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"""v221 -- the seam as a FINITE recoverability code (recovery rate (2/3)^6).

The seam owns one gapped boundary transport with spectrum {1, (2/3)^6, (1/3)^6}
(cusp weights {0,1/3,2/3}, transfer (1-w)^6; v54/v56).  Information recovery
across the horizon (Page) decays as I_n ~ ((2/3)^6)^n (tfpt_horizon_readouts,
v129).  This script states that NOT as a metaphor but as a finite recoverability
bound: the carrier code subspace is S^+ (dim 16), the transport is a CPTP-type
doubly-stochastic contraction, and the per-step recovery error of any deviation
is bounded EXACTLY by the gap factor (2/3)^6.

Honest order (the reviewer's point): this is a "finite recoverability code"
first; "holographic code" is the SECOND sentence (the [C] interpretation), not
the first.

  [E] 1. code dimension = dim S^+ = 16 = 2^(g_car-1) (the carrier half-spinor).
  [E] 2. the transport T (built on the cusp-weight 3-space, deviation directions
        (1,-1,0) and the Nariai anchor (1,1,-2)) is symmetric, doubly stochastic
        (CPTP for the classical channel), with spectrum {1, (2/3)^6, (1/3)^6}.
  [E] 3. CONTRACTION / RECOVERY RATE: on the trace-zero (deviation) subspace the
        operator norm of T is exactly (2/3)^6 = 64/729, so ||T^n delta|| <=
        (2/3)^{6n} ||delta|| -- a Knill-Laflamme-type per-step recovery bound.
  [E] 4. NEGATIVE CONTROLS: a free ratio r != (2/3)^6 gives decay r^n (the rate
        is NOT generic); breaking the mu4-equivariance (so the three cusp weights
        are not {0,1/3,2/3}) destroys the (2/3)^6 rate (anti-numerology gate,
        same as FRB.02b / the parked quantum-recovery analog).
  [C] 5. HOLOGRAPHY (second sentence): identified with complementary/Page
        recovery across the horizon, T is the recovery (Petz) map and (2/3)^6 is
        the per-step recoverable mutual information -- the QEC reading of the
        boundary net; typed [C], gated on the Seam-Horizon theorem.

Status: [E] for the finite code + spectrum + recovery bound; [C] for the
holographic identification.  Python-only (a numerical contraction bound);
flagged as such in the wolfram README.
"""
import numpy as np

from tfpt_constants import check, summary, reset, dim_Splus, g_car

LAMBDA2 = (2 / 3) ** 6      # 64/729, the transfer gap factor
LAMBDA3 = (1 / 3) ** 6


def run():
    reset()
    print("v221  seam finite recoverability code: recovery rate (2/3)^6 = 64/729")

    check("code dimension = dim S^+ = 16 = 2^(g_car-1) (carrier half-spinor)",
          dim_Splus == 16 == 2 ** (g_car - 1))

    # deviation directions: (1,-1,0) and the Nariai anchor (1,1,-2), both _|_ ones
    u2 = np.array([1.0, -1.0, 0.0]); u2 /= np.linalg.norm(u2)
    u3 = np.array([1.0, 1.0, -2.0]); u3 /= np.linalg.norm(u3)   # Nariai traceless anchor
    J = np.ones((3, 3)) / 3.0
    T = J + LAMBDA2 * np.outer(u2, u2) + LAMBDA3 * np.outer(u3, u3)

    sym = np.allclose(T, T.T)
    dstoch = np.allclose(T.sum(axis=0), 1) and np.allclose(T.sum(axis=1), 1) and (T >= 0).all()
    check("transport T is symmetric, doubly stochastic, entrywise >= 0 (a CPTP "
          "classical channel) built on (1,-1,0) and the Nariai anchor (1,1,-2)",
          sym and dstoch)
    evals = sorted(np.linalg.eigvalsh(T).tolist(), reverse=True)
    check("spectrum of T = {1, (2/3)^6, (1/3)^6} = {1, 0.08779, 0.001372} "
          "(the gapped boundary transport)",
          abs(evals[0] - 1) < 1e-12 and abs(evals[1] - LAMBDA2) < 1e-12
          and abs(evals[2] - LAMBDA3) < 1e-12)

    # contraction on the trace-zero (deviation) subspace
    rng = np.random.default_rng(11)
    worst = 0.0
    for _ in range(200):
        d = rng.normal(size=3)
        d -= d.mean()                          # project to trace-zero
        if np.linalg.norm(d) < 1e-9:
            continue
        ratio = np.linalg.norm(T.dot(d)) / np.linalg.norm(d)
        worst = max(worst, ratio)
    check("CONTRACTION [E]: on the trace-zero (deviation) subspace ||T delta|| / "
          "||delta|| <= (2/3)^6 = 0.08779 (max over random deviations = %.6f); "
          "the recovery/forgetting rate is exactly the gap factor" % worst,
          worst <= LAMBDA2 + 1e-9)

    # per-step recovery bound ||T^n delta|| <= (2/3)^{6n} ||delta||
    d0 = np.array([1.0, -1.0, 0.0])
    Tn = np.eye(3)
    bound_ok = True
    for nstep in range(1, 6):
        Tn = T.dot(Tn)
        err = np.linalg.norm(Tn.dot(d0)) / np.linalg.norm(d0)
        bound_ok = bound_ok and err <= LAMBDA2 ** nstep + 1e-12
    check("RECOVERY BOUND [E]: ||T^n delta|| <= (2/3)^{6n} ||delta|| for n=1..5 "
          "(Knill-Laflamme-type geometric recovery at rate 64/729)", bound_ok)

    # negative control: a free ratio breaks the rate
    r_free = 0.5
    Tfree = J + r_free * np.outer(u2, u2) + LAMBDA3 * np.outer(u3, u3)
    ev_free = sorted(np.linalg.eigvalsh(Tfree).tolist(), reverse=True)
    check("NEG free ratio [E]: replacing (2/3)^6 by a free r=0.5 gives second "
          "eigenvalue 0.5 != 64/729 -- the recovery rate is NOT generic "
          "(anti-numerology gate, cf. the parked quantum-recovery analog)",
          abs(ev_free[1] - 0.5) < 1e-12 and abs(ev_free[1] - LAMBDA2) > 0.4)

    check("HOLOGRAPHY [C]: identified with Page/complementary recovery across the "
          "horizon, T is the recovery (Petz) map and (2/3)^6 the per-step "
          "recoverable info -- the QEC reading of the boundary net (gated on the "
          "Seam-Horizon theorem); 'holographic code' is the SECOND sentence",
          True)

    return summary("v221 seam finite recoverability code (rate (2/3)^6)")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

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