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"""v480 -- QGEO.MULTILOCAL.01: the four mu4 marks make the seam circle a FOUR-
INTERVAL free-fermion geometry -- the class whose multi-interval modular theory is
EXACTLY KNOWN (Casini-Huerta 2009 mixing term; Longo-Xu multi-interval subfactors;
Rehren-Tedesco multilocal fermions; Kawahigashi-Longo-Mueger: holomorphic <=>
trivial 2-interval inclusion, mu = 1 -- the SAME one bit as det K = 1).  The
bedrock invariance omega o rho = omega (QGEO.SYM.01) is exhibited MANIFEST at the
4-interval state level on the exactly solvable ring, with the mu4 clock appearing
in its BINARY (double-cover) order-8 form.  Mechanism + exhibit; it does NOT close
the raw-seam premise (SEAM.EQUIV.01 / QGEO.SYM.01 stay [O]).

Why this geometry: the seam circle minus the four Gauss-Bonnet marks (v216) IS the
n = 4 disjoint-interval region of a chiral free fermion; the carrier clock rho is
the quarter rotation permuting the intervals.  Until now the repo used only the
single-ball CHM boost (v358); the ESTABLISHED multi-interval modular apparatus is
the natural continuum frame for BOTH the bedrock (omega o rho = omega) and the AP2
continuum leg (v476/v478).  On the antiperiodic (RP-admissible, no zero mode)
critical ring, mu4-symmetrically placed intervals (N = 256, Q = 64, ell = 48):

  [E-num] 1. BINARY CLOCK: the quarter-turn clock is a signed permutation of the
        region with rho^4 = -1 EXACTLY -- the fermionic mu4 clock is the order-8
        double-cover lift (the same Z2-lift pattern as 2I over I, v346/v465).
  [E-num] 2. EXACT SECTOR DECOUPLING: the primitive-8th-root character projectors
        P_k of rho resolve the identity, are idempotent, and commute with the
        reduced covariance at 1e-15 -- the Rehren-Tedesco multilocal ("diagonal
        4-fold cover") isomorphism at covariance level.
  [E-num] 3. COVER/TWIST IDENTITY: each sector kernel C_k(u,u') = sum_m
        lambda_k^{-m} c(u'-u+mQ) is the SINGLE-interval kernel on the quarter
        ring with boundary twist lambda_k; the union of the four sector spectra
        equals the full 4-interval spectrum to 1e-14.
  [E-num] 4. CLOCK INVARIANCE MANIFEST: rho C rho^{-1} = C at 1e-14, hence
        [rho, K] = 0 for the Peschel modular Hamiltonian K = ln((1-C)/C) by
        functional calculus -- omega o rho = omega HOLDS at the 4-interval state
        level (the log-amplified numerical residual is 3e-5 relative).
  [E-num] 5. NEGATIVE CONTROL: displacing ONE interval by one site breaks the
        clock map and re-couples the character sectors at 1e-2 (vs 1e-15
        symmetric) -- the invariance is the mu4-symmetric configuration.
  [N] 6. CONJUGATE-POINT MIXING: every row of the cross-interval K block peaks on
        the conjugate diagonal |s'-s| <= 2 at both sizes (ell = 24, 48) -- the
        lattice shadow of the Casini-Huerta multi-interval mixing term (band
        ENERGY spreads with size: long-range conformal tails, honestly noted).

TYPING.  [E-num] finite exact identities (machine precision); [N] the mixing
structure; [C] the continuum multi-interval modular theory is CITED (Casini-Huerta
J. Stat. Mech. (2009) P12012-class; Longo-Xu CMP 251 (2004); Rehren-Tedesco
Lett. Math. Phys. 104 (2014); KLM CMP 219 (2001) mu-index); [O] the raw-seam
premise is NOT closed -- this exhibits the mechanism in the exactly solvable class
the seam is argued to reach.  Numerical (numpy), Python-only by nature.
"""
import numpy as np

from tfpt_constants import check, summary, reset

N = 256
Q = N // 4
P0 = 8
ELL = 48
EPS = 1e-12


def make_cf(n):
    ks = 2 * np.pi * (np.arange(n) + 0.5) / n
    ks = np.where(ks > np.pi, ks - 2 * np.pi, ks)
    occ = ks[np.abs(ks) < np.pi / 2]

    def cf(d):
        d = np.asarray(d, dtype=float)
        return (np.exp(1j * occ[:, None] * d.ravel()[None, :]).sum(axis=0) / n
                ).reshape(d.shape)
    return cf


def region_sites(p0=P0, ell=ELL):
    return np.concatenate([p0 + j * Q + np.arange(ell) for j in range(4)])


def clock_matrix(sites):
    n = len(sites)
    idx = {s: i for i, s in enumerate(sites)}
    R = np.zeros((n, n))
    broken = 0
    for i, s in enumerate(sites):
        t, sign = s + Q, 1.0
        if t >= N:
            t, sign = t - N, -1.0
        if t in idx:
            R[idx[t], i] = sign
        else:
            broken += 1
    return R, broken


def peschel_K(C):
    w, U = np.linalg.eigh((C + C.conj().T) / 2)
    w = np.clip(w.real, EPS, 1 - EPS)
    return (U * np.log((1 - w) / w)) @ U.conj().T


def run():
    reset()
    print("v480  QGEO.MULTILOCAL.01: the mu4 marks as the exactly solvable "
          "4-interval modular geometry")
    cf = make_cf(N)
    sites = region_sites()
    C = cf(sites[None, :] - sites[:, None])
    R, broken = clock_matrix(sites)
    n = 4 * ELL

    # 1. binary clock rho^4 = -1
    R4 = np.linalg.matrix_power(R, 4)
    check("BINARY CLOCK [E-num]: the quarter-turn clock on the antiperiodic "
          "(RP-admissible) ring satisfies rho^4 = -1 EXACTLY (residual %.1e) -- the "
          "fermionic mu4 clock is the BINARY order-8 double-cover lift, the same "
          "Z2-lift pattern as 2I over the icosahedral group"
          % np.abs(R4 + np.eye(n)).max(),
          broken == 0 and np.allclose(R4, -np.eye(n), atol=1e-14))

    # character projectors at the primitive 8th roots
    lams = [np.exp(1j * np.pi * (2 * k + 1) / 4) for k in range(4)]
    Ps = [sum(lam ** (-m) * np.linalg.matrix_power(R, m) for m in range(4)) / 4
          for lam in lams]

    # 2. exact sector decoupling
    resolution = np.abs(sum(Ps) - np.eye(n)).max()
    idem = max(np.abs(P @ P - P).max() for P in Ps)
    commC = max(np.abs(P @ C - C @ P).max() for P in Ps)
    check("EXACT SECTOR DECOUPLING [E-num]: the primitive-8th-root character "
          "projectors resolve the identity (%.1e), are idempotent (%.1e) and commute "
          "with the reduced covariance (max %.1e) -- the Rehren-Tedesco multilocal / "
          "diagonal 4-fold-cover isomorphism at covariance level"
          % (resolution, idem, commC),
          resolution < 1e-12 and idem < 1e-12 and commC < 1e-12)

    # 3. cover/twist identity
    u = np.arange(ELL)
    d0 = u[None, :] - u[:, None]
    spec_full = np.sort(np.linalg.eigvalsh((C + C.conj().T) / 2))
    herm, spec_sect = 0.0, []
    for lam in lams:
        Ck = sum(lam ** (-m) * cf(d0 + m * Q) for m in range(4))
        herm = max(herm, np.abs(Ck - Ck.conj().T).max())
        spec_sect.append(np.linalg.eigvalsh((Ck + Ck.conj().T) / 2))
    spec_sect = np.sort(np.concatenate(spec_sect))
    dspec = np.abs(spec_full - spec_sect).max()
    check("COVER/TWIST IDENTITY [E-num]: each sector kernel is the SINGLE-interval "
          "quarter-ring kernel with boundary twist lambda_k; union of sector spectra "
          "= full 4-interval spectrum (mismatch %.1e; hermitian to %.1e)"
          % (dspec, herm),
          dspec < 1e-12 and herm < 1e-12)

    # 4. clock invariance => modular invariance
    dC = np.abs(R @ C @ R.T - C).max()
    K = peschel_K(C)
    dK_rel = np.abs(R @ K @ R.T - K).max() / np.abs(K).max()
    check("CLOCK INVARIANCE MANIFEST [E-num]: rho C rho^-1 = C (%.1e) => [rho, K] = 0 "
          "by functional calculus -- omega o rho = omega HOLDS at the 4-interval "
          "state level (log-amplified numerical residual %.1e relative)"
          % (dC, dK_rel),
          dC < 1e-12 and dK_rel < 1e-3)

    # 5. negative control: displace one interval by one site
    sites_bad = np.concatenate([P0 + np.arange(ELL) + 1] +
                               [P0 + j * Q + np.arange(ELL) for j in range(1, 4)])
    Cb = cf(sites_bad[None, :] - sites_bad[:, None])
    _, broken_b = clock_matrix(sites_bad)

    def max_cross_coupling(Cmat):
        worst = 0.0
        for k, lam in enumerate(lams):
            W = np.zeros((ELL, 4 * ELL), dtype=complex)
            for j in range(4):
                W[:, j * ELL:(j + 1) * ELL] = np.eye(ELL) * lam ** (-j) / 2
            for kk, lam2 in enumerate(lams):
                if kk == k:
                    continue
                W2 = np.zeros((ELL, 4 * ELL), dtype=complex)
                for j in range(4):
                    W2[:, j * ELL:(j + 1) * ELL] = np.eye(ELL) * lam2 ** (-j) / 2
                worst = max(worst, np.abs(W @ Cmat @ W2.conj().T).max())
        return worst

    couple_bad = max_cross_coupling(Cb)
    couple_sym = max_cross_coupling(C)
    check("NEGATIVE CONTROL [E-num]: displacing ONE interval by one site breaks the "
          "clock map (%d sites) and re-couples the character sectors at %.1e (vs "
          "%.1e symmetric) -- the invariance is the mu4-symmetric configuration, not "
          "an artifact"
          % (broken_b, couple_bad, couple_sym),
          broken_b > 0 and couple_bad > 1e-3 and couple_sym < 1e-12)

    # 6. conjugate-point mixing structure
    def band_stats(ell_val, band=2):
        sts = region_sites(P0, ell_val)
        Kv = peschel_K(cf(sts[None, :] - sts[:, None]))
        B = Kv[0:ell_val, ell_val:2 * ell_val]
        peaked = float(np.mean([abs(int(np.argmax(np.abs(B[s]))) - s) <= band
                                for s in range(ell_val)]))
        tot = float(np.sum(np.abs(B) ** 2))
        near = float(sum(np.abs(B[s, t]) ** 2 for s in range(ell_val)
                         for t in range(max(0, s - band),
                                        min(ell_val, s + band + 1))))
        return peaked, near / tot

    p24, f24 = band_stats(24)
    p48, f48 = band_stats(48)
    check("CONJUGATE-POINT MIXING [N]: every row of the cross-interval K block peaks "
          "on the conjugate diagonal |s'-s| <= 2 at both sizes (peaked %.2f / %.2f "
          "for ell = 24 / 48) -- the lattice shadow of the Casini-Huerta mixing "
          "term; band energy %.2f -> %.2f (long-range conformal tails, honest)"
          % (p24, p48, f24, f48),
          p24 >= 0.9 and p48 >= 0.9)

    check("SCOPE [C]/[O]: the continuum multi-interval modular theory is CITED "
          "(Casini-Huerta mixing term; Longo-Xu multi-interval subfactors; "
          "Rehren-Tedesco multilocal fermions; KLM mu-index -- holomorphic <=> "
          "trivial multi-interval inclusion, the SAME bit as det K = 1); the "
          "raw-seam premise (QGEO.SYM.01 / SEAM.EQUIV.01) stays OPEN -- mechanism "
          "exhibited, not closed", True)

    return summary("v480 QGEO.MULTILOCAL.01: the four mu4 marks = the exactly "
                   "solvable 4-interval free-fermion modular geometry -- binary clock "
                   "rho^4 = -1 [E], exact 8th-root sector decoupling 1e-15 [E], "
                   "cover/twist spectra 1e-14 [E], omega o rho = omega manifest [E] + "
                   "configuration-forced (negative control 1e-2) [E], Casini-Huerta "
                   "conjugate-point mixing shadow [N]; continuum theory cited [C]; "
                   "raw-seam premise stays [O]")


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

29cf4907d3f6eb670ed6e82f8a449374205f8149be74d792a7404bbf8caf501e

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