| """ | |
| 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 | |