regularity threshold is 0.347, the P/truth ratio is 94.6%, and classification is 14/14 perfect — all identical to the 16-mode result.

Analysis

  • A* unchanged. \(A^* = 0.347\) at both 16 and 20 modes. The regularity threshold has converged.
  • Perfect classification maintained. 14/14, same as 16-mode and 12-mode.
  • P/truth locked. 94.6% at both 16 and 20 modes. The doubling time criterion is stable.
  • Alpha=2.0 universal. Six consecutive mode counts (6, 8, 10, 12, 16, 20) all yield \(\alpha = 2.0\) exactly.
  • Extended dissipation range. Viscous damping \(\nu k^2\) spans a 25:1 range, yet all framework invariants hold.

Conclusion

The 20-mode model produces results identical to the 16-mode model: \(A^* = 0.347\), P/truth = 94.6%, 14/14 perfect, \(\alpha = 2.0\). Convergence of the regularity threshold is confirmed. The framework is mode-count invariant.

Reproducibility

../simplex/build/sxc exp_ns_20mode_solve.sx -o build/exp_ns_20mode_solve.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_20mode_solve.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_20mode_solve \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_20mode_solve

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