sync: update from blackroad-operator 2026-03-14

Synced from BlackRoad-OS-Inc/blackroad-operator/orgs/personal/lucidia
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RoadChain-Full: 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

codex/mirror/mirror_mechanics.py

@@ -0,0 +1,113 @@
+"""
+mirror_mechanics.py
+
+This module implements the mirror operator \u03a8' and breath operator \u2102
+for the harmonic oscillator. It provides a basic demonstration of
+oscillator dynamics and how positive and negative frequency components
+are defined.
+
+Functions:
+    mirror_split(signal) -> (pos, neg)
+    breath_step(q, p, omega=1.0, dt=0.01) -> (q_new, p_new)
+    run_oscillator(steps=1000, dt=0.01, omega=1.0) -> (qs, ps)
+
+Example:
+    if __name__ == "__main__":
+        qs, ps = run_oscillator()
+        pos, neg = mirror_split(qs)
+"""
+import numpy as np
+try:
+    from scipy.signal import hilbert
+except ImportError:
+    hilbert = None
+
+def mirror_split(signal: np.ndarray):
+    """
+    Split a real-valued signal into its positive and negative frequency components.
+
+    Parameters
+    ----------
+    signal : np.ndarray
+        Real-valued time series.
+
+    Returns
+    -------
+    pos : np.ndarray
+        The positive frequency component (analytic signal divided by 2).
+    neg : np.ndarray
+        The negative frequency component.
+    """
+    if hilbert is None:
+        raise ImportError(
+            "scipy is required for mirror_split; install scipy to use this function"
+        )
+    analytic = hilbert(signal)
+    pos = analytic / 2.0
+    neg = np.conj(analytic) - pos
+    return pos, neg
+
+def breath_step(q: float, p: float, omega: float = 1.0, dt: float = 0.01):
+    """
+    Perform a single leap-frog (symplectic) update for a harmonic oscillator.
+
+    Parameters
+    ----------
+    q : float
+        Position.
+    p : float
+        Momentum.
+    omega : float, optional
+        Oscillator frequency (default 1.0).
+    dt : float, optional
+        Time step (default 0.01).
+
+    Returns
+    -------
+    q_new : float
+        Updated position.
+    p_new : float
+        Updated momentum.
+    """
+    p_half = p - 0.5 * dt * (omega ** 2) * q
+    q_new = q + dt * p_half
+    p_new = p_half - 0.5 * dt * (omega ** 2) * q_new
+    return q_new, p_new
+
+def run_oscillator(steps: int = 1000, dt: float = 0.01, omega: float = 1.0):
+    """
+    Run a harmonic oscillator using the breath operator.
+
+    Parameters
+    ----------
+    steps : int, optional
+        Number of time steps (default 1000).
+    dt : float, optional
+        Time step (default 0.01).
+    omega : float, optional
+        Oscillator frequency (default 1.0).
+
+    Returns
+    -------
+    qs : np.ndarray
+        Array of positions over time.
+    ps : np.ndarray
+        Array of momenta over time.
+    """
+    q, p = 1.0, 0.0
+    qs, ps = [], []
+    for _ in range(steps):
+        qs.append(q)
+        ps.append(p)
+        q, p = breath_step(q, p, omega, dt)
+    return np.array(qs), np.array(ps)
+
+if __name__ == "__main__":
+    # Simple demonstration: simulate and split into mirror components
+    qs, ps = run_oscillator(steps=1024, dt=0.01, omega=1.0)
+    if hilbert is not None:
+        pos, neg = mirror_split(qs)
+        print(f"First few positive components: {pos[:5]}")
+        print(f"First few negative components: {neg[:5]}")
+    else:
+        print("Scipy not installed; cannot compute mirror components.")