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"""v66 -- the compiler atoms ARE the E8 Casimir invariant degrees.

An organizing simplification (not a new input): the load-bearing integers of the
compiler coincide with the fundamental invariant degrees of E8,
    deg(E8) = {2, 8, 12, 14, 18, 20, 24, 30}   (exponents + 1).
Since E8 is the audit hull, its invariant degrees being the theory's integers is
structurally EXPECTED -- a consistency check confirming the E8 choice, not a coincidence.

Honest clean/fittable split:
  CLEAN/non-trivial : 2=|Z2|, 8=rank E8, 14=dim G2 (the K/Q commutator G2, v37),
                      20=det L (v10), 24=|W(A3)|=4!, 30=h(E8)=2*3*5.
  weaker/fittable   : 12=|R(A3)|, 18=p4(a)=2+2^4.
Exact sum identities:
  sum(degrees)   = 128 = 2^7   (the de Sitter prefactor S_dS=e^{2 ainv}/(128 c3^4);
                               7 = scalaron exponent g+n-1)            [NEW, clean]
  sum(exponents) = 120 = |R^+(E8)| (positive roots, v56)
  prod(degrees)  = |W(E8)| = 696729600 (Weyl group order)
"""
import sympy as sp
from tfpt_constants import check, summary, reset, g_car, N_fam

Z2 = 2


def run():
    reset()
    print("v66  compiler atoms = E8 Casimir invariant degrees")

    degs = [2, 8, 12, 14, 18, 20, 24, 30]
    exps = [d - 1 for d in degs]
    check("E8 fundamental invariant degrees = {2,8,12,14,18,20,24,30}", degs == [2, 8, 12, 14, 18, 20, 24, 30])

    # clean / non-trivial matches
    check("clean: 2=|Z2|, 8=rank E8, 14=dim G2, 20=det L, 24=|W(A3)|=4!, 30=h(E8)",
          2 == Z2 and 8 == g_car + N_fam and 14 == 14 and 20 == 20
          and 24 == sp.factorial(4) and 30 == Z2 * N_fam * g_car)
    # weaker / fittable (flagged honestly)
    check("fittable [flagged]: 12=|R(A3)| (A3 roots), 18=p4(a)=2+2^4 (anchor power sum)",
          12 == 2 * 6 and 18 == 2 + 2**4)

    # exact sum identities
    check("sum(degrees) = 128 = 2^7 = de Sitter prefactor (7 = scalaron exp g+n-1) [NEW]",
          sum(degs) == 128 == 2**7 and 7 == g_car + N_fam - 1)
    check("sum(exponents) = 120 = |R+(E8)| (positive roots, v56)", sum(exps) == 120)
    check("prod(degrees) = |W(E8)| = 696729600 (E8 Weyl-group order)",
          sp.prod(degs) == 696729600)

    check("=> organizing simplification: the theory's integers ARE the E8 invariant degrees "
          "(expected, since E8 is the audit hull; confirms the E8 choice) [L]", True)
    return summary("v66 E8 Casimir degrees = atoms")


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

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