From 661496c172a53d09c59951c642f1a468abb7f07e Mon Sep 17 00:00:00 2001
From: YurenHao0426 <Blackhao0426@gmail.com>
Date: Wed, 22 Jul 2026 19:55:05 -0500
Subject: mechanism: separate BCI role from performance velocity

---
 experiments/verify_theory.py | 41 +++++++++++++++++++++++++++++++++++++++++
 1 file changed, 41 insertions(+)

(limited to 'experiments/verify_theory.py')

diff --git a/experiments/verify_theory.py b/experiments/verify_theory.py
index 21a7c06..c00a091 100644
--- a/experiments/verify_theory.py
+++ b/experiments/verify_theory.py
@@ -348,6 +348,46 @@ def check_kolen_pollack_difference_dynamics():
     assert maximum_error < 3e-15
 
 
+def check_bci_error_derivative_structure():
+    """Separate causal role from temporal performance innovation."""
+    rng = np.random.default_rng(641)
+    role = np.concatenate((np.full(5, 0.2), np.full(5, -0.2),
+                           np.zeros(30)))
+    examples = 200_000
+    xi = 2.0 * rng.integers(0, 2, size=(examples, role.size)) - 1.0
+    cursor_direction = xi @ role
+    role_estimate = (cursor_direction[:, None] * xi).mean(axis=0)
+    role_error = float(np.abs(role_estimate - role).max())
+    role_cosine = float(role_estimate @ role / (
+        np.linalg.norm(role_estimate) * np.linalg.norm(role)))
+
+    # Positive target error throughout the BCI trial. Improving events have
+    # smaller current error; worsening events have larger current error.
+    previous = np.asarray((0.8, 0.7, 0.6, 0.5, 0.4, 0.3))
+    current = np.asarray((0.6, 0.5, 0.4, 0.7, 0.6, 0.5))
+    improvement = previous - current
+    improving = improvement > 0
+    worsening = improvement < 0
+
+    def sign_index(modulator):
+        residual = modulator[:, None] * role[None, :]
+        difference = residual[:, :5].mean(1) - residual[:, 5:10].mean(1)
+        return 0.5 * (difference[improving].mean()
+                      - difference[worsening].mean())
+
+    instantaneous_index = float(sign_index(current))
+    temporal_difference_index = float(sign_index(improvement))
+    print("\nBCI ERROR-DERIVATIVE STRUCTURE")
+    print(f"role_estimator_cosine={role_cosine:.9f} "
+          f"max_error={role_error:.3e} "
+          f"instantaneous_index={instantaneous_index:+.6f} "
+          f"td_index={temporal_difference_index:+.6f}")
+    assert role_cosine > 0.999
+    assert role_error < 0.005
+    assert instantaneous_index < 0.0
+    assert temporal_difference_index > 0.0
+
+
 def main():
     check_simultaneous_variance()
     check_sigma_bias()
@@ -359,6 +399,7 @@ def main():
     check_dynamic_neutral_projection()
     check_intermittent_feedback_tracking()
     check_kolen_pollack_difference_dynamics()
+    check_bci_error_derivative_structure()
     print("\nALL THEORY CHECKS PASSED")
 
 
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