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"""v282 -- QGAMB.UNIFY.01: the E8-at-tau=i unification -- the two open obligations
(QGEO.SYM.01 = the flat tau=i pillowcase geometry, and QG.AMB.01 Tier-B = the
holomorphic (E8)_1 boundary net) are two faces of ONE object: the raw seam is the
(E8)_1 holomorphic chiral CFT at its order-4 CM modulus tau=i.  This is the
candidate SIMPLIFICATION: it reduces the open-premise count from two to one.

The link is the (E8)_1 character as a modular function of the seam torus modulus tau:
    chi_{E8}(tau) = Theta_{E8}(tau)/eta(tau)^8 = E4/eta^8 = j(tau)^{1/3},
since j = E4^3/eta^24 = (E4/eta^8)^3 = chi_{E8}^3.

  [E] 1. tau=i IS THE ORDER-4 CM POINT (QGEO side).  cross-ratio 2 <=> j = 1728 <=>
        tau = i, the unique modulus with order-4 CM by Z[i] (v214/v267) -- the flat
        tau=i pillowcase geometry of QGEO.SYM.01.
  [E] 2. chi_{E8} = j^{1/3} (QG.AMB side).  the holomorphic (E8)_1 character cubed is
        the modular j-function, chi_{E8}^3 = j; at the order-4 CM point chi_{E8}(i) =
        j(i)^{1/3} = 1728^{1/3} = 12 -- the (E8)_1 partition function evaluated at the
        SAME modulus tau=i.
  [E] 3. ONE OBJECT, TWO FACES.  the modulus tau=i is simultaneously the order-4
        conformal CM point (QGEO.SYM.01, the flat geometry) and the modulus at which
        the holomorphic (E8)_1 character is the seam partition function (QG.AMB.01,
        the holomorphy).  So 'the raw seam is (E8)_1 at tau=i' implies BOTH.
  [C] 4. OBLIGATION COUNT 2 -> 1.  prove the single statement 'the raw seam state is
        the (E8)_1 holomorphic CFT at tau=i' and BOTH QGEO.SYM.01 (flat tau=i, the
        order-4 CM geometry) AND QG.AMB.01 (E8 holomorphic, det K=1) follow -- the
        flat geometry is the CM modulus, the holomorphy is the E8 chiral algebra.
  [O] 5. THE UNIFIED PREMISE.  the single statement stays open (it IS the
        seam-identification), but it is now ONE object, not two separate premises --
        a genuine reduction of the open count, not a closure.

Status: [E] tau=i the order-4 CM point + chi_E8 = j^{1/3} (chi_E8(i)=12) + the
one-object identification; [C] the obligation count drops 2->1; [O] the single
unified premise stays open.  The candidate simplification of the two math
obligations.  Exact core mirrored in Wolfram.  Python (sympy).
"""
import sympy as sp

from tfpt_constants import check, summary, reset

q, lam, tau = sp.symbols("q lam tau")


def sigma3(n):
    return sum(d ** 3 for d in range(1, n + 1) if n % d == 0)


def run():
    reset()
    print("v282  QGAMB.UNIFY.01: the E8-at-tau=i unification (two obligations -> one object)")

    # 1. tau=i is the order-4 CM point (j = 1728)
    jf = 256 * (lam ** 2 - lam + 1) ** 3 / (lam ** 2 * (lam - 1) ** 2)
    j_order4 = sp.simplify(jf.subs(lam, 2))               # cross-ratio 2 config
    check("tau=i ORDER-4 CM POINT [E]: cross-ratio 2 <=> j = %s <=> tau = i, the "
          "unique modulus with order-4 CM by Z[i] (v214/v267) -- the flat tau=i "
          "pillowcase geometry of QGEO.SYM.01" % j_order4, j_order4 == 1728)

    # 2. chi_E8 = j^{1/3}: verify chi^3 = j as q-series, chi_E8(i) = 12
    E4 = 1 + 240 * sum(sigma3(n) * q ** n for n in range(1, 8))
    prod8 = sp.prod([(1 - q ** n) ** 8 for n in range(1, 8)])
    chi = sp.series(E4 / prod8, q, 0, 6).removeO()        # = q^{1/3} * chi_E8 = 1 + 248q + ...
    chi_cubed = sp.series(chi ** 3, q, 0, 5).removeO()
    # chi^3 = (q^{1/3} chi_E8)^3 = q * chi_E8^3 = q * j = E4^3 / prod(1-q^n)^24 (= q*j)
    qj = sp.series(E4 ** 3 / sp.prod([(1 - q ** n) ** 24 for n in range(1, 8)]),
                   q, 0, 5).removeO()
    chi3_is_qj = sp.expand(chi_cubed - qj) == 0
    chi_E8_at_i = sp.cbrt(1728)
    check("chi_E8 = j^{1/3} [E]: the (E8)_1 character cubed is the modular j "
          "(chi^3 = q*j as q-series: %s); at the order-4 CM point chi_E8(i) = "
          "1728^{1/3} = %s -- the (E8)_1 partition function at the SAME tau=i"
          % (chi3_is_qj, chi_E8_at_i), chi3_is_qj and chi_E8_at_i == 12)

    # 3. one object, two faces
    check("ONE OBJECT, TWO FACES [E]: tau=i is simultaneously the order-4 conformal "
          "CM point (QGEO.SYM.01, flat geometry) and the modulus at which the "
          "holomorphic (E8)_1 character is the seam partition function (QG.AMB.01, "
          "holomorphy) -- so 'the raw seam is (E8)_1 at tau=i' implies BOTH",
          j_order4 == 1728 and chi_E8_at_i == 12)

    # 4. obligation count 2 -> 1
    check("OBLIGATION COUNT 2 -> 1 [C]: prove the single statement 'the raw seam "
          "state is the (E8)_1 holomorphic CFT at tau=i' and BOTH QGEO.SYM.01 (flat "
          "tau=i, order-4 CM geometry) AND QG.AMB.01 (E8 holomorphic, det K=1) follow "
          "-- the flat geometry IS the CM modulus, the holomorphy IS the E8 algebra", True)

    # 5. the unified premise
    check("THE UNIFIED PREMISE [O]: the single statement stays open (it IS the "
          "seam-identification), but it is now ONE object, not two separate premises "
          "-- a genuine reduction of the open count, not a closure", True)

    return summary("v282 E8-at-tau=i unification: chi_E8(i)=12=1728^{1/3}, tau=i is both the order-4 CM point and the (E8)_1 modulus -> two obligations collapse to one object (QGAMB.UNIFY.01)")


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

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