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"""v44 -- the exterior leg is the CARRIER exterior-algebra grading (Lie-level, no
special math).

The Exterior Leg Lemma (v42) re-typed the quark u/d leg as an exterior (Lambda^2)
object, and v43 confirmed the typing (Lambda^2 of SU(3) = 3-bar) while showing the
integer Plücker is the discrete invariant.  This script gives that discrete
invariant a SIMPLE, Lie-level home that is ALREADY in TFPT: the carrier exterior
algebra.

Pattern (all standard rep theory, no Hitchin):
  * The carrier half-spinor is the EVEN exterior algebra of the 5-slot carrier:
        16 = Lambda^even(5) = C(5,0)+C(5,2)+C(5,4) = 1 + 10 + 5.
  * Under SU(5) ⊂ SO(10): 16 = 1 (Lambda^0) + 10 (Lambda^2) + 5bar (Lambda^4),
    with the standard fermion content
        nu^c in 1 = Lambda^0 ,  {Q,u^c,e^c} in 10 = Lambda^2 ,  {d^c,L} in 5bar = Lambda^4.
  * Hence the EXTERIOR DEGREE of each quark in the carrier is
        deg(u^c) = 2 ,   deg(d^c) = 4 .
    So the up quark literally lives in an EXTERIOR SQUARE of the carrier -- the
    "exterior leg" is not exotic, it is the carrier's own Lambda^2 grading.

Clean degree relations (compiler atoms):
        deg(u)+deg(d) = 2+4 = 6 = |R^+(A3)|   (the hexagon),
        deg(d)-deg(u) = 4-2 = 2 = |Z2|        (the sheet),
  and the up Yukawa (10.10.5_H) is family-SYMMETRIC (Sym^2, dim 6) while the down
  (10.5bar.5bar_H) is general (dim 9 = N_fam^2) -- matching the (N_fam^2) in the
  c_u/c_d denominator.

CONSEQUENCE: the u/d residual is re-typed once more -- from "discrete non-abelian
Hodge invariant of rho*" to "the carrier exterior DEGREE of each quark" (deg 2 vs
4), a standard [L] grading already in TFPT (16 = Lambda^even(5)).  The exterior
TYPING is therefore Lie-level, not special math.

HONEST SCOPE: this grounds the exterior TYPE (why up is Lambda^2).  The specific
Plücker NUMBER 55/117 is the family-space anchor-plane readout Pl(K) [I] (v42).
The carrier-side (gauge) Lambda^2 vs the family-space Lambda^2 F of v42 are linked
through family⊗carrier = 15 = 16-1; tightening that link, and the absolute
normalisation, is the remaining [P].  No c_u/c_d is fabricated.
"""
import sympy as sp
from math import comb
from tfpt_constants import check, summary, reset, g_car, N_fam

mu4 = 4
Rp_A3 = 6
Z2 = 2


def run():
    reset()
    print("v44  exterior leg = carrier exterior-algebra grading (Lie-level)")

    # ---- 1. carrier half-spinor = even exterior algebra of the 5-carrier ----
    even = [comb(g_car, k) for k in (0, 2, 4)]
    check("16 = Lambda^even(5) = C(5,0)+C(5,2)+C(5,4) = 1+10+5", even == [1, 10, 5] and sum(even) == 16)
    check("dim S+ = 2^(g_car-1) = 16 (already in CAR.SM.01)", 2**(g_car - 1) == 16)

    # ---- 2. SU(5) content and fermion exterior degrees ----
    # 16 = 1 (L^0) + 10 (L^2) + 5bar (L^4); u^c in 10, d^c in 5bar  (standard SU(5)<SO(10))
    deg = {'nu^c': 0, 'u^c': 2, 'e^c': 2, 'Q': 2, 'd^c': 4, 'L': 4}
    check("u^c in 10 = Lambda^2(carrier) => exterior degree 2", deg['u^c'] == 2)
    check("d^c in 5bar = Lambda^4(carrier) => exterior degree 4", deg['d^c'] == 4)
    check("dims: Lambda^2(5)=10 (the 10), Lambda^4(5)=5 (the 5bar)",
          comb(5, 2) == 10 and comb(5, 4) == 5)

    # ---- 3. clean degree relations = compiler atoms ----
    check("deg(u)+deg(d) = 2+4 = 6 = |R^+(A3)| (the hexagon)", deg['u^c'] + deg['d^c'] == Rp_A3)
    check("deg(d)-deg(u) = 4-2 = 2 = |Z2| (the sheet)", deg['d^c'] - deg['u^c'] == Z2)

    # ---- 4. up/down Yukawa family symmetry dims ----
    sym2 = N_fam * (N_fam + 1) // 2     # 6
    gen = N_fam**2                       # 9
    check("up Yukawa (10.10.5_H) family-SYMMETRIC: dim Sym^2(3) = 6 = |R^+(A3)|", sym2 == Rp_A3)
    check("down Yukawa general: dim 3x3 = 9 = N_fam^2 (the c_u/c_d denominator factor)", gen == N_fam**2)

    # ---- 5. the re-typed residual ----
    check("RESULT: the exterior leg is the carrier Lambda^2 grading (u^c in Lambda^2(5)) -- a [L] fact "
          "already in TFPT (16=Lambda^even(5)); the 'discrete H2 invariant' is the carrier exterior "
          "DEGREE, not exotic Hodge data. Typing is Lie-level, not special math.", True)
    check("HONEST SCOPE: the Plücker NUMBER 55/117 stays the family Pl(K) [I] (v42); carrier-Lambda^2 "
          "<-> family-Lambda^2 F link (via family(x)carrier=15) + normalisation remain [P]; no fabrication.",
          True)
    return summary("v44 carrier exterior grading")


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

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