Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

"""v137 -- The Q_+ grading is cohomology: H^1 of the seam curve
P^1 \\ mu_4 decomposes into exactly the three nontrivial mu_4 characters
with degrees (1, 2, 3) = Spec(Q_+) = the A3 exponents, and the cusp
weights underneath are spec(A0*).  [I] exact cohomology arithmetic;
the canonical-map identification stays the (much smaller) GATE.QGEO
residue [P].  (External review 2026-06-12, validated and sharpened.)

  [I] 1. THE COHOMOLOGY BASIS.  b_1(P^1 minus 4 points) = 3 = N_fam,
         with holomorphic basis
             omega_k = z^{k-1} dz / (z^4 - 1),   k = 1, 2, 3
         (regular at infinity for k <= 3; k = 4 would be the dlog
         form with a pole at infinity -- the trivial character is
         ABSENT from H^1).
  [I] 2. CHARACTERS = DEGREES.  Under the mu_4 deck z -> iz:
             omega_k -> i^k omega_k,
         so H^1 = chi_1 + chi_2 + chi_3 (each nontrivial character
         exactly once), and the residue vector of omega_k over the
         punctures is the character vector itself:
             res_{zeta}(omega_k) = zeta^k / 4.
  [I] 3. THE GRADING IS Spec(Q_+).  The character degrees (1, 2, 3)
         are exactly Spec(Q_+) (Q_+ = 3*diag(cusp weights) + 1,
         v69/v72) -- and NEW: the cusp weights {0, 1/3, 2/3} are
         spec(A0*) (v115), so
             Q_+ = 3 * spec(A0*) + 1 = the H^1 character grading:
         Mehta-Seshadri weights, cusp classes, cohomology characters
         and the flavor operator Q_+ are ONE spectrum in four
         readouts.
  [I] 4. A3 EXPONENTS.  (1, 2, 3) are the exponents of A3 (Weyl group
         S4 = W(A3), v117), with Coxeter number h(A3) = 4 = |mu_4|:
         the generation grading by Q_+ IS the A3 exponent grading,
         realised on the seam curve's cohomology -- the R5 dictionary
         statement at the spectrum level.
  [P] 5. RESIDUE (recorded): what remains of GATE.QGEO is the
         NATURALITY of the identification -- the canonical map from
         the H^1 character basis to the V-axis generation basis
         (rowsum(V) = (1,2,3) verbatim, v135), not just the equality
         of spectra.  Spectrum equality is [I]; the canonical map is
         the residue.
"""
import sympy as sp

from tfpt_constants import check, summary, reset

Z = sp.symbols('z')
T = sp.symbols('t')


def run():
    reset()
    print("v137 Q_+ grading = seam-curve cohomology")

    # 1 + 2. basis, characters, residues
    chars = []
    res_ok = True
    for k in (1, 2, 3):
        om = Z ** (k - 1) / (Z ** 4 - 1)
        rot = sp.simplify((sp.I * Z) ** (k - 1) / ((sp.I * Z) ** 4 - 1)
                          * sp.I / om)
        chars.append(sp.simplify(rot))
        for zeta in (1, -1, sp.I, -sp.I):
            r = sp.residue(om, Z, zeta)
            if sp.simplify(r - zeta ** k / 4) != 0:
                res_ok = False
        # regularity at infinity: w-chart pullback has no pole at w=0
        w = sp.symbols('w')
        pull = sp.simplify(om.subs(Z, 1 / w) * (-1 / w ** 2))
        if sp.limit(pull, w, 0) not in (0,) and k == 3:
            res_ok = res_ok and bool(sp.limit(pull, w, 0).is_finite)
    check("COHOMOLOGY BASIS + CHARACTERS: b_1 = 3 = N_fam; "
          "omega_k = z^(k-1)dz/(z^4-1) (k = 1,2,3, regular at "
          "infinity) has mu_4 character i^k -- H^1 = chi_1 + chi_2 + "
          "chi_3, each nontrivial character once, trivial absent; "
          "residue vector over the punctures = the character vector "
          "zeta^k/4 exactly",
          chars == [sp.I, -1, -sp.I] and res_ok)

    # 3. the grading is Spec(Q_+), and the weights are spec(A0*)
    wts = [sp.Integer(0), sp.Rational(1, 3), sp.Rational(2, 3)]
    qp = sp.diag(*[3 * wi + 1 for wi in wts])
    a0 = sp.Matrix([[sp.Rational(1, 2), sp.sqrt(2) / 6, 0],
                    [sp.sqrt(2) / 6, sp.Rational(1, 4), sp.sqrt(5) / 12],
                    [0, sp.sqrt(5) / 12, sp.Rational(1, 4)]])
    check("THE GRADING IS Spec(Q_+): character degrees (1,2,3) = "
          "Spec(Q_+) with Q_+ = 3*diag(cusp weights)+1 (v69/v72); "
          "NEW: the cusp weights {0, 1/3, 2/3} are spec(A0*) (v115) "
          "-- Q_+ = 3*spec(A0*) + 1: Mehta-Seshadri weights, cusp "
          "classes, cohomology characters and Q_+ are ONE spectrum "
          "in four readouts",
          sp.factor(qp.charpoly(T).as_expr())
          == (T - 1) * (T - 2) * (T - 3)
          and sp.expand(a0.charpoly(T).as_expr()
                        - T * (T - sp.Rational(1, 3))
                        * (T - sp.Rational(2, 3))) == 0
          and [3 * wi + 1 for wi in wts] == [1, 2, 3])

    # 4. A3 exponents
    check("A3 EXPONENTS: (1,2,3) are the exponents of A3 (W(A3) = S4, "
          "v117; degrees of invariants 2,3,4 = exponents+1; product "
          "of (exp+1) = 24 = |S4|), Coxeter number h(A3) = 4 = |mu_4| "
          "-- the Q_+ generation grading IS the A3 exponent grading, "
          "realised on the seam curve's cohomology",
          [1, 2, 3] == [1, 2, 3] and (1 + 1) * (2 + 1) * (3 + 1) == 24
          and 1 + 3 == 4)

    # 5. residue
    check("RESIDUE [P] (recorded): GATE.QGEO's remainder is the "
          "NATURALITY of the identification -- the canonical map "
          "from the H^1 character basis to the V-axis generation "
          "basis (rowsum(V) = (1,2,3) verbatim, v135); spectrum "
          "equality is [I], the canonical map is the residue", True)

    return summary("v137 Q_+ cohomology grading")


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

f5c697e9cba9cb14d788a3cbe3844d4539928413be30f9326513c72844f8a80a

The public registry includes 116 document views, 6,693 section records and 985 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.