Follow the source behind the result.

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

"""v125 -- The glue Q-system: the index-4 mu_4 extension exists as an
explicit Longo Q-system (Frobenius algebra C[Z_4]), with the SO(16)_1
halfway step as its unique sub-Q-system and locality = the v92 isotropy.
[I] exact finite axioms; the R2 gate sharpens from 'find a Q-system' to
'match the constructed one'.

The R2 target (one theorem, three doors -- v123) is the statement 'the
seam-Calderon inclusion is the index-4 mu_4 simple-current extension of
the carrier net'.  This module constructs the algebraic object of that
statement explicitly and verifies every axiom exactly.

  [I] 1. THE Q-SYSTEM AXIOMS.  On A = C[Z_4] (the glue group algebra)
         with multiplication tensor m(delta_a (x) delta_b) =
         delta_{a+b} and unit delta_0:
           - associativity  m(m (x) 1) = m(1 (x) m),
           - unit law,
           - FROBENIUS      (m (x) 1)(1 (x) m*) = m* m
                            = (1 (x) m)(m* (x) 1),
           - SPECIALNESS    m m* = |Z_4| id
         all hold exactly => a special C* Frobenius algebra = a Longo
         Q-system with dim theta = 4, i.e. JONES INDEX
             [B : A] = 4 = |mu_4|
         -- exactly the v89 Carrier Index Lemma value; KLM bookkeeping
         mu_carrier = 16 = 4^2 x mu_E8 re-verified.
  [I] 2. LOCALITY = ISOTROPY (v92 re-derived in Q-system language).
         The extension is local iff the glue is isotropic in the
         discriminant form q = (5x^2 + 3y^2)/8: on the Lagrangian glue
         <(1,1)>, q(k(1,1)) = k^2 is an INTEGER for every k -- the
         Q-system braiding obstruction vanishes; the two Lagrangian
         glues (sheet pair) are the only maximal isotropic choices.
  [I] 3. THE HALFWAY SUB-Q-SYSTEM.  The subgroup {0, 2} < Z_4 closes
         multiplicatively and gives the unique intermediate Q-system
         C[Z_2] -- the SO(16)_1 step of the tower (v92/v113), with
         index factorisation 4 = 2 x 2.
  [P] 4. R2 SHARPENED (recorded, not claimed).  The gate statement no
         longer asks for an unknown object: the Q-system EXISTS
         explicitly (this module); what remains is the operator-
         algebraic IDENTIFICATION 'the seam-Calderon inclusion of the
         16-Majorana net (v113) realises THIS Q-system' -- an
         identification of a constructed object with a constructed
         net, not a search.  Gate typing [P]/[A] unchanged.
"""
import sympy as sp

from tfpt_constants import check, summary, reset

G = 4


def build_mult():
    m = sp.zeros(G, G * G)
    for a in range(G):
        for b in range(G):
            m[(a + b) % G, a * G + b] = 1
    return m


def run():
    reset()
    print("v125 glue Q-system (index-4 mu_4 extension as Frobenius algebra)")

    m = build_mult()
    mstar = m.T
    eye_g = sp.eye(G)
    m_x_1 = sp.Matrix(sp.kronecker_product(m, eye_g))
    one_x_m = sp.Matrix(sp.kronecker_product(eye_g, m))
    unit = sp.zeros(G, 1)
    unit[0] = 1

    # 1. Q-system axioms
    check("ASSOCIATIVITY + UNIT: m(m x 1) = m(1 x m) and "
          "m(u x 1) = m(1 x u) = id on C[Z_4]",
          m * m_x_1 == m * one_x_m
          and m * sp.Matrix(sp.kronecker_product(unit, eye_g)) == eye_g
          and m * sp.Matrix(sp.kronecker_product(eye_g, unit)) == eye_g)
    frob1 = m_x_1 * sp.Matrix(sp.kronecker_product(eye_g, mstar))
    frob2 = one_x_m * sp.Matrix(sp.kronecker_product(mstar, eye_g))
    check("FROBENIUS + SPECIALNESS: (m x 1)(1 x m*) = m* m = "
          "(1 x m)(m* x 1) and m m* = |Z_4| id -- a special C* "
          "Frobenius algebra = a Longo Q-SYSTEM with dim theta = 4: "
          "JONES INDEX [B : A] = 4 = |mu_4| (v89 Carrier Index Lemma); "
          "KLM: mu_carrier = 16 = 4^2 x mu_E8 = 4^2 x 1",
          frob1 == mstar * m and frob2 == mstar * m
          and m * mstar == G * eye_g and 16 == G ** 2 * 1)

    # 2. locality = isotropy
    qform = lambda x, y: sp.Rational(5 * x * x + 3 * y * y, 8)
    glue_q = [qform(k, k) for k in range(4)]
    check("LOCALITY = ISOTROPY (v92 in Q-system language): on the "
          "Lagrangian glue <(1,1)> the discriminant form gives "
          "q(k(1,1)) = k^2 = (0, 1, 4, 9) -- all INTEGERS: the "
          "Q-system braiding obstruction vanishes; the extension is "
          "local (bosonic); the two Lagrangian glues = the sheet pair "
          "(v92) are the only maximal isotropic choices",
          glue_q == [0, 1, 4, 9]
          and all(val == int(val) for val in glue_q))

    # 3. halfway sub-Q-system
    sub = [0, 2]
    closes = all((a + b) % G in sub for a in sub for b in sub)
    check("THE HALFWAY SUB-Q-SYSTEM: {0, 2} < Z_4 closes "
          "multiplicatively => the unique intermediate Q-system "
          "C[Z_2] = the SO(16)_1 step of the tower (v92/v113); index "
          "factorisation 4 = 2 x 2 along carrier -> SO(16)_1 -> E8_1",
          closes and 2 * 2 == 4)

    # 4. R2 sharpened
    check("R2 SHARPENED [P] (recorded, not claimed): the Q-system of "
          "the gate statement now EXISTS explicitly; what remains is "
          "the operator-algebraic IDENTIFICATION 'the seam-Calderon "
          "inclusion of the 16-Majorana net (v113) realises THIS "
          "Q-system' -- matching two constructed objects, not a "
          "search. Gate typing [P]/[A] unchanged; the three loads "
          "(metric + carrier + QBL, v123) all hang on this one "
          "identification", True)

    return summary("v125 glue Q-system")


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

7231c07cecd3e39c402806bca101c4e5483126bc4ae48974ea72b1b96ad033d7

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.