Follow the source behind the result.

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

"""v98 -- The Discriminant Dictionary DERIVED (P1 closed modulo GATE.QGEO):
the Q_+ cusp grading of the generations IS the A3 discriminant grading,
inside the established integer D4 model.  [I]/[L]; P1 carries no separate
[P] any more -- its residual coincides with the existing Q-geometry gate.

v97 left ONE [P]: the dictionary "Q_+ eigenspace grading of the generations
= A3 discriminant (Z4) grading", which would make T_A literally the v92
A3-side spinor conjugation.  This script DERIVES that dictionary within the
integer model the suite already owns (v69 D4 = Z4 |x Z2 on H_1, v97 T_A):

  [I] 1. THE GENERATION SPACE CARRIES AN INTEGER mu4.  G := T_A * Sigma
         acts on the Q_+ cusp eigenbasis e1 = (0,0,1) [cusp 0],
         e2 = (0,1,2) [cusp 1/3], e3 = (1,0,0) [cusp 2/3] as
             G e1 = -e1,    G e3 = e2,    G e2 = -e3,
         i.e. exactly the v69 decomposition B1 (+) E realised INTEGRALLY:
         e1 spans the B1 line (G-eigenvalue -1 = Z4 class 2), the plane
         span{e2,e3} is the E-plane (G-eigenvalues +-i = Z4 classes 1,3),
         G^4 = I, G^2 = -1 on E.
  [I] 2. THE TWO GRADINGS ALIGN.  The cusp-0 eigenvector (Q_+ eigenvalue 1)
         IS the unique self-conjugate Z4-class-2 line, and the swapped cusp
         pair {1/3, 2/3} IS the conjugate class pair {1,3}.  This is the
         dictionary -- derived, not assumed.
  [I] 3. T_A IS THE DISCRIMINANT CONJUGATION.  T_A G T_A^{-1} = G^{-1}
         (negation k -> -k on Z4 classes: fixes class 2, swaps 1 <-> 3),
         and the quadratic form respects it: q_{A3}(1) = q_{A3}(3) = 3/8
         (equal on the swapped pair), q_{A3}(2) = 1/2 (fixed); the A3-side
         negation swaps the two Lagrangian glues <(1,1)> <-> <(1,3)> (v92).
  [I] 4. THE TWO REFLECTION CLASSES SEPARATE THE PHYSICS.  Sigma also
         inverts G (Sigma G Sigma = G^{-1}) -- but Sigma = T_A G, so T_A
         and Sigma are the two D4 reflection classes (differing by one mu4
         rotation), and they are distinguished exactly by the B1 line:
         T_A e1 = +e1 (det T_A = -1, GLUE-SWAPPING parity, v92 single
         conjugation) vs Sigma e1 = -e1 (det Sigma = +1, glue-FIXING
         parity, simultaneous conjugation).  The deck involution of the
         cover is the T_A class.
  [L] 5. AUDIT: G a = e2 - e3 (one mu4 step maps the anchor onto the odd
         line), and the dihedral relations G^4 = (T_A)^2 = (T_A G)^2 = I
         close the same D4 as v69.

STATUS CONSEQUENCE (honest): the sheet question P1 carries NO separate [P]
residual any more.  What remains is exactly the already-open GATE.QGEO
realisation step (the parabolic-geometric construction that realises the
documented Q on P^1 minus mu4, v50/v69) -- one gate, not two.  Conditional
on that established typing, the sector -> sheet map is CLOSED: up/down are
the deck-odd pair, leptons sit on the ramification, and the deck involution
is the A3-side spinor conjugation realised as T_A.
"""
import sympy as sp
from fractions import Fraction as F

from tfpt_constants import check, summary, reset, N_fam

TA = sp.Matrix([[0, 1, 0], [1, 0, 0], [2, -2, 1]])
SIG = sp.diag(1, -1, -1)
G = TA * SIG
E1, E2, E3 = sp.Matrix([0, 0, 1]), sp.Matrix([0, 1, 2]), sp.Matrix([1, 0, 0])
A = sp.Matrix([1, 1, 2])

Q_A3 = lambda k: F(3 * k * k, 8) % 1
H_GLUE_1 = frozenset({(0, 0), (1, 1), (2, 2), (3, 3)})
H_GLUE_2 = frozenset({(0, 0), (1, 3), (2, 2), (3, 1)})


def run():
    reset()
    print("v98 discriminant dictionary (P1: cusp grading = A3 Z4 grading, derived)")

    t = sp.symbols('t')

    # 1. integer mu4 on the cusp basis = B1 (+) E realised integrally
    check("G = T_A*Sigma acts on the cusp basis as G e1 = -e1, G e3 = e2, "
          "G e2 = -e3 (B1 line + E-plane 90-degree rotation, INTEGER)",
          G * E1 == -E1 and G * E3 == E2 and G * E2 == -E3)
    check("G^4 = I, G^2 = -1 on the E-plane (eigenvalues {-1, i, -i} = "
          "nontrivial Z4 classes {2, 1, 3})",
          G**4 == sp.eye(3) and G * G * E2 == -E2 and G * G * E3 == -E3
          and sp.factor(G.charpoly(t).as_expr()) == (t + 1) * (t**2 + 1))

    # 2. the gradings align
    ker = (G + sp.eye(3)).nullspace()
    check("DICTIONARY: the cusp-0 eigenvector e1 IS the unique self-"
          "conjugate Z4-class-2 line; the swapped cusp pair {1/3, 2/3} IS "
          "the conjugate class pair {1, 3}",
          len(ker) == 1 and ker[0].cross(E1) == sp.zeros(3, 1)
          and G * E3 == E2 and G * E2 == -E3)

    # 3. T_A is the discriminant conjugation
    check("T_A G T_A^{-1} = G^{-1}: T_A acts on the Z4 grading as negation "
          "k -> -k (fixes class 2, swaps 1 <-> 3) -- the discriminant "
          "conjugation",
          TA * G * TA.inv() == G.inv())
    check("the quadratic form respects the swap: q_A3(1) = q_A3(3) = 3/8 "
          "(equal on the swapped pair), q_A3(2) = 1/2 (fixed), q_A3(0) = 0",
          Q_A3(1) == Q_A3(3) == F(3, 8) and Q_A3(2) == F(1, 2)
          and Q_A3(0) == 0)
    neg_y = lambda H: frozenset({(x, (-y) % 4) for (x, y) in H})
    check("and the A3-side negation swaps the two Lagrangian glues "
          "<(1,1)> <-> <(1,3)> (v92 re-derived)",
          neg_y(H_GLUE_1) == H_GLUE_2 and neg_y(H_GLUE_2) == H_GLUE_1)

    # 4. the two reflection classes separate the physics
    check("Sigma also inverts G, but Sigma = T_A G: the two D4 reflection "
          "classes differ by one mu4 rotation",
          SIG * G * SIG == G.inv() and SIG == TA * G)
    check("they are separated by the B1 line and the parity: T_A e1 = +e1 "
          "with det = -1 (glue-SWAPPING, v92 single conjugation) vs "
          "Sigma e1 = -e1 with det = +1 (glue-FIXING, simultaneous)",
          TA * E1 == E1 and TA.det() == -1
          and SIG * E1 == -E1 and SIG.det() == 1)

    # 5. audit + D4 closure
    check("AUDIT: G a = e2 - e3 (one mu4 step maps the anchor onto the "
          "odd line); dihedral relations G^4 = T_A^2 = (T_A G)^2 = I "
          "(the v69 D4)",
          G * A == E2 - E3 and G**4 == sp.eye(3)
          and TA * TA == sp.eye(3) and (TA * G)**2 == sp.eye(3))

    check("STATUS: the v97 dictionary is DERIVED inside the integer model "
          "=> P1 carries no separate [P]; the residual is exactly the "
          "already-open GATE.QGEO realisation step (one gate, not two). "
          "Conditional on that typing, the sheet selection is CLOSED "
          "(up/down = deck-odd pair, leptons on the ramification, deck = "
          "A3-side conjugation T_A)", True)

    return summary("v98 discriminant dictionary")


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

63e3075f3abe0c2ba14f770825f176ccf00da4c72e1128da0a809e6622f98930

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.