"""
Apply the EXACT Casimir-based isotypic projectors (projectors_exact.npz)
to the two example states, at both cuts, replacing the earlier
Monte-Carlo-based ||A_j||_* estimates with machine-precision values.

Uses T1, T2, C12 (saved by recoupling_check.py) so that "Example 1"
(pure psi1) and "Example 2" (Xi at alpha) are both obtained from the
SAME three fixed tensors, no new quantum simulation.
"""
import numpy as np

T1 = np.load("T1.npy")
T2 = np.load("T2.npy")
C12 = np.load("C12.npy")
overlap = 0.25  # <psi1|psi2>, real (checked earlier)

P = np.load("projectors_exact.npz")

def state_tensor(alpha):
    N2 = 1 + np.sin(2*alpha)*overlap
    return (np.cos(alpha)**2*T1 + np.sin(alpha)**2*T2
            + np.cos(alpha)*np.sin(alpha)*C12) / N2

def report(label, alpha):
    T = state_tensor(alpha)
    M_c1 = T.reshape(27,27)   # cut ABC|DEF
    M_c2 = T.reshape(9,81)    # cut AB|CDEF

    print(f"\n=== {label} (alpha={alpha}) ===")
    print("-- cut ABC|DEF --")
    tot = 0.0
    for j in range(4):
        Pj = P[f"P3_{j}"]
        block = Pj @ M_c1 @ Pj
        nn = np.linalg.svd(block, compute_uv=False).sum()
        Aj = nn/(2*j+1)
        tot += (2*j+1)*Aj
        print(f"   j={j}: ||A_j||_* = {Aj:.6f}")
    raw_nn = np.linalg.svd(M_c1, compute_uv=False).sum()
    print(f"   sum_j (2j+1)||A_j||_* = {tot:.6f}   vs. ||M||_* direct = {raw_nn:.6f}")

    print("-- cut AB|CDEF --")
    tot = 0.0
    for j in range(3):
        Pj_src = P[f"P2_{j}"]
        Pj_tgt = P[f"P4_{j}"]
        block = Pj_src @ M_c2 @ Pj_tgt
        nn = np.linalg.svd(block, compute_uv=False).sum()
        Aj = nn/(2*j+1)
        tot += (2*j+1)*Aj
        print(f"   j={j}: ||A_j||_* = {Aj:.6f}")
    raw_nn = np.linalg.svd(M_c2, compute_uv=False).sum()
    print(f"   sum_j (2j+1)||A_j||_* = {tot:.6f}   vs. ||M||_* direct = {raw_nn:.6f}")

report("Example 1 (pure psi1, three aligned singlets)", 0.0)
report("Example 2 (superposition)", np.pi/5)