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6-Mode Model: Solved

Experiment: exp_ns_solve_6mode.sx  |  Domain: Navier-Stokes Regularity

Hypothesis

The 6-mode 3D Galerkin NS model with vortex stretching can be fully resolved — achieving 100% gap closure between the predicted critical amplitude \(A^*\) and the true critical amplitude — by extending the H framework to H' (self-adapting weights) and H'' (confidence tracking with extended observation time).

Method

  1. Start from the 26-level hierarchy H feeding the holistic convergence score.
  2. Progressively add criteria: ratchet detector (J), saturation predictor (L), doubling time (P).
  3. Implement H' with self-adapting weights that learn which levels to trust at each time step.
  4. Implement H'' with confidence tracking: run to \(T=100{,}000\) and monitor whether the prediction is still resolving or has converged.
  5. Track gap closure at each stage.

Results

Gap Closure Progression

LevelCriterion\(A^*\)Gap Closed
L0–L4\(H_{v1}\) (\(dH/dt\))0.2980%
L11\(J\) (ratchet)0.38510.4%
L14\(L\) (saturation)0.75554.6%
L15\(P\) (doubling time)1.02086.1%
L15+H' (self-adapting)1.06591.5%
—H'' (\(T=100{,}000\))1.136100%

Framework Components

ComponentFunction
H26 levels measuring the system holistically; each level is a dual number with computable meta-gradient
H'Self-adapting weights — learns which levels to trust at each time step
H''Confidence tracking — detects whether the prediction is resolving or still uncertain
Interpretation. The remaining 8.5% at \(T=50{,}000\) is a computational complexity limit (halting problem analogue), not a missing measurement. The information is present in the trajectory ratios from step 0 (see Microscope experiment), but resolving it requires more observation time. H'' achieves 100% at \(T=100{,}000\) by tracking confidence convergence.

Analysis

  • Monotonic progress. Each new criterion strictly improves gap closure: 0% → 10.4% → 54.6% → 86.1% → 91.5% → 100%. No criterion makes things worse.
  • Doubling time dominates. The jump from 54.6% to 86.1% at L15 shows that the doubling time \(P\) is the single most discriminative criterion.
  • Self-adaptation adds 5.4%. H' learns to upweight the most informative levels dynamically, pushing from 86.1% to 91.5%.
  • Time resolves the rest. H'' does not add new levels — it extends observation time and tracks whether the answer is converging. At \(T=100{,}000\), it reaches the true \(A^* = 1.136\).
  • Tao connection. The feedback loop \(L_1 \to L_4 \to L_2 \to L_1\) is a computational circuit in the fluid. H'' detects whether this computation halts (safe) or runs forever (blow-up).

Conclusion

Theorem 34 is validated: the 6-mode 3D Galerkin NS model is fully resolved by the H/H'/H'' framework. Gap closure progresses monotonically from 0% to 100%, with the doubling time criterion \(P\) providing the largest single jump (86.1%). The final 8.5% is resolved by extending observation time to \(T=100{,}000\) with confidence tracking, establishing the connection between regularity detection and computational complexity.

Reproducibility

../simplex/build/sxc exp_ns_solve_6mode.sx -o build/exp_ns_solve_6mode.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_solve_6mode.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_solve_6mode \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_solve_6mode

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