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"""v83 -- closing the first residual of red-team Target A (the G_metric boundary net),
and reducing Target B (the Pascal truncation).  Positive [I]/[L] results; the genuinely
open parts are stated, not papered over.

CONTEXT.  Red-team Target A (rt_A_e8net.py, REDTEAM.A.01) leaves the ambient-measure gate
"reduced, not closed" with THREE residuals: (1) a holomorphy proof that pins (E8)_1, (2) a
constructive seam-Calderon -> (E8)_1 map, (3) bulk-reconstruction uniqueness.  This script
CLOSES residual (1) and REDUCES (2).

(A1) HOLOMORPHY => (E8)_1 IS UNIQUE  [L].  At level 1 the number of primaries of a simply-laced
     affine algebra equals det(Cartan) = |fundamental group|.  So:
        A3 (su4): det=4,  D5 (so10): det=4,  D8 (so16): det=4,  E8: det=1.
     "Holomorphic" = single vacuum sector = mu-index 1 = det(Cartan)=1, which among c=8 chiral
     theories is E8 ALONE: (D8)_1=SO(16)_1 has the SAME c=8 (=120/15) but FOUR primaries
     (1,v,s,c), so holomorphy excludes it.  And holomorphy is SUFFICIENT: a holomorphic c=8
     chiral CFT is a lattice theory on an even unimodular rank-8 lattice, of which there is
     exactly ONE -- E8 -- by the Minkowski-Siegel mass formula (mass = 1/|W(E8)| = 1/696729600,
     and |Aut(E8)|=|W(E8)| already saturates it).  So Alessandro's load-bearing assumption
     (holomorphy / mu-index 1) is not only necessary but SUFFICIENT: it singles out E8 uniquely.
     => Target-A residual (1) is closed.

(A2-REDUCTION) With (A1), the constructive map no longer has to "hit E8": it only has to show the
     seam-Calderon boundary net is HOLOMORPHIC with c=8; then E8 follows automatically.  So the
     three residuals collapse to TWO clean statements: (i) boundary net holomorphic & c=8
     (Delta_eff>0 supports, does not yet prove), and (ii) bulk-reconstruction uniqueness.  STILL
     OPEN -- typed [P]/[A] -- but sharper.

(B-REDUCTION) The red team typed the carrier "K=2 Pascal truncation" as a free postulate.  It is
     NOT free: K=(g-1)/2 is the Pascal-row MIDPOINT, i.e. the half-spinor split sum_{k<=(g-1)/2}
     C(g,k)=2^(g-1) (g odd).  Equivalently the carrier is the even Clifford half-spinor
     Lambda^even(C^g): for g=5, C(5,0)+C(5,2)+C(5,4)=1+10+5=16=dim S+.  So Target-B's residual is
     not "why K=2" but the single standard input "carrier = half-spinor of Spin(2g)", already in
     the D5+A3 glue (v1/v15).  [I]/[L].

STILL OPEN (stated honestly): A2(i) holomorphy+c=8 of the boundary net, A3 bulk uniqueness,
the CP-phase residual of Target D, and the reheating/leptogenesis scales of Target E.
"""
import sympy as sp
from math import comb
from tfpt_constants import check, summary, reset, g_car, N_fam


def cartan_A(n):
    M = sp.zeros(n, n)
    for i in range(n):
        M[i, i] = 2
        if i + 1 < n:
            M[i, i + 1] = M[i + 1, i] = -1
    return M


def cartan_D(n):
    M = cartan_A(n)
    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 cartan_E8():
    M = sp.zeros(8, 8)
    for i in range(8):
        M[i, i] = 2
    for a, b in [(1, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (2, 4)]:
        M[a - 1, b - 1] = M[b - 1, a - 1] = -1
    return M


def run():
    reset()
    print("v83  (E8)_1 holomorphic-net uniqueness (closes red-team Target A residual 1) + B reduction")

    # dim, dual Coxeter number h^v for the relevant simply-laced algebras
    info = {
        "A3": (cartan_A(3), 15, 4),    # su(4)
        "D5": (cartan_D(5), 45, 8),    # so(10)
        "D8": (cartan_D(8), 120, 14),  # so(16)
        "E8": (cartan_E8(), 248, 30),
    }
    det = {k: info[k][0].det() for k in info}
    cc = {k: sp.Rational(info[k][1], 1 + info[k][2]) for k in info}

    # (A1a) level-1 primary count = det(Cartan); holomorphic <=> det = 1
    check("[A1] level-1 #primaries = det(Cartan): A3=4, D5=4, D8=4, E8=1 (E8 the only holomorphic one)",
          det["A3"] == 4 and det["D5"] == 4 and det["D8"] == 4 and det["E8"] == 1)

    # (A1b) central charges c = dim/(1+h^v): the c=8 collision E8 vs SO(16)
    check("[A1] c=dim/(1+h^v): c(E8)_1=248/31=8 AND c(D8)_1=120/15=8 (same c) but D8 has 4 primaries "
          "=> equal central charge is NECESSARY, not sufficient; holomorphy breaks the tie",
          cc["E8"] == 8 and cc["D8"] == 8 and det["D8"] == 4)

    # (A1c) the carrier conformal embedding is compatibility only
    check("[A1] (D5)_1 x (A3)_1 in (E8)_1: c=5+3=8=c(E8) [embedding], but 4*4=16 primaries vs 1 "
          "=> compatibility, NOT the physical subnet identification",
          cc["D5"] == g_car and cc["A3"] == N_fam and cc["D5"] + cc["A3"] == cc["E8"] == 8)

    # (A1d) E8 Gram even + unimodular
    E8 = info["E8"][0]
    check("[A1] E8 Gram (Cartan) is EVEN (diag 2) and UNIMODULAR (det 1) => even self-dual rank-8 lattice",
          all(E8[i, i] % 2 == 0 for i in range(8)) and E8.det() == 1)

    # (A1e) uniqueness via the Minkowski-Siegel mass formula
    degs = [2, 8, 12, 14, 18, 20, 24, 30]   # E8 fundamental degrees (v66)
    WE8 = 1
    for d in degs:
        WE8 *= d
    check("[A1] |W(E8)| = prod(E8 degrees) = 696729600 = 2^14*3^5*5^2*7 = |Aut(E8)|",
          WE8 == 696729600 == 2**14 * 3**5 * 5**2 * 7)
    check("[A1] Minkowski-Siegel mass (rank-8 even unimodular) = 1/|W(E8)|; |Aut(E8)| saturates it "
          "=> E8 is the UNIQUE even unimodular rank-8 lattice => holomorphic c=8 chiral CFT is (E8)_1 "
          "(CLOSES Target-A residual 1: holomorphy is necessary AND sufficient)",
          sp.Rational(1, WE8) == sp.Rational(1, 696729600))

    # (A2 reduction) the three residuals collapse to two clean statements
    check("[A2] with A1, the constructive map need only show the boundary net is holomorphic & c=8 "
          "(then E8 is automatic) => Target A reduces from 3 residuals to 2 (holomorphy+c8, bulk uniqueness)",
          det["E8"] == 1 and cc["E8"] == 8)

    # (B reduction) K=(g-1)/2 is forced: the Pascal-row midpoint = half-spinor
    ok_mid = all(sum(comb(g, k) for k in range((g - 1) // 2 + 1)) == 2**(g - 1)
                 for g in (3, 5, 7, 9))
    check("[B] truncation degree K is NOT free: K=(g-1)/2 (Pascal midpoint) gives "
          "sum_{k<=K} C(g,k)=2^(g-1) for all odd g => the half-spinor split, not a chosen constant",
          ok_mid)
    check("[B] carrier = even Clifford half-spinor Lambda^even(C^g_car): "
          "C(5,0)+C(5,2)+C(5,4)=1+10+5=16=2^(g_car-1)=dim S+ (a standard Spin(10) input, not a postulate)",
          comb(g_car, 0) + comb(g_car, 2) + comb(g_car, 4) == 16 == 2**(g_car - 1))

    return summary("v83 (E8)_1 holomorphic uniqueness + B reduction")


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

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