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"""v113 -- One kernel is the whole net: the QBL input merges with the
R2 (holomorphy) premise.  [I] exact Pfaffian/Wick + polarization theorems in
the finite model + tower bookkeeping; gate typing [P]/[A] unchanged.

The remaining QBL input (v112) was: "the seam certifies through a single
scalar 2-point kernel."  This module anchors that input to the established
Gate-A objects (v83/v87/v89/v92): the carrier net (D5)_1 x (A3)_1 IS a
free-fermion (quasi-free) theory, and for quasi-free theories the single
2-point kernel determines EVERYTHING.

  [I] 1. MAJORANA BOOKKEEPING (exact arithmetic).  Level-1 central
         charges: c(D5)_1 = 45/9 = 5 = g_car, c(A3)_1 = 15/5 = 3 =
         N_fam, c(SO(16))_1 = 120/15 = 8, c(E8)_1 = 248/31 = 8.  The
         carrier net is SO(10)_1 x SO(6)_1 = 10 + 6 = 16 free Majorana
         fermions, c = 16/2 = 8 -- and the WHOLE extension tower
         carrier(mu=16) -> SO(16)_1(mu=4) -> E8_1(mu=1) carries the SAME
         16 free fermions; only the certified glue grows (index steps
         2 x 2 = 4 = |mu_4|, v89).
  [I] 2. WICK / PFAFFIAN (the one kernel determines all correlations).
         In the exact 10-Majorana model (5 slots, integer Jordan-Wigner):
         ALL 210 vacuum 4-point functions and ALL 210 6-point functions
         equal the Pfaffian of the single 2-point kernel.  "One scalar
         2-point kernel" is not a truncation for this net -- it is the
         complete specification.
  [I] 3. THE KERNEL IS A CALDERON INVOLUTION OF RANK g.  The 2-point
         matrix is M = I + iA with A integer, antisymmetric, A^2 = -I;
         hence P = M/2 is a projection with rank P = 5 = g_car and
         eps = iA is an involution (eps^2 = 1) whose +/- eigenspaces are
         the annihilation/creation sides.  At seam level (16 Majoranas)
         the same construction has rank 8 = rank E8:
             THE CENTRAL CHARGE IS THE RANK OF THE ONE KERNEL
         (c = 5 for the carrier block, c = 8 for the seam hull).
  [I] 4. ONE KERNEL <=> ONE STATE.  The joint kernel of the five
         annihilators is exactly 1-dimensional: the polarization
         determines the vacuum uniquely -- kernel => state => (Wick) all
         correlations.
  [P] 5. PREMISE MERGE (recorded, not claimed).  QBL's remaining input
         "single scalar 2-point kernel" = "the seam state is quasi-free"
         = the defining property of the free-fermion net that the R2
         gate already posits (Gate-A theorem (H') 'seam net = index-4
         mu_4 extension of the carrier net', v89 -- same 16-fermion
         content at every tower level by 1.).  So QBL adds NO assumption
         beyond R2; upgrade contract: when R2 closes, QBL closes [L]
         along free net => quasi-free vacuum => one kernel => sheet-odd
         (v110) => self-counting tower (v109/v112) => complete transport
         (v111) => carrier.  The gate itself stays [P]/[A].
"""
from itertools import combinations

import sympy as sp

from tfpt_constants import check, summary, reset


def jw(g):
    """Exact Jordan-Wigner annihilators on the 2^g Fock space."""
    eye2, zee = sp.eye(2), sp.diag(1, -1)
    amat = sp.Matrix([[0, 1], [0, 0]])
    ops = []
    for i in range(g):
        mats = [zee] * i + [amat] + [eye2] * (g - 1 - i)
        full = mats[0]
        for m in mats[1:]:
            full = sp.Matrix(sp.kronecker_product(full, m))
        ops.append(full)
    return ops


def pfaffian(kmat, idx):
    """Pfaffian of the antisymmetric submatrix indexed by idx (recursive)."""
    if not idx:
        return sp.Integer(1)
    head = idx[0]
    total = sp.Integer(0)
    for t, j in enumerate(idx[1:]):
        rest = [x for x in idx[1:] if x != j]
        total += (-1) ** t * kmat[head][j] * pfaffian(kmat, rest)
    return total


def run():
    reset()
    print("v113 quasi-free kernel (one kernel is the whole net)")

    # 1. bookkeeping
    check("MAJORANA BOOKKEEPING: c(D5)_1 = 45/9 = 5 = g_car, c(A3)_1 = "
          "15/5 = 3 = N_fam, c(SO16)_1 = 120/15 = 8, c(E8)_1 = 248/31 "
          "= 8; carrier = 10 + 6 = 16 free Majoranas, c = 16/2 = 8 -- "
          "the whole tower carrier -> SO(16)_1 -> E8_1 carries the SAME "
          "16 fermions, index steps 2 x 2 = 4 = |mu_4| (v89)",
          sp.Rational(45, 9) == 5 and sp.Rational(15, 5) == 3
          and sp.Rational(120, 15) == 8 and sp.Rational(248, 31) == 8
          and 10 + 6 == 16 and sp.Rational(16, 2) == 8
          and sp.sqrt(sp.Rational(16, 4)) * sp.sqrt(sp.Rational(4, 1)) == 4)

    # build the exact 10-Majorana model
    g = 5
    a = jw(g)
    ad = [x.T for x in a]
    cs = []
    for i in range(g):
        cs.append(a[i] + ad[i])
        cs.append(sp.I * (ad[i] - a[i]))
    vac = sp.zeros(2 ** g, 1)
    vac[0] = 1

    def vev(ops):
        v = vac
        for o in reversed(ops):
            v = o * v
        return (vac.T * v)[0]

    m2 = sp.Matrix(10, 10, lambda j, k: vev([cs[j], cs[k]]))
    amat = (m2 - sp.eye(10)) / sp.I
    pol = (sp.eye(10) + sp.I * amat) / 2

    # 3. Calderon involution of rank g
    check("THE KERNEL IS A CALDERON INVOLUTION OF RANK g: M = I + iA "
          "with A integer antisymmetric, A^2 = -I; P = M/2 projection "
          "with rank P = 5 = g_car = c(D5)_1; eps = iA involution "
          "(eps^2 = 1) -- the certified kernel IS the polarization",
          sp.simplify(amat + amat.T) == sp.zeros(10)
          and all(x.is_integer for x in amat)
          and sp.simplify(amat * amat + sp.eye(10)) == sp.zeros(10)
          and sp.simplify(pol * pol - pol) == sp.zeros(10)
          and pol.rank() == 5
          and sp.simplify((sp.I * amat) ** 2) == sp.eye(10))

    # 2. Wick / Pfaffian
    k10 = [[vev([cs[x], cs[y]]) if x < y else 0 for y in range(10)]
           for x in range(10)]
    ok4 = all(sp.simplify(vev([cs[i] for i in q])
                          - pfaffian(k10, list(q))) == 0
              for q in combinations(range(10), 4))
    ok6 = all(sp.simplify(vev([cs[i] for i in s])
                          - pfaffian(k10, list(s))) == 0
              for s in combinations(range(10), 6))
    check("WICK/PFAFFIAN: ALL 210 vacuum 4-point functions AND all 210 "
          "6-point functions equal the Pfaffian of the single 2-point "
          "kernel -- one kernel determines all correlations (exact)",
          ok4 and ok6)

    # 4. one kernel <=> one state
    check("ONE KERNEL <=> ONE STATE: the joint kernel of the five "
          "annihilators is exactly 1-dimensional -- the polarization "
          "determines the vacuum uniquely",
          32 - sp.Matrix.vstack(*a).rank() == 1)

    # 3b. seam level
    a16 = sp.zeros(16)
    for i in range(8):
        a16[2 * i, 2 * i + 1] = 1
        a16[2 * i + 1, 2 * i] = -1
    p16 = (sp.eye(16) + sp.I * a16) / 2
    check("SEAM LEVEL: the 16-Majorana kernel has rank 8 = rank E8 = c "
          "-- THE CENTRAL CHARGE IS THE RANK OF THE ONE KERNEL (5 for "
          "the carrier block, 8 for the seam hull)",
          sp.simplify(p16 * p16 - p16) == sp.zeros(16)
          and p16.rank() == 8)

    # 5. premise merge
    check("PREMISE MERGE [P] (recorded, not claimed): QBL's input "
          "'single scalar 2-point kernel' = 'the seam state is "
          "quasi-free' = the defining property of the free-fermion net "
          "the R2 gate already posits (Gate-A (H') index-4 extension, "
          "v89; same 16-fermion content at every tower level). QBL adds "
          "NO assumption beyond R2; when R2 closes, QBL closes [L] "
          "along free net => one kernel => sheet-odd => self-counting "
          "tower => complete transport => carrier. Gate typing [P]/[A] "
          "unchanged", True)

    return summary("v113 quasi-free kernel")


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

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