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10-Mode Model: Framework Confirmed

Experiment: exp_ns_10mode_solve.sx  |  Domain: Navier-Stokes Regularity

Hypothesis

Extending from 8 to 10 modes (5 velocity + 5 vorticity, wavenumbers \(k=1,2,3\) with two cascade stages) will maintain the framework's structural properties. The doubling time criterion \(P\) should continue to improve as the dynamics become richer.

Method

  1. Implement the 10-mode model with 5 velocity and 5 vorticity modes at wavenumbers \(k=1,2,3\), including two cascade stages for energy transfer.
  2. Run the full 26-level H/H'/H'' hierarchy with doubling time criterion \(P\) as the primary discriminator.
  3. Sweep amplitudes to find \(A^*(\text{truth})\) and \(A^*(P)\). Evaluate classification accuracy across 14 test amplitudes.
  4. Check feedback loop, scaling law exponent \(\alpha\), and positive \(A^*\) invariant.

Results

Critical Amplitude

QuantityValueNote
\(A^*(\text{truth})\)0.302Up from 0.290 in 8-mode — more modes stabilise
\(A^*(P)\)0.290Doubling time criterion estimate
\(P/\text{truth}\)96.1%Best yet: 86.1% → 95.5% → 96.1%

Classification

MetricResult
Score13/14
Feedback loop\(L_1 \to L_4 \to L_2 \to L_1\) present at every checkpoint
Scaling law \(\alpha\)2.0 exactly
Framework improvement trend. The P/truth ratio has improved steadily across mode counts: 86.1% (6-mode) → 95.5% (8-mode) → 96.1% (10-mode). The framework becomes MORE accurate as the system grows. \(\alpha = 2.0\) continues to hold exactly, confirming the scaffold bound transfers.

Analysis

  • A* increases. \(A^* = 0.302\) vs 0.290 in 8-mode. More modes means more stability, and the framework captures this.
  • Best P/truth yet. 96.1% accuracy — the criterion is sharpening with richer dynamics.
  • Structural invariance. Feedback loop, scaling law (\(\alpha = 2.0\)), and positive \(A^*\) all transfer unchanged from 6-mode and 8-mode.
  • Two cascade stages. The energy cascade from \(k=1 \to k=2 \to k=3\) introduces more complex nonlinear interactions, yet the framework handles them without modification.

Conclusion

The 10-mode model confirms the framework's mode-count invariance with the best P/truth ratio yet (96.1%). The \(A^*\) increase from 0.290 to 0.302 shows that more modes stabilise the system, and the doubling time criterion tracks this precisely.

Reproducibility

../simplex/build/sxc exp_ns_10mode_solve.sx -o build/exp_ns_10mode_solve.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_10mode_solve.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_10mode_solve \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_10mode_solve

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