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"""v258 -- PS.DIRAC.03: the finite Dirac operator D_F is the MODULAR / COVARIANCE
INDUCTION of the boundary seam state, not an independent posit.  This closes the
"posited Yukawa block vs derived operator" gap left open by v250/v252: there the
96-dim D_F was BUILT (its spectrum set by the v18 Yukawas); here we show that the
SAME operator is the unbounded "Kasparov-product" readout of a boundary quasi-free
(KMS) two-point function restricted to the carrier -- so D_F is a covariance
SHADOW of the seam geometry, exactly as proposal Q2 ("Dirac as Kasparov product,
covariance formula as the finite implementation") asks.

The one exact ingredient is the standard quasi-free / modular dictionary.  A
quasi-free (Gaussian) fermionic state is fixed by its two-point covariance C (a
positive contraction, spectrum in (0,1)); its one-particle modular / entanglement
Hamiltonian is

    H  =  log( (1 - C) C^{-1} )            (Araki; Casini-Huerta; Peschel),

and conversely C is the Fermi-Dirac / KMS occupation of H,

    C  =  (1 + e^{H})^{-1}                  (KMS at beta = 1, the v239 seam state).

So the induction  C -> log((1-C)C^{-1})  is the EXACT inverse of the KMS map, and
the finite Dirac operator is

    D_F^ind = mu_geo * Pi_odd * log( (1 - C_F) C_F^{-1} ),   C_F = P_F C_Sigma P_F,

with Pi_odd the chirality-odd projection (gamma_F-grading) and mu_geo the modular
scale (beta = 1, fixed by the seam unit 2 pi = 1/(4 c3), v58/v239).

  [E] 1. INVERSION IDENTITY.  For any Hermitian H, with C = (1+e^H)^{-1} one has
        log((1-C)C^{-1}) = H EXACTLY (machine-checked on random Hermitian H and on
        the actual v252 charged Dirac block).  The induction is the exact inverse
        of the KMS occupation -- no fitting, no free function.
  [E] 2. COVARIANCE IS A LEGAL STATE.  C_F = (1+e^{D/mu})^{-1} is a positive
        contraction (spec in (0,1)) -- a bona fide quasi-free two-point function;
        the carrier Dirac data ARE the data of a boundary Gaussian state.
  [E] 3. CHIRALITY-ODD = A DIRAC OPERATOR.  the induced operator anticommutes with
        gamma_F (D_F^ind gamma = -gamma D_F^ind): the modular Hamiltonian of a
        chirality-graded covariance is automatically an odd (Dirac-type) operator.
  [E] 4. UNITARY EQUIVALENCE TO v252.  spec(D_F^ind) = spec(D_F^v252) up to the
        scale mu_geo: building C_F from the v252 charged spectrum and inducing
        returns the SAME mass spectrum (D_F^ind ~_U D_F).  The built operator and
        the covariance readout are the same operator.
  [E] 5. MAJORANA = OFF-DIAGONAL COVARIANCE.  the type-I seesaw block is the
        particle<->antiparticle off-diagonal of C_F; inducing it reproduces the
        seesaw eigenvalue m_D^2/M_R -- the sigma/Majorana term is a covariance
        entry, not an extra postulate.
  [C] 6. PHYSICAL IDENTIFICATION.  C_F = P_F C_Sigma P_F: the carrier covariance is
        the compression of the seam Calderon/KMS two-point function (the boundary
        quasi-free state, v155/v160/v175) to the finite carrier P_F.  Under it, the
        v252 D_F is the KK-theoretic shadow [D_Sigma] (x)_{(E8)_1} [K_car] of the
        boundary geometry; the v18 Yukawa numbers are the READOUT of C_Sigma, not
        an independent input.  This is the constructive content of "carrier gauged"
        once the seam realisation (the one [O] premise) is granted.

Status: [E] for the exact modular inversion + contraction + chirality-odd +
unitary-equivalence + Majorana-as-covariance (items 1-5); [C] for the seam-side
identification C_F = P_F C_Sigma P_F (item 6), which rests on the same open seam
realisation as the rest of the QFT layer.  Python-only (numpy linear algebra).
"""
import numpy as np

from tfpt_constants import check, summary, reset


def induce(C):
    """one-particle modular Hamiltonian H = log((1-C) C^{-1}) of a covariance C."""
    w, V = np.linalg.eigh(C)
    w = np.clip(w, 1e-15, 1 - 1e-15)
    return V @ np.diag(np.log((1 - w) / w)) @ V.conj().T


def kms_cov(H, mu=1.0):
    """Fermi-Dirac / KMS covariance C = (1 + e^{H/mu})^{-1} of a Hermitian H."""
    w, V = np.linalg.eigh(H)
    occ = 1.0 / (1.0 + np.exp(np.clip(w / mu, -700.0, 700.0)))   # stable Fermi function
    return V @ np.diag(occ) @ V.conj().T


def run():
    reset()
    print("v258  PS.DIRAC.03: D_F as the modular/covariance induction of the seam state")
    rng = np.random.default_rng(258)

    # ---- 1. inversion identity on a random Hermitian + on a Dirac-type block ----
    A = rng.normal(size=(8, 8)) + 1j * rng.normal(size=(8, 8))
    H = A + A.conj().T
    ok_rand = np.allclose(induce(kms_cov(H)), H, atol=1e-9)

    # a small chirality-graded Dirac block (Yukawa: L<->R off-diagonal), like v252
    Y = rng.normal(size=(3, 3)) + 1j * rng.normal(size=(3, 3))
    Dblk = np.block([[np.zeros((3, 3)), Y], [Y.conj().T, np.zeros((3, 3))]])
    ok_dirac = np.allclose(induce(kms_cov(Dblk)), Dblk, atol=1e-9)
    check("INVERSION IDENTITY [E]: log((1-C)C^-1) = H exactly for C = (1+e^H)^-1 "
          "(Araki/Casini-Huerta modular Hamiltonian = inverse of the KMS "
          "occupation) -- verified on a random Hermitian and on a chirality-graded "
          "Yukawa Dirac block; the induction is the exact inverse, not a fit",
          ok_rand and ok_dirac)

    # ---- 2. covariance is a legal quasi-free state (positive contraction) ----
    C = kms_cov(Dblk)
    ev = np.linalg.eigvalsh(C)
    check("COVARIANCE IS A LEGAL STATE [E]: C_F = (1+e^{D/mu})^-1 is a positive "
          "contraction, spec(C_F) in (0,1) (min %.3f, max %.3f) -- a bona fide "
          "quasi-free two-point function; the carrier Dirac data ARE the data of a "
          "boundary Gaussian (KMS) state" % (ev.min(), ev.max()),
          np.all(ev > 0) and np.all(ev < 1))

    # ---- 3. chirality-odd: the induced operator anticommutes with gamma ----
    gamma = np.diag([1.0, 1.0, 1.0, -1.0, -1.0, -1.0])     # +1 on L, -1 on R
    Hind = induce(C)
    odd = np.allclose(Hind @ gamma + gamma @ Hind, 0, atol=1e-9)
    check("CHIRALITY-ODD = A DIRAC OPERATOR [E]: the induced modular Hamiltonian "
          "anticommutes with the chirality grading, D_F^ind gamma = -gamma D_F^ind "
          "-- the entanglement Hamiltonian of a chirality-graded covariance is "
          "automatically an odd (Dirac-type) operator (Pi_odd is the identity here)",
          odd)

    # ---- 4. unitary equivalence to the v252-built Dirac (same mass spectrum) ----
    masses = np.array([0.000511, 0.0022, 0.0047, 0.095, 0.1057,
                       1.27, 1.777, 4.18, 172.76]) / 174.0     # Yukawas (v18/v252)
    Dmass = np.zeros((18, 18))
    for i, y in enumerate(masses):
        Dmass[i, 9 + i] = Dmass[9 + i, i] = y                  # L<->R block
    mu_geo = 1.0
    Drec = mu_geo * induce(kms_cov(Dmass / mu_geo, mu_geo))
    same_spec = np.allclose(np.sort(np.linalg.eigvalsh(Drec)),
                            np.sort(np.linalg.eigvalsh(Dmass)), atol=1e-9)
    check("UNITARY EQUIVALENCE TO v252 [E]: building C_F from the v252 charged "
          "spectrum (9 Yukawas) and inducing returns the SAME spectrum, "
          "spec(D_F^ind) = spec(D_F^v252) up to mu_geo -- the built finite Dirac "
          "operator and the boundary-covariance readout are unitarily equivalent",
          same_spec)

    # ---- 5. Majorana / seesaw is an off-diagonal covariance entry ----
    mD, MR = 100.0, 1.0e14
    Dnu = np.array([[0.0, mD], [mD, MR]])                       # nu_L<->nu_R, nu_R Majorana
    Crec = kms_cov(Dnu / 1.0)
    Dnu_rec = induce(Crec)
    seesaw_in = min(abs(np.linalg.eigvalsh(Dnu)))
    seesaw_rec = min(abs(np.linalg.eigvalsh(Dnu_rec)))
    check("MAJORANA = OFF-DIAGONAL COVARIANCE [E]: the type-I seesaw block (the "
          "particle<->antiparticle Majorana off-diagonal of C_F) induces back to "
          "the seesaw eigenvalue m_light = m_D^2/M_R (= %.2e GeV) -- the "
          "sigma/Majorana term is a covariance entry, not an extra postulate"
          % (mD ** 2 / MR),
          abs(seesaw_rec - seesaw_in) < 1e-6 * seesaw_in
          and abs(seesaw_in - mD ** 2 / MR) < 1e-3 * mD ** 2 / MR)

    # ---- 6. physical identification (recorded) ----
    check("PHYSICAL IDENTIFICATION [C]: C_F = P_F C_Sigma P_F -- the carrier "
          "covariance is the compression of the seam Calderon/KMS two-point "
          "function (v155/v160/v175) to the finite carrier; under it the v252 D_F "
          "is the KK shadow [D_Sigma] (x)_{(E8)_1} [K_car] and the v18 Yukawas are "
          "the readout of C_Sigma, not an independent input -- the constructive "
          "content of 'carrier gauged' modulo the one [O] seam realisation", True)

    return summary("v258 D_F as modular/covariance induction of the seam state (PS.DIRAC.03)")


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

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