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"""v254 -- PS.NCG.FLUCT.01: the INNER FLUCTUATIONS of the finite Dirac operator
Omega^1_D(A_F) computed explicitly, closing the open obligation v250 #5 (Higgs
content 10/16, NOT 126) and the NO-JUNK obligation, and fixing the sigma quantum
numbers (v250 #4, the rep-theory half).

Inner fluctuation:  D -> D_A = D + A + eps' J A J^-1,  A = sum_i pi(a_i)[D_F, pi(b_i)]
                    in  Omega^1_D = span{ pi(a)[D_F, pi(b)] : a,b in A_F }.

Because the finite D_F is purely chirality-ODD (Yukawa L<->R + Majorana), every
1-form is odd: the FINITE fluctuations are SCALARS (the gauge bosons come from the
continuum Dirac [D_M, a], not from D_F).  The scalar content is exactly the Higgs
sector, and the no-126 / no-junk statement becomes a support-containment check.

  [E] 1. FINITE FLUCTUATIONS ARE SCALARS.  Omega^1_{D_F} is purely chirality-odd
        (gamma-anticommuting); its gamma-even (gauge) part is zero -- the finite
        inner fluctuations are scalars only.
  [E] 2. NO JUNK, NO 126.  for the SM algebra C(+)H(+)M_3, every 1-form has support
        EXACTLY inside the Higgs-doublet pattern (chirality-odd, colour-diagonal,
        no quark<->lepton mixing) -- i.e. exactly the one SM Higgs doublet
        (1,2)_{1/2}; no colour/weak triplet, no (10,1,3)/126-type scalar, no junk.
  [E] 3. sigma NEEDS "BEYOND FIRST ORDER".  with the SM algebra the sigma (nu_R nu_R^c
        Majorana) direction is NOT generated by the inner fluctuations (consistent
        with first order HOLDING, v252); with the Pati-Salam algebra it IS generated
        -- sigma is precisely the Chamseddine-Connes-van Suijlekom beyond-first-order
        scalar (v251/v252).
  [E] 4. sigma QUANTUM NUMBERS.  sigma is an SM singlet (1,1,0) carrying B-L = 2:
        nu_R has B-L = -1, so the Majorana coupling sigma*(nu_R nu_R) is gauge
        invariant only for B-L(sigma) = +2 -- sigma gives the nu_R Majorana mass and
        breaks B-L (= the Pati-Salam (4,1,2)->SM step).
  [C] 5. sigma = SCALARON.  sigma is thus a real SM-singlet scalar at the B-L/PS
        scale; its identification with the R+R^2 scalaron (M_PS = kappa M_s, v249/
        v253) is consistent in quantum numbers and scale; the gravitational coupling
        proper remains [C]/[O].

Status: [E] items 1-4 (finite fluctuations are the SM Higgs doublet with no junk/
no 126; sigma is the beyond-first-order scalar with B-L=2); [C] sigma=scalaron.
Reuses the explicit v252 triple.  Python-only (numpy).
"""
import numpy as np

import v252_full_finite_triple as t
from tfpt_constants import check, summary, reset

I2 = np.eye(2, dtype=complex)


def sm_basis():
    """basis of A_F = C (+) H (+) M_3(C) as 96x96 representatives."""
    sx = np.array([[0, 1], [1, 0]], complex)
    sy = np.array([[0, -1j], [1j, 0]])
    sz = np.array([[1, 0], [0, -1]], complex)
    z2, z3 = np.zeros((2, 2), complex), np.zeros((3, 3), complex)
    B = [t.rep(1.0, z2, z3)]
    for q in (I2, 1j * sx, 1j * sy, 1j * sz):
        B.append(t.rep(0.0, q, z3))
    for k in range(3):
        for l in range(3):
            E = np.zeros((3, 3), complex); E[k, l] = 1.0
            B.append(t.rep(0.0, z2, E))
    return B


def ps_basis():
    z2, z3 = np.zeros((2, 2), complex), np.zeros((3, 3), complex)
    qb = [t.quaternion(1, 0), t.quaternion(1j, 0), t.quaternion(0, 1), t.quaternion(0, 1j), z2]
    B = []
    for qL in qb:
        for qR in qb:
            B.append(t.rep_PS(qL, qR, z3))
    for k in range(3):
        for l in range(3):
            E = np.zeros((3, 3), complex); E[k, l] = 1.0
            B.append(t.rep_PS(z2, z2, E))
    return B


def run():
    reset()
    print("v254  PS.NCG.FLUCT.01: inner fluctuations Omega^1_D -> SM Higgs doublet (no 126, no junk) + sigma")

    NH, NP, IDX, NG, P = t.NH, t.NP, t.IDX, t.NG, t.P
    gamma = t.grading()
    rng = np.random.default_rng(254)

    def Y():
        return rng.normal(size=(3, 3)) + 1j * rng.normal(size=(3, 3))
    MR0 = rng.normal(size=(3, 3)); MR = MR0 + MR0.T
    D = t.full_D(Y(), Y(), Y(), Y(), MR)
    B = sm_basis()

    # 1. finite fluctuations are scalars (odd), gauge part 0
    odd_norm = even_norm = 0.0
    for a in B:
        for b in B:
            om = a @ (D @ b - b @ D)
            even_norm += np.linalg.norm(0.5 * (om + gamma @ om @ gamma))
            odd_norm += np.linalg.norm(0.5 * (om - gamma @ om @ gamma))
    check("FINITE FLUCTUATIONS ARE SCALARS [E]: Omega^1_{D_F} is purely "
          "chirality-odd (||even/gauge part|| = %.2e vs ||odd/scalar|| = %.1f) -- the "
          "finite inner fluctuations are scalars only; gauge bosons come from the "
          "continuum [D_M,a]" % (even_norm, odd_norm),
          even_norm < 1e-9 and odd_norm > 1.0)

    # 2. no junk, no 126: support within the SM Higgs-doublet pattern
    def sector(name):
        return "lep" if name in ("nu", "e") else "q"
    allowed = np.zeros((NH, NH), bool)
    for blk in range(2):
        off = blk * NP
        for ni, (nm_i, ch_i, c_i, g_i) in enumerate(P):
            for nj, (nm_j, ch_j, c_j, g_j) in enumerate(P):
                if ch_i != ch_j and c_i == c_j and sector(nm_i) == sector(nm_j):
                    allowed[off + ni, off + nj] = True
    nuR = [IDX[("nu", "R", 0, g)] for g in range(NG)]
    for i in nuR:
        for j in nuR:
            allowed[i, NP + j] = True; allowed[NP + j, i] = True
    union = np.zeros((NH, NH), bool)
    for a in B:
        for b in B:
            union |= (np.abs(a @ (D @ b - b @ D)) > 1e-9)
    extra = int((union & ~allowed).sum())
    check("NO JUNK, NO 126 [E]: for the SM algebra every 1-form has support EXACTLY "
          "inside the Higgs-doublet pattern (chirality-odd, colour-diagonal, no "
          "quark<->lepton mixing) -- the one SM Higgs doublet (1,2)_{1/2}; %d junk "
          "entries (no triplet, no (10,1,3)/126-type scalar)" % extra,
          extra == 0)

    # 3. sigma needs beyond first order
    maj = np.zeros((NH, NH), complex)
    for ii, i in enumerate(nuR):
        for jj, j in enumerate(nuR):
            maj[i, NP + j] = MR[ii, jj]; maj[NP + j, i] = np.conj(MR[ii, jj])

    def in_span(basis_):
        forms = [(a @ (D @ b - b @ D)).reshape(-1) for a in basis_ for b in basis_]
        F = np.array(forms).T
        sol, *_ = np.linalg.lstsq(F, maj.reshape(-1), rcond=None)
        return np.linalg.norm(F @ sol - maj.reshape(-1)) / np.linalg.norm(maj) < 1e-6
    sigma_sm = in_span(B)
    sigma_ps = in_span(ps_basis())
    check("sigma NEEDS BEYOND FIRST ORDER [E]: the sigma (nu_R Majorana) direction "
          "is NOT in Omega^1_D for the SM algebra (%s, consistent with first order "
          "holding) but IS for the Pati-Salam algebra (%s) -- sigma is the CCvS "
          "beyond-first-order scalar (v251/v252)"
          % ("absent" if not sigma_sm else "present",
             "present" if sigma_ps else "absent"),
          (not sigma_sm) and sigma_ps)

    # 4. sigma quantum numbers: SM singlet (1,1,0), B-L = 2
    from fractions import Fraction as Fr
    BL_nuR = Fr(0) - Fr(1)                 # B=0, L=1 -> B-L = -1
    BL_sigma = -2 * BL_nuR                 # sigma*(nu_R nu_R) invariant -> B-L(sigma) = +2
    check("sigma QUANTUM NUMBERS [E]: sigma is an SM singlet (1,1,0) with B-L = %s "
          "(nu_R has B-L = %s, so sigma*(nu_R nu_R) is invariant only for B-L=+2) -- "
          "sigma gives the nu_R Majorana mass and breaks B-L (the PS (4,1,2)->SM step)"
          % (BL_sigma, BL_nuR),
          BL_sigma == Fr(2) and BL_nuR == Fr(-1))

    # 5. sigma = scalaron (consistency)
    check("sigma = SCALARON [C]: sigma is therefore a real SM-singlet scalar at the "
          "B-L/PS scale; its identification with the R+R^2 scalaron (M_PS = kappa "
          "M_s, v249/v253) is consistent in quantum numbers and scale; the "
          "gravitational coupling proper remains [C]/[O]", True)

    return summary("v254 inner fluctuations: SM Higgs doublet (no 126/no junk) + beyond-first-order sigma (PS.NCG.FLUCT.01)")


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

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