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8-Mode Model: Framework Verified

Experiment: exp_ns_8mode_solve.sx  |  Domain: Navier-Stokes Regularity

Hypothesis

The H/H'/H'' framework developed on the 6-mode 3D Galerkin NS model transfers to the 8-mode model (4 velocity + 4 vorticity modes, wavenumbers \(k=1,2,3\) with forward cascade) without modification. Every structural feature — feedback loop, scaling law, doubling time criterion, positive \(A^*\) — should survive the transition.

Method

  1. Implement the 8-mode model with coupled velocity and vorticity modes and forward energy cascade.
  2. Run the full 26-level 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 16 test amplitudes.
  4. Check for feedback loop \(L_1 \to L_4 \to L_2 \to L_1\) at every checkpoint. Measure scaling law exponent \(\alpha\).
  5. Vary time horizon from \(T=30{,}000\) to \(T=80{,}000\) to test stability of \(P/\text{truth}\) ratio.

Results

Critical Amplitude

QuantityValueNote
\(A^*(\text{truth})\)0.290Down from 1.136 in 6-mode — more fragile system
\(A^*(P)\)0.277Doubling time criterion estimate
\(P/\text{truth}\)95.5%vs 86.1% in 6-mode — IMPROVED

Classification

MetricResult
Score16/16 perfect
Feedback loop\(L_1 \to L_4 \to L_2 \to L_1\) present at every checkpoint
Scaling law \(\alpha\)2.0 exactly

Time Horizon Stability

Time Horizon\(P/\text{truth}\)
\(T = 30{,}000\)93%
\(T = 50{,}000\)94%
\(T = 80{,}000\)95%

The ratio is stable at 93–95% across all time horizons, confirming the framework does not depend on a specific integration length.

Interpretation. The framework is mode-count invariant. The 8-mode system has sharper dynamics (more decisive blow-up, cleaner safe trajectories), making the criterion MORE accurate, not less. The lower \(A^*\) (0.290 vs 1.136) reflects a more fragile system, but the doubling time \(P\) captures this perfectly.

Analysis

  • Perfect classification (16/16). Every amplitude correctly identified as safe or blow-up.
  • Improved accuracy. \(P/\text{truth} = 95.5\%\) in 8-mode vs 86.1% in 6-mode. More modes means sharper dynamics, which the framework exploits.
  • Structural invariance. Feedback loop, scaling law (\(\alpha = 2.0\)), and positive \(A^*\) all transfer unchanged.
  • Time horizon independence. The 93–95% band across \(T=30{,}000\) to \(T=80{,}000\) shows the criterion converges, not drifts.

Conclusion

Theorem 35 is validated: the H/H'/H'' framework transfers to the 8-mode model with improved performance. The doubling time criterion \(P\) achieves 16/16 perfect classification with \(P/\text{truth} = 95.5\%\). The feedback loop, scaling law, and time horizon stability all confirm the framework is mode-count invariant.

Reproducibility

../simplex/build/sxc exp_ns_8mode_solve.sx -o build/exp_ns_8mode_solve.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_8mode_solve.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_8mode_solve \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_8mode_solve

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