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debe354 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | """
Non-Commutative Torus T²_{89/2462}: Explicit Matrix Representation
U (Clock): diagonal phase rotation, U_kk = ω^k
V (Shift): cyclic permutation, V|k⟩ = |k+1 mod Q⟩
Weyl relation: VU = ω UV where ω = exp(2πi × 89/2462)
"""
import numpy as np
from .sovereign_shift import THETA_NUM, THETA_DEN, Q, THETA
def clock_matrix(q: int = Q, p: int = THETA_NUM) -> np.ndarray:
"""U (Scaling Flow): q×q diagonal with U_kk = exp(2πi·p·k/q)."""
omega = np.exp(2j * np.pi * p / q)
return np.diag([omega**k for k in range(q)])
def shift_matrix(q: int = Q) -> np.ndarray:
"""V (Lateral Displacement): q×q cyclic shift."""
V = np.zeros((q, q), dtype=complex)
for k in range(q - 1):
V[k + 1, k] = 1.0
V[0, q - 1] = 1.0
return V
def verify_weyl(U: np.ndarray, V: np.ndarray, q: int = Q, p: int = THETA_NUM) -> float:
"""Verify VU = exp(2πiθ) UV. Returns error norm."""
omega = np.exp(2j * np.pi * p / q)
return float(np.linalg.norm(V @ U - omega * (U @ V)))
def verify_periods(U: np.ndarray, V: np.ndarray, q: int = Q) -> tuple:
"""Verify V^q = I and U^q = I."""
I = np.eye(q, dtype=complex)
v_err = float(np.linalg.norm(np.linalg.matrix_power(V, q) - I))
u_err = float(np.linalg.norm(np.linalg.matrix_power(U, q) - I))
return v_err, u_err
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