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Level 10: Trajectory Acceleration
Experiment: exp_ns_level10.sx | Domain: Navier-Stokes Regularity | Theorem: 24
Overview
The viscosity spectrum has a resonance at period ~10,000. The trajectory through the spectrum — how fast \(\nu\) changes — determines whether you hit or skip the resonance. Level 10 makes the trajectory acceleration \(d^2\nu / dt^2\) a dual number, giving \(\partial(\text{enstrophy response}) / \partial(\text{acceleration})\) and \(\partial(\text{blow-up time}) / \partial(\text{acceleration})\). This is a learnable control law for navigating viscosity space.
Key Results
| Measurement | Value | Significance |
|---|
| Meta-gradient sign change | At resonance period | \(\partial T / \partial(\text{accel})\) switches from positive to negative |
| Optimal acceleration | 0.1 | Maximises blow-up delay (best control) |
| Positive acceleration effect | Sweeps past resonance | Fast sweep skips dangerous zone |
| Negative acceleration effect | Lingers near resonance | Slow-down amplifies enstrophy peak |
Chirp Sweep
A chirp sweep (linearly increasing frequency) through the viscosity spectrum reveals the resonance structure. As the sweep frequency passes through the natural frequency at period ~10,000, the enstrophy response peaks sharply and then decays. The meta-gradient changes sign precisely at this crossing:
| Sweep Phase | Meta-Gradient | Action |
|---|
| Below resonance (slow sweep) | Positive | Accelerate to approach resonance |
| At resonance | Zero (sign change) | Transition point |
| Above resonance (fast sweep) | Negative | Decelerate — resonance is behind |
Interpretation. The meta-gradient sign change at resonance is the key result. It tells the dual agent: "accelerate to skip past dangerous zones, decelerate in safe zones." The optimal acceleration of 0.1 represents a gentle push that navigates the resonance without dwelling in it. This completes the 10-level differentiable hierarchy: from system state (L0) to trajectory control (L10).
The Complete Hierarchy
| Levels | What They Measure |
|---|
| L0–L3 | System state and convergence |
| L4–L7 | Stability margin and perturbation probing |
| L8 | Simulation parameters as dual numbers |
| L9 | Physics coupling as dual numbers |
| L10 | Trajectory through parameter space |
Reproducibility
../simplex/build/sxc exp_ns_level10.sx -o build/exp_ns_level10.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_level10.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_ns_level10 \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_ns_level10
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