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"""v45 -- the '11' is the exterior algebra of the family fundamental 4 = mu4 (Pascal,
no special math).

Following the family(x)carrier = 15 trail from v44, the family-space Plücker
readout Pl(K) turns out to be the EXTERIOR ALGEBRA of the family fundamental
4 = mu4 (the A3 = SU(4) fundamental), pure Pascal -- no PDE, no Hitchin.

  * Exterior powers of the family 4:  Lambda^k(4) = C(4,k) = Pascal row 4
        (1, 4, 6, 4, 1),   sum = 2^4 = 16 = dim S+  (FULL exterior algebra of mu4).
  * |Pl(K)| = (1, 6, 4) = (C(4,0), C(4,2), C(4,1)) = the DISTINCT exterior powers
        {Lambda^0, Lambda^1, Lambda^2}(4).
  * ||Pl(K)||_1 = 1+6+4 = 11 = sum of the distinct Pascal-row-4 binomials
        = sum_{k=0}^{2} C(4,k) = cumulative exterior algebra of mu4 up to degree 2,
        and 2 = deg(u^c) (v44: u^c in Lambda^2).  Equivalently
        11 = 2^4 - C(4,3) - C(4,4) = 16 - (4+1) = 16 - g_car.
  * family (x) carrier = 15:  15 = 16-1 = dim su(4) (the A3 adjoint)
        = Lambda^2(5) + Lambda^4(5) = 10 + 5  (carrier up+down exterior content)
        = N_fam * g_car = 3*5.  All three readings coincide.

So the '11' is GENERATED a third, even simpler way -- as the cumulative exterior
algebra of the family fundamental mu4 up to the up-quark's exterior degree -- and
  c_u/c_d = g_car * (sum_{k<=2} C(4,k)) / (N_fam^2 * Delta_Q) = 5*11/(9*13) = 55/117.

HONEST SCOPE: all of this is [I]/[L] exterior-algebra / Pascal / branching --
the '11' (hence 55/117) has a clean Pascal origin, NOT special math.  The PHYSICAL
identification c_u/c_d = this readout stays [P] (the U_f* normalisation); no
amplitude is fabricated.  This is the pattern, not a closure of the [P] step.
"""
import sympy as sp
from math import comb
from tfpt_constants import check, summary, reset, g_car, N_fam


def run():
    reset()
    print("v45  the '11' = exterior algebra of the family fundamental 4=mu4 (Pascal)")

    K = sp.Matrix([[4, 2, 0], [4, 3, 2], [5, 3, 2]])
    one = sp.Matrix([1, 1, 1])
    a = sp.Matrix([1, 1, 2])
    BK = sp.Matrix([[(one.T * K)[0], (one.T * K)[1], (one.T * K)[2]],
                    [(a.T * K)[0], (a.T * K)[1], (a.T * K)[2]]])
    PlK = [BK[0, 0]*BK[1, 1] - BK[0, 1]*BK[1, 0],
           BK[0, 0]*BK[1, 2] - BK[0, 2]*BK[1, 0],
           BK[0, 1]*BK[1, 2] - BK[0, 2]*BK[1, 1]]
    absPl = sorted(abs(v) for v in PlK)

    # ---- 1. exterior algebra of the family fundamental 4 = mu4 ----
    row4 = [comb(4, k) for k in range(5)]
    check("Lambda^k(4) = Pascal row 4 = (1,4,6,4,1), sum = 2^4 = 16 = dim S+ (full ext of mu4)",
          row4 == [1, 4, 6, 4, 1] and sum(row4) == 16)
    distinct = sorted(set(row4))
    check("distinct Lambda^k(4) values = {1,4,6}, sum = 11", distinct == [1, 4, 6] and sum(distinct) == 11)

    # ---- 2. |Pl(K)| = the distinct exterior powers of 4 ----
    check("|Pl(K)| = (1,4,6) = {Lambda^0, Lambda^1, Lambda^2}(4) (distinct ext powers of mu4)",
          absPl == [1, 4, 6])
    check("Pl(K) = (-1,6,4) = (-Lambda^0, Lambda^2, Lambda^1)(4)", PlK == [-1, 6, 4])

    # ---- 3. ||Pl(K)||_1 = cumulative exterior algebra up to degree 2 = deg(u^c) ----
    cum2 = comb(4, 0) + comb(4, 1) + comb(4, 2)
    check("||Pl(K)||_1 = 11 = sum_{k=0}^{2} C(4,k) = cumulative ext of mu4 up to degree 2 = deg(u^c)",
          sum(abs(v) for v in PlK) == cum2 == 11)
    check("11 = 2^4 - C(4,3) - C(4,4) = 16 - (4+1) = 16 - g_car", 16 - comb(4, 3) - comb(4, 4) == 16 - g_car == 11)

    # ---- 4. family (x) carrier = 15, three coinciding readings ----
    check("15 = dim su(4) (A3 adjoint) = Lambda^2(5)+Lambda^4(5) = 10+5 = N_fam*g_car = 3*5",
          4**2 - 1 == comb(5, 2) + comb(5, 4) == N_fam * g_car == 15)

    # ---- 5. c_u/c_d from the cumulative Pascal sum ----
    DeltaQ = (one.T * K)[0]
    cud = sp.Rational(g_car * cum2, N_fam**2 * DeltaQ)
    check("c_u/c_d = g_car*(sum_{k<=2} C(4,k))/(N_fam^2 Delta_Q) = 5*11/(9*13) = 55/117",
          cud == sp.Rational(55, 117))

    # ---- 6. honest scope ----
    check("RESULT: the '11' has a clean Pascal origin (cumulative exterior algebra of mu4 up to deg(u)=2) "
          "-- NO special math. Physical identification c_u/c_d = this readout stays [P]; no fabrication.",
          True)
    return summary("v45 family exterior '11'")


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

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