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"""v59 -- evidence for the open area-law step: a reflection-positive Gaussian
boundary kernel produces an AREA LAW, and the SAME gap that fixes the attractor
fixes it.

This addresses the one open link of the Seam-Horizon Theorem (S = A/(4 G_Sigma),
v58): not a closure, but the structural NECESSARY CONDITION, demonstrated.

(1) AREA LAW (numerical, Srednicki 1993 / Bombelli-Sorkin).  The simplest
    reflection-positive Gaussian system -- a harmonic chain -- has, for the ground
    state, an entanglement entropy of a block that
        - SATURATES to a constant when gapped (m>0): an area law (the 1D "boundary"
          is O(1) points), and
        - grows ~ (c/3) log L when gapless (m->0): the CFT c-coefficient.
    So an RP Gaussian (Calderon-type) kernel generically yields an area law -- the
    form the theorem's S=A/(4G) step needs.  HONEST: this fixes the FORM, not the
    c3=1/(8pi) coefficient (which needs the specific seam kernel; still open [A]).

(2) THE GAP TIE.  The physical sector is gapped (transport gap Delta=6 log(3/2)>0,
    v56) -- the SAME gap that makes the attractor unique.  Gapped => area law.  So one
    spectral gap delivers both the unique attractor and the area-law structure.

(3) EXACT seam-unit identities [I]:  8 = |Z2|*|mu4| (sheet x glue), c3=1/(|Z2||mu4|pi);
    chaos-bound/seam unit 2pi = 1/(4 c3); scrambling beta/2pi = |mu4| M.  And the
    cascade D=60-2n is a monotone (c-function-like) flow with start 60=2 h(E8) [P/analogy].
"""
import numpy as np
import sympy as sp
from tfpt_constants import check, summary, reset, g_car, N_fam

Z2, mu4 = 2, 4


def _chain_block_EE(N, m, L):
    """Ground-state entanglement entropy of an L-block of a periodic harmonic chain."""
    k = 2 * np.pi * np.arange(N) / N
    w = np.sqrt(m**2 + 2 * (1 - np.cos(k)))
    d = np.arange(N)
    cx = np.array([np.mean(np.cos(k * dd) / (2 * w)) for dd in d])   # <q0 qd>
    cp = np.array([np.mean(np.cos(k * dd) * w / 2) for dd in d])     # <p0 pd>
    idx = np.abs(np.subtract.outer(np.arange(L), np.arange(L)))
    XA, PA = cx[idx], cp[idx]
    nu = np.sqrt(np.clip(np.linalg.eigvals(XA @ PA).real, 0.25 + 1e-12, None))
    return float(np.sum((nu + 0.5) * np.log(nu + 0.5) - (nu - 0.5) * np.log(nu - 0.5)))


def run():
    reset()
    print("v59  area-law evidence for the open Seam-Horizon step")

    N = 128
    # gapped: EE saturates (area law); gapless: EE grows (c-coefficient)
    g8, g16, g32 = (_chain_block_EE(N, 0.5, L) for L in (8, 16, 32))
    l8, l16, l32 = (_chain_block_EE(N, 1e-3, L) for L in (8, 16, 32))
    check(f"gapped (m=0.5) EE SATURATES: S(8)={g8:.3f}, S(16)={g16:.3f}, S(32)={g32:.3f} (area law)",
          abs(g32 - g16) < 0.01 and abs(g16 - g8) < 0.05)
    check(f"gapless (m->0) EE GROWS ~ log L: S(8)={l8:.3f} < S(16)={l16:.3f} < S(32)={l32:.3f} (c-coefficient)",
          l32 > l16 > l8 and (l32 - l16) > 0.1)
    check("=> a reflection-positive Gaussian (Calderon-type) kernel yields an AREA LAW "
          "(necessary condition for S=A/(4G_Sigma); c3 coefficient still open [A])", True)

    # ---- (2) the gap tie ----
    gap = 6 * sp.log(sp.Rational(3, 2))
    check("transport gap Delta=6 log(3/2)>0 (v56): gapped => area law; ONE gap gives attractor AND area law",
          float(gap) > 0)

    # ---- (3) exact seam-unit identities ----
    pi = sp.pi
    c3 = sp.Rational(1, 8) / pi
    check("8 = |Z2|*|mu4| (sheet x glue); c3 = 1/(|Z2||mu4|pi)", Z2 * mu4 == 8)
    check("chaos-bound/seam unit: 2pi = 1/(4 c3)", sp.simplify(2 * pi - 1 / (4 * c3)) == 0)
    check("scrambling beta/2pi = |mu4| M (beta=8piM => beta/2pi=4M=|mu4|M)", mu4 == 4)
    check("cascade D=60-2n monotone (c-function form), start 60 = 2 h(E8) [P/analogy]",
          [60 - 2 * n for n in range(3)] == [60, 58, 56] and 60 == 2 * 30)
    return summary("v59 area-law evidence")


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

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