"""
Sovereign Shift: θ = 89/2462
The fundamental non-commutativity parameter of the Quantum AP Orchestrator.
"""

import math

THETA_NUM = 89
THETA_DEN = 2462
THETA = THETA_NUM / THETA_DEN

Q = THETA_DEN        # Total agents / lateral displacement period
N_ACTIVE = 1024      # Active agent subset
THRESHOLD = N_ACTIVE / 2.0  # Dream Cycle trigger: 512

OMEGA = complex(math.cos(2 * math.pi * THETA), math.sin(2 * math.pi * THETA))


def gcd(a: int, b: int) -> int:
    while b:
        a, b = b, a % b
    return a


def verify():
    """Verify θ = 89/2462 is in lowest terms and 89 is prime."""
    assert gcd(THETA_NUM, THETA_DEN) == 1, "Not coprime"
    assert THETA_DEN == 2 * 1231, "Factorization mismatch"
    assert all(THETA_NUM % d != 0 for d in range(2, THETA_NUM)), "89 not prime"
    return True


def weyl_bound() -> float:
    """Weyl estimate for max hallucination magnitude: √(N·log Q)."""
    return math.sqrt(N_ACTIVE * math.log(Q))


def snr_db() -> float:
    """Signal-to-noise ratio in dB: 20·log10(N / √(N·log Q))."""
    return 20 * math.log10(N_ACTIVE / weyl_bound())


if __name__ == "__main__":
    verify()
    print(f"θ = {THETA_NUM}/{THETA_DEN} = {THETA:.10f}")
    print(f"Q = {Q}, N = {N_ACTIVE}, τ = {THRESHOLD:.0f}")
    print(f"ω = exp(2πiθ) = {OMEGA:.6f}")
    print(f"Weyl bound: √(N·log Q) = {weyl_bound():.2f} ≈ 89")
    print(f"SNR: {snr_db():.2f} dB")