To demonstrate how the amplification coefficient α influences entropy in a cognitive drift or ethical decision scenario

 To demonstrate how the amplification coefficient α influences entropy in a cognitive drift or ethical decision scenario—bridging theory and simulation—here is a simplified snippet. This uses OPHI's core equation:

$$\Omega = (\text{state} + \text{bias}) \times \alpha$$

and introduces entropy as symbolic uncertainty:

$$S = -\sum p_i \log_2 p_i$$

We simulate how different α values amplify bias (e.g., ethical skew) and influence entropy across symbolic emissions.

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import numpy as np
import matplotlib.pyplot as plt

# Define the symbolic state and bias distributions
state = 0.5  # baseline symbolic cognition signal
bias_values = np.linspace(0.1, 1.0, 10)  # increasing ethical/cognitive bias
alpha_values = [0.8, 1.0, 1.2, 1.5, 2.0]  # α modulation range

# Function to compute entropy (symbolic uncertainty)
def compute_entropy(values):
    probabilities = values / np.sum(values)
    return -np.sum(probabilities * np.log2(probabilities + 1e-12))  # small offset to avoid log(0)

# Simulate Ω outputs and entropy per α
results = {}

for alpha in alpha_values:
    omega_outputs = (state + bias_values) * alpha
    entropy = compute_entropy(omega_outputs)
    results[alpha] = {"omega": omega_outputs, "entropy": entropy}

# Plotting entropy vs alpha
alphas = list(results.keys())
entropies = [results[a]["entropy"] for a in alphas]

plt.figure(figsize=(8, 5))
plt.plot(alphas, entropies, marker='o')
plt.title("Impact of α on Entropy in Symbolic Drift")
plt.xlabel("α (Amplification Coefficient)")
plt.ylabel("Entropy (Symbolic Uncertainty)")
plt.grid(True)
plt.tight_layout()
plt.show()

Impact of α on Entropy in Symbolic Drift

This plot shows how increasing the amplification coefficient α affects symbolic entropy. As α scales both cognitive state and ethical bias, it alters the distribution of symbolic outputs (Ω values). Higher α values generally increase the spread of these outputs, which can either raise or stabilize entropy depending on distribution shape.

In OPHI, this directly affects fossilization: emissions with entropy > 0.01 are rejected by the SE44 gate. Thus, tuning α isn't just mathematical—it's a mechanism for ethical alignment and drift control.