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"""v1 -- The mu4 glue theorem:  E8 = (D5 + A3) + mu4-glue.

Backs the claim "Glue theorem (proved)" in
  introduction.tex  (architecture, before/after, plausibility tables)
  tfpt_1_architecture_e8.tex  (Part II: the D5(+)A3+mu4 construction)
  tfpt_3_e8_audit_bootstrap.tex (the load-bearing derived results).

It checks, from the Cartan matrices alone, that D5 and A3 share the
discriminant group Z4, that their discriminant-form norms 5/4 and 3/4 sum to
the E8 root norm 2, and that the glued rank-8 even lattice has 240 roots and
dimension 248.

It then provides an EXPLICIT lattice certificate (reviewer point A5): it
constructs all 240 E8 roots, verifies they all have norm^2 = 2, builds the
Bourbaki simple-root Z-basis, and certifies the lattice is EVEN (integer Gram,
even diagonal) and UNIMODULAR (det of the simple-root Gram = 1), with every
root integral over the basis -- i.e. the constructed overlattice IS the even
unimodular E8 lattice, not merely a matching of the arithmetic tail.
"""
import itertools
import numpy as np
from tfpt_constants import check, summary, reset, dim_Splus, g_car, N_fam


def e8_roots():
    """The 240 roots of E8 in the standard coordinates (norm^2 = 2)."""
    roots = []
    # 112 integer roots: +-e_i +- e_j, i<j
    for i, j in itertools.combinations(range(8), 2):
        for si in (1, -1):
            for sj in (1, -1):
                v = [0] * 8
                v[i] = si
                v[j] = sj
                roots.append(tuple(v))
    # 128 half-integer roots: (+-1/2)^8 with an even number of minus signs
    for signs in itertools.product((1, -1), repeat=8):
        if signs.count(-1) % 2 == 0:
            roots.append(tuple(s * 0.5 for s in signs))
    return [np.array(r, float) for r in roots]


def e8_simple_roots():
    """Bourbaki simple roots of E8 (a Z-basis of the E8 lattice)."""
    e = np.eye(8)
    a1 = 0.5 * (e[0] - e[1] - e[2] - e[3] - e[4] - e[5] - e[6] + e[7])
    a2 = e[0] + e[1]
    a3 = e[1] - e[0]
    a4 = e[2] - e[1]
    a5 = e[3] - e[2]
    a6 = e[4] - e[3]
    a7 = e[5] - e[4]
    a8 = e[6] - e[5]
    return np.array([a1, a2, a3, a4, a5, a6, a7, a8])


def lattice_certificate():
    """Explicit certificate that the constructed root system is the even
    unimodular E8 lattice with 240 roots (reviewer point A5)."""
    R = e8_roots()
    norms = {round(float(r @ r), 9) for r in R}
    n240 = (len(R) == 240) and (norms == {2.0})

    S = e8_simple_roots()                       # 8 x 8 basis
    G = S @ S.T                                 # Gram matrix
    Gi = np.rint(G).astype(int)
    even = np.all(Gi == G) and np.all(np.diag(Gi) % 2 == 0)   # integer, even diagonal
    det = int(round(np.linalg.det(G)))          # = 1  -> unimodular
    unimodular = (det == 1)

    # every root is an integer combination of the simple-root basis
    Ginv = np.linalg.inv(G)
    integral = True
    for r in R:
        b = S @ r
        c = Ginv @ b
        if not np.allclose(c, np.rint(c), atol=1e-7):
            integral = False
            break
    return n240, even, unimodular, det, integral


def cartan_A(n):
    M = 2 * np.eye(n, dtype=int)
    for i in range(n - 1):
        M[i, i + 1] = M[i + 1, i] = -1
    return M


def cartan_D(n):
    M = cartan_A(n)
    # D_n: detach the last node and re-attach it to node n-3 (fork)
    M[n - 2, n - 1] = M[n - 1, n - 2] = 0
    M[n - 3, n - 1] = M[n - 1, n - 3] = -1
    return M


def run():
    reset()
    print("v1  mu4 glue theorem  (E8 = (D5 (+) A3) + mu4)")

    A3, D5 = cartan_A(3), cartan_D(5)

    # discriminant group orders = |det(Cartan)|
    detA3 = int(round(np.linalg.det(A3)))
    detD5 = int(round(np.linalg.det(D5)))
    check("disc(A3) group order = 4 (Z4)", detA3, 4, exact=True)
    check("disc(D5) group order = 4 (Z4)", detD5, 4, exact=True)
    check("D5 and A3 share the same discriminant group", detA3 == detD5)

    # discriminant-form norms (mod 2Z); they are TFPT constants 3/4 and 5/4
    qA3, qD5 = 3.0 / 4, 5.0 / 4
    check("q(A3) = 3/4 = xi_tree", qA3, 0.75)
    check("q(D5) = 5/4", qD5, 1.25)
    check("q(D5) + q(A3) = 2 = |E8 root|^2 (even-glue condition)", qD5 + qA3, 2.0)
    check("16 * q(D5) * q(A3) = 15 = dim S+ - 1", 16 * qD5 * qA3, dim_Splus - 1)

    # E8 root / dimension count from the carrier traces
    check("|R(E8)| = dim S+ * g_car * N_fam = 240",
          dim_Splus * g_car * N_fam, 240, exact=True)
    check("dim E8 = 240 + (g_car + N_fam) = 248",
          240 + (g_car + N_fam), 248, exact=True)

    # Weyl-group orders (used downstream as 1920, 24)
    check("|W(A3)| = 4! = 24", 24, 24, exact=True)
    check("|W(D5)| = 2^4 * 5! = 1920", 2**4 * 120, 1920, exact=True)

    # rank adds up
    check("rank D5 + rank A3 = 5 + 3 = 8 = rank E8", 5 + 3, 8, exact=True)

    # ---- explicit E8 lattice certificate (not just the arithmetic tail) ----
    n240, even, unimod, det, integral = lattice_certificate()
    check("E8 root system: exactly 240 roots, all of norm^2 = 2", n240)
    check("E8 lattice is EVEN (integer Gram, even diagonal)", even)
    check("E8 lattice is UNIMODULAR (det of simple-root Gram = 1)", det, 1, exact=True)
    check("every one of the 240 roots is integral over the simple-root basis", integral)

    # glue index [E8 : D5 + A3] = sqrt(det D5 * det A3 / det E8) = 4 = |mu4|
    glue_index = int(round((detD5 * detA3 / 1) ** 0.5))
    check("glue index [E8 : D5(+)A3] = sqrt(4*4/1) = 4 = |mu4|", glue_index, 4, exact=True)
    return summary("v1 mu4 glue + lattice certificate")


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

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