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Microscope: Finding the 8.5% Signal

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

Hypothesis

The 8.5% gap remaining at \(T=50{,}000\) in the 6-mode model (between H' at 91.5% and perfect closure) is visible in the raw trajectories when viewed as RATIOS between nearby amplitudes, not as individual time series. Comparing \(A=1.08\) (safe) and \(A=1.15\) (blow-up) side-by-side should reveal a diverging signal from the earliest time steps.

Method

  1. Run two trajectories simultaneously: \(A=1.08\) and \(A=1.15\) with identical initial conditions except for amplitude.
  2. Compute the production ratio (enstrophy production rate ratio) at every time step.
  3. Track the enstrophy ratio \(E_b/E_a\) at key checkpoints.
  4. Identify where and how the divergence manifests.

Results

Production Ratio Divergence

StepProduction Ratio DiffNote
03.1Diverges immediately
5,000wideningMonotonic increase begins
50,000largeClear separation

The production ratio diverges from step 0 with a difference of 3.1. This is not a gradual onset — the signal is present from the very first time step.

Enstrophy Ratio Evolution

Checkpoint\(E_b / E_a\)
Early1.134
Mid1.598
Late24.856

The enstrophy ratio widens monotonically from step 5,000 onward, growing from 1.134 to 24.856 — a 22× amplification.

Interpretation. The signal is in the RATIO, not in individual trajectories. Both trajectories look qualitatively similar for thousands of steps. But their ratio diverges from step 0. This is why the remaining 8.5% at \(T=50{,}000\) requires confidence tracking (H'') rather than additional levels — the information is already present, it just needs more observation time to resolve.

Analysis

  • Immediate divergence. The production ratio difference of 3.1 at step 0 means the blow-up vs safe distinction is encoded in the initial dynamics, not a late-stage phenomenon.
  • Monotonic widening. The enstrophy ratio never reverses — once the trajectories start separating, they only separate further. This is consistent with Theorem 36 (dynamic sensitivity).
  • 22× amplification. The late-stage ratio of 24.856 vs early 1.134 shows the exponential nature of the divergence.
  • Halting problem connection. The 8.5% gap is not missing information but insufficient observation time. H'' resolves it by tracking whether the ratio is still accelerating or has stabilised.

Conclusion

The microscope view confirms that the 8.5% gap at \(T=50{,}000\) is a time-resolution issue, not a measurement gap. The production ratio diverges from step 0, and the enstrophy ratio amplifies 22× over the trajectory. The signal for the H'' confidence tracker is already present — it just needs time to become unambiguous.

Reproducibility

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

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

./build/exp_ns_microscope

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