Experiment: Timescale Separation

Hypothesis

Composed subsystems operating at different timescales maintain proper separation: fast variables converge to their quasi-static equilibria before slow variables take significant steps. The separation ratio \( \eta_{\text{fast}} / \eta_{\text{slow}} \gg 1 \) must be maintained throughout training.

Method

Setup: Three test configurations.

Parameters:

Two-timescale: fast \( \eta = 0.1 \), slow \( \eta = 0.001 \), ratio = 100
Three-timescale: fast \( \eta = 0.1 \), medium \( \eta = 0.01 \), slow \( \eta = 0.001 \)
Insufficient separation: \( \eta_{\text{fast}} / \eta_{\text{slow}} = 2 \) (deliberate failure case)
Trials: 99 per configuration

Procedure: Run the composed system and measure whether the fast variable has converged (residual below threshold) before each slow variable update. Report the fraction of slow updates that satisfy the separation criterion.

Results

Configuration	Variable	Separation Rate	Status
Two-timescale	w (fast/slow)	99/99 (100%)	Pass
Three-timescale	w (fast)	100%	Pass
Three-timescale	v (slow)	95.9%	Pass
Insufficient separation	w	0.1%	Correctly detected

Analysis

Two-timescale (100%): With a 100x learning rate ratio, every slow update occurs after the fast variable has equilibrated. This is the standard regime assumed by Theorem 1.
Three-timescale (100% / 95.9%): Adding a third timescale introduces more complex dynamics. The fast variable still achieves perfect separation. The slow variable achieves 95.9% — the 4.1% gap occurs during transient phases where the medium variable has not fully equilibrated.
Insufficient separation (0.1%): With only a 2x ratio, the system correctly fails the separation test. This is the expected negative control: insufficient timescale gaps lead to coupled dynamics that violate the quasi-static assumption.

The insufficient-separation test is critical: it confirms the diagnostic is not vacuously true.

Conclusion

Pass — Timescale separation is maintained at 100% for two-timescale and 95.9%+ for three-timescale systems. Insufficient separation is correctly detected at 0.1%. Theorem 1 timescale condition is validated.

Reproducibility

../simplex/build/sxc exp_timescale.sx -o build/exp_timescale.ll
clang -O2 build/exp_timescale.ll ../simplex/runtime/standalone_runtime.c \
  -o build/exp_timescale -lm -lssl -lcrypto -L$(brew --prefix openssl)/lib
./build/exp_timescale