theory: expose intermittent feedback tracking lag

1 files changed, 26 insertions, 0 deletions

diff --git a/experiments/verify_theory.py b/experiments/verify_theory.py
index ba07487..8054b0d 100644
--- a/experiments/verify_theory.py
+++ b/experiments/verify_theory.py
@@ -196,6 +196,31 @@ def check_innovation_identification():
     assert abs(retained_fraction - (1.0 - predictable_fraction)) < 0.01
 
 
+def check_intermittent_feedback_tracking():
+    eta_m = 0.1
+    cadence = 16
+    per_step_change = 0.002
+    block_change = cadence * per_step_change
+
+    # Constant collinear forward-weight motion attains the norm bound.
+    error = 0.0
+    for _ in range(1000):
+        error = (1.0 - eta_m) * error - block_change
+    steady_state = -block_change / eta_m
+    assert abs(error - steady_state) < 1e-14
+
+    # Once forward motion stops, endpoint tracking can look excellent despite
+    # the large error accumulated during the task-active trajectory.
+    error_before_quiet_tail = abs(error)
+    for _ in range(80):
+        error *= 1.0 - eta_m
+    expected_endpoint = error_before_quiet_tail * (1.0 - eta_m) ** 80
+    print("\nINTERMITTENT FEEDBACK TRACKING")
+    print(f"steady_error={abs(steady_state):.6f} bound={cadence * per_step_change / eta_m:.6f} "
+          f"endpoint_after_quiet_tail={abs(error):.9e}")
+    assert abs(abs(error) - expected_endpoint) < 1e-15
+
+
 def main():
     check_simultaneous_variance()
     check_sigma_bias()
@@ -203,6 +228,7 @@ def main():
     check_conditional_projection()
     check_predictor_timescale()
     check_innovation_identification()
+    check_intermittent_feedback_tracking()
     print("\nALL THEORY CHECKS PASSED")