From 7455c126ab8a5cbfd3bb1ebe8b029a04f858e824 Mon Sep 17 00:00:00 2001
From: YurenHao0426 <Blackhao0426@gmail.com>
Date: Wed, 22 Jul 2026 15:18:17 -0500
Subject: theory: formalize reciprocal KP tracking dynamics

---
 experiments/verify_theory.py | 37 +++++++++++++++++++++++++++++++++++++
 1 file changed, 37 insertions(+)

(limited to 'experiments/verify_theory.py')

diff --git a/experiments/verify_theory.py b/experiments/verify_theory.py
index 8054b0d..cc9b6c3 100644
--- a/experiments/verify_theory.py
+++ b/experiments/verify_theory.py
@@ -221,6 +221,42 @@ def check_intermittent_feedback_tracking():
     assert abs(abs(error) - expected_endpoint) < 1e-15
 
 
+def check_kolen_pollack_difference_dynamics():
+    rng = np.random.default_rng(601)
+    eta = 0.07
+    momentum = 0.9
+    decay = 1e-3
+    forward = rng.normal(size=(7, 5))
+    reciprocal = rng.normal(size=(7, 5))
+    forward_momentum = rng.normal(size=(7, 5))
+    reciprocal_momentum = rng.normal(size=(7, 5))
+    maximum_error = 0.0
+    for _ in range(40):
+        # Both paths independently obtain the same local activity product.
+        correlation = rng.normal(size=(7, 5))
+        difference = reciprocal - forward
+        momentum_difference = reciprocal_momentum - forward_momentum
+        predicted_momentum_difference = (
+            momentum * momentum_difference - decay * difference)
+        predicted_difference = difference + eta * predicted_momentum_difference
+
+        forward_momentum = (momentum * forward_momentum
+                            + correlation - decay * forward)
+        reciprocal_momentum = (momentum * reciprocal_momentum
+                               + correlation - decay * reciprocal)
+        forward = forward + eta * forward_momentum
+        reciprocal = reciprocal + eta * reciprocal_momentum
+        maximum_error = max(
+            maximum_error,
+            float(np.abs((reciprocal_momentum - forward_momentum)
+                         - predicted_momentum_difference).max()),
+            float(np.abs((reciprocal - forward)
+                         - predicted_difference).max()))
+    print("\nKOLEN-POLLACK DIFFERENCE DYNAMICS")
+    print(f"task-correlation cancellation error={maximum_error:.3e}")
+    assert maximum_error < 3e-15
+
+
 def main():
     check_simultaneous_variance()
     check_sigma_bias()
@@ -229,6 +265,7 @@ def main():
     check_predictor_timescale()
     check_innovation_identification()
     check_intermittent_feedback_tracking()
+    check_kolen_pollack_difference_dynamics()
     print("\nALL THEORY CHECKS PASSED")
 
 
-- 
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