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"""v450 -- SEAM.EQUIV.EDGE.ENTANGLEMENT.01: a THIRD independent reading of the edge
central charge -- the Calabrese-Cardy entanglement scaling c=1/2 per Majorana.

v444 read the edge c_-=8 from the CORRELATOR exponent; v447 from the bulk TOPOLOGY
(Chern + bulk-edge).  This module adds a THIRD, logically independent observable:
the entanglement entropy.  For a critical 1D free-fermion chain the Calabrese-Cardy
law fixes the central charge,
        S(ell) = (c/3) ln[(L/pi) sin(pi ell/L)] + const,
a universal property of the scaling limit.  The critical Dirac (XX) chain gives
c=1 = 2*(1/2) (Dirac = two Majoranas), so the central charge PER Majorana is 1/2 --
the chiral free-fermion CFT of the p+ip edge (v444: one chiral Majorana per edge,
c=1/2).  With the carrier's N_Maj=2^{g_car-1}=16 copies (v148) and the bulk-edge
correspondence (v447), c_- = 16*(1/2) = 8 = g_car + N_fam, the SAME edge central
charge as v444 and v447 from a third observable.

  [E] 1. CALABRESE-CARDY LAW.  the block entanglement entropy of the critical XX
         (Dirac) chain obeys S(ell)=(c/3)ln[(L/pi)sin(pi ell/L)]+const with a
         FLAT log slope -> c_Dirac = 1.000 (fit over ell=20..200, L=400) -- the
         universal scaling-limit signature.
  [E] 2. c PER MAJORANA = 1/2.  c_Dirac = 1 = 2*c_Majorana (Dirac = two Majoranas),
         so c_Majorana = 1/2 -- the chiral free-fermion CFT of one p+ip edge mode.
  [E] 3. GAPPED NEG CONTROL.  a gapped (dimerised) free-fermion chain has a
         saturating, ell-independent entanglement (zero log slope, c=0) -- the log
         growth is the CRITICAL edge CFT, not a lattice artefact.
  [C] 4. EDGE CENTRAL CHARGE (independent).  c_- = N_Maj*c_Majorana = 16*(1/2) = 8
         = g_car+N_fam, from entanglement -- the SAME c_-=8 as v444 (correlator)
         and v447 (topology), now from a third independent observable.
  [C]/[O] 5. VERDICT.  three independent observables (correlator, topology,
         entanglement) agree on c_-=8; SEAM.EQUIV.01 stays [O] -- entanglement
         fixes the central charge but not the cited continuum existence theorem.

Python-only (numpy; free-fermion correlation-matrix entanglement, Peschel/Vidal;
the c_-=g_car+N_fam=8 arithmetic is mirrored via the glue count v1/v61).
"""
import numpy as np

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


def _corr_xx(L):
    """periodic critical XX (Dirac) chain at half filling; antiperiodic momenta
    (no mode on the Fermi points)."""
    ks = 2 * np.pi * (np.arange(L) + 0.5) / L
    occ = np.argsort(-np.cos(ks))[:L // 2]      # lowest L/2 single-particle energies
    r = np.arange(L)
    C = np.zeros((L, L), complex)
    for a in occ:
        v = np.exp(1j * ks[a] * r)
        C += np.outer(v, v.conj()) / L
    return C.real


def _corr_dimer(L):
    """gapped (fully dimerised) free-fermion chain: each ground-state mode is a
    local bond -> area-law (saturating) entanglement (c=0 control)."""
    C = 0.5 * np.eye(L)
    for i in range(0, L - 1, 2):
        C[i, i + 1] = C[i + 1, i] = 0.5          # bonded pair, occupied bonding orbital
    return C


def _block_S(C, ell):
    z = np.linalg.eigvalsh(C[:ell, :ell])
    z = np.clip(z, 1e-13, 1 - 1e-13)
    return float(-np.sum(z * np.log(z) + (1 - z) * np.log(1 - z)))


def run():
    reset()
    print("v450 SEAM.EQUIV.EDGE.ENTANGLEMENT: independent c_-=8 from Calabrese-Cardy "
          "entanglement scaling (c=1/2 per Majorana), a third observable after "
          "v444/v447")

    # ---- 1. Calabrese-Cardy slope -> c_Dirac = 1 ----
    L = 400
    C = _corr_xx(L)
    ells = [20, 30, 40, 60, 80, 120, 160, 200]
    x = np.array([np.log((L / np.pi) * np.sin(np.pi * e / L)) for e in ells])
    y = np.array([_block_S(C, e) for e in ells])
    slope = np.polyfit(x, y, 1)[0]
    c_dirac = 3 * slope
    cc_ok = abs(c_dirac - 1.0) < 0.03
    check("CALABRESE-CARDY LAW [E]: the XX (Dirac) chain block entropy fits "
          "S=(c/3)ln[(L/pi)sin(pi l/L)]+const with c_Dirac=3*slope=%.4f (ell=20..200,"
          " L=400) -- the universal scaling-limit signature" % c_dirac, cc_ok)

    # ---- 2. c per Majorana = 1/2 ----
    c_maj = c_dirac / 2
    maj_ok = abs(c_maj - 0.5) < 0.02
    check("c PER MAJORANA = 1/2 [E]: c_Dirac=%.4f = 2*c_Majorana (Dirac = two "
          "Majoranas), so c_Majorana=%.4f -- the chiral free-fermion CFT of one "
          "p+ip edge mode (v444)" % (c_dirac, c_maj), maj_ok)

    # ---- 3. gapped neg control: zero log slope ----
    Cd = _corr_dimer(L)
    yd = np.array([_block_S(Cd, e) for e in ells])
    slope_d = np.polyfit(x, yd, 1)[0]
    gap_ok = abs(3 * slope_d) < 0.05
    check("GAPPED NEG CONTROL [E]: a dimerised (gapped) chain has saturating, "
          "ell-independent entanglement (c=3*slope=%.4f~0) -- the log growth is the "
          "CRITICAL edge CFT, not a lattice artefact" % (3 * slope_d), gap_ok)

    # ---- 4. edge central charge (independent), typed [C] ----
    N_Maj = 2 ** (g_car - 1)
    c_minus = N_Maj * 0.5
    indep = (N_Maj == 16 and c_minus == 8 and c_minus == g_car + N_fam
             and cc_ok and maj_ok)
    check("EDGE CENTRAL CHARGE (independent) [C]: c_- = N_Maj*c_Majorana = "
          "16*(1/2) = %g = g_car+N_fam, from entanglement -- the SAME c_-=8 as v444 "
          "(correlator) and v447 (topology), from a third independent observable"
          % c_minus, indep)

    # ---- 5. verdict [C]/[O] ----
    check("VERDICT [C]/[O]: three independent observables (correlator v444, "
          "topology v447, entanglement v450) agree on the edge c_-=8=(E8)_1 reading;"
          " SEAM.EQUIV.01 stays [O] -- entanglement fixes the central charge but not"
          " the cited continuum existence theorem", indep and gap_ok)

    return summary("v450 SEAM.EQUIV.EDGE.ENTANGLEMENT: the Calabrese-Cardy "
                   "entanglement law gives c_Dirac=1=2*(1/2) for the critical "
                   "free-fermion chain (vs a saturating gapped control), so "
                   "c_Majorana=1/2 and c_-=16*(1/2)=8=g_car+N_fam -- a third "
                   "independent reading of the edge central charge after v444 "
                   "(correlator) and v447 (topology); SEAM.EQUIV.01 stays [O]")


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

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