Follow the source behind the result.

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

"""v419 -- The seam mu4 IS the carrier's Galois group: a positive resolution of
the v409 / RES.COXETER.SYMMETRY.01 asymmetry.  In the cyclotomic picture the
flavor clocks (sheet/family/carrier/CP) are powers of zeta30 (the cyclic side,
order h(E8)=30), but the seam mu4 (i) sits on the Galois side (Z/30)^x (order
rank E8 = 8).  v409 flagged this as the "prime-2 is the Galois bridge" asymmetry.
This shows WHY -- and what the seam mu4 actually IS.  [E] number theory + the
explicit Frobenius operator; [C] the architecture reading.

  [E] 1. 30 IS SQUAREFREE => NO ORDER-4 ON THE CYCLIC CLOCK.  h(E8)=30=2*3*5 is
         squarefree, so the cyclic clock Z/30 has elements only of orders
         {1,2,3,5,6,10,15,30} -- there is NO order-4 element.  The seam mu4
         (order 4) therefore CANNOT be a rotation hand; it is forced onto the
         Galois side.
  [E] 2. THE GALOIS GROUP SPLITS BY PRIME.  By CRT
             (Z/30)^x = (Z/2)^x x (Z/3)^x x (Z/5)^x = 1 x Z/2 x Z/4 = mu4 x Z2,
         order phi(30) = 8 = rank E8.  The mu4 (Z/4) factor IS (Z/5)^x -- the
         CARRIER prime 5; the Z2 factor IS (Z/3)^x -- the FAMILY prime 3 (= CP
         conjugation, v316).  The order-4 elements {7,13,17,23} are exactly the
         generators of (Z/5)^x mod 5.
  [E] 3. THE SEAM mu4 = Gal(Q(zeta5)/Q) = (Z/5)^x (the carrier's automorphisms),
         realised on the carrier clock C5 (v418) by the Frobenius zeta5->zeta5^2:
         the explicit order-4 operator G has
             G C5 G^-1 = C5^2,   G^4 = I,   G^2 C5 G^-2 = C5^-1 (= complex conj).
         So the square seam (atom 2, mu4) and the carrier pentagon (atom 5, mu5)
         are ONE object: the seam IS the carrier's Galois group.
  [C] 4. RESOLUTION OF RES.COXETER.SYMMETRY.01.  v409 closed the asymmetry as a
         lemma ("prime-2 necessary but not autonomous, |mu4|=2^2 the Galois
         bridge, carries no contraction rate").  This gives the positive content:
         the 2^2=4 is the ORDER of (Z/5)^x (so prime-2 and prime-5 meet here),
         and "no contraction rate" is because the seam is an AUTOMORPHISM
         (|lambda|=1, finite order), not a rotation/transfer hand -- consistent
         with the clock No-Go.  Q(zeta30) carries BOTH E8 invariants: the cyclic
         hands (h=30) and the Galois gears (rank=8 = mu4 x Z2).

Mirrored in wolfram/tfpt_readouts_extension.wl (exact integer/Galois facts).
"""
import sympy as sp

from tfpt_constants import check, summary, reset, rankE8, g_car, N_fam

x = sp.symbols('x')
Phi5 = x**4 + x**3 + x**2 + x + 1
C5 = sp.Matrix([[0, 0, 0, -1], [1, 0, 0, -1], [0, 1, 0, -1], [0, 0, 1, -1]])


def mult_order(u, n):
    o, y = 1, u % n
    while y != 1:
        y = (y * u) % n
        o += 1
    return o


def run():
    reset()
    print("v419 the seam mu4 is the carrier's Galois group Gal(Q(zeta5)) = (Z/5)^x")

    # ---- 1. 30 squarefree => no order-4 on the cyclic clock ----
    orders30 = {30 // sp.gcd(k, 30) for k in range(30)}
    check("30 SQUAREFREE => NO ORDER-4 CYCLIC [E]: h(E8)=30=2*3*5 squarefree, so "
          "Z/30 has element orders {1,2,3,5,6,10,15,30} -- NO order 4; the seam "
          "mu4 cannot be a rotation hand, it is forced onto the Galois side",
          all(e == 1 for e in sp.factorint(30).values())
          and orders30 == {1, 2, 3, 5, 6, 10, 15, 30}
          and 4 not in orders30)

    # ---- 2. the Galois group splits by prime: mu4 x Z2 = (Z/5)^x x (Z/3)^x ----
    units = [u for u in range(1, 30) if sp.gcd(u, 30) == 1]
    ord4 = [u for u in units if mult_order(u, 30) == 4]
    check("GALOIS SPLITS BY PRIME [E]: (Z/30)^x = (Z/2)^x x (Z/3)^x x (Z/5)^x = "
          "1 x Z/2 x Z/4 = mu4 x Z2, order phi(30)=8=rank E8; the mu4 (Z/4) "
          "factor IS (Z/5)^x (carrier 5), the Z2 IS (Z/3)^x (family 3 = CP "
          "conj); order-4 elements {7,13,17,23} are (Z/5)^x generators",
          len(units) == sp.totient(30) == 8 == rankE8
          and sp.totient(5) == 4 and sp.totient(3) == 2 and sp.totient(2) == 1
          and ord4 == [7, 13, 17, 23]
          and all(mult_order(u % 5, 5) == 4 for u in ord4)
          and sp.is_primitive_root(2, 5))      # (Z/5)^x is cyclic, gen 2

    # ---- 3. the seam mu4 = Gal(Q(zeta5)) via the explicit Frobenius G ----
    def col(e):
        p = sp.Poly(sp.rem(x**e, Phi5, x), x)
        return sp.Matrix([p.coeff_monomial(x**j) for j in range(4)])
    G = sp.Matrix.hstack(*[col(2 * k) for k in range(4)])      # sigma: x -> x^2
    check("SEAM mu4 = Gal(Q(zeta5)) [E]: the Frobenius zeta5->zeta5^2 is the "
          "explicit order-4 operator G with G C5 G^-1 = C5^2, G^4 = I, and "
          "G^2 C5 G^-2 = C5^-1 (complex conjugation) -- the seam mu4 acts on the "
          "carrier clock C5 (v418) as its Galois automorphism (square seam = "
          "carrier-pentagon automorphisms)",
          sp.simplify(G * C5 * G.inv() - C5**2) == sp.zeros(4)
          and G**4 == sp.eye(4)
          and sp.simplify(G**2 * C5 * (G**2).inv() - C5**4) == sp.zeros(4)
          and G != sp.eye(4) and G**2 != sp.eye(4))

    # ---- 4. resolution of RES.COXETER.SYMMETRY.01 (typing) ----
    check("RESOLVES RES.COXETER.SYMMETRY.01 [C]: the 2^2=4 is the ORDER of "
          "(Z/5)^x (prime-2 and prime-5 meet); 'no contraction rate' (v409) is "
          "because the seam is an AUTOMORPHISM (|lambda|=1, finite order=4), not "
          "a rotation/transfer hand; Q(zeta30) carries BOTH E8 invariants -- "
          "cyclic hands h=30 and Galois gears rank=8=mu4 x Z2",
          2**2 == 4 == sp.totient(5)
          and rankE8 == g_car + N_fam == 8
          and 30 == 2 * 3 * 5)

    return summary("v419 seam mu4 = Gal(Q(zeta5)) = (Z/5)^x (resolves the "
                   "v409 cyclic/Galois asymmetry)")


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

cc6152ad7ea68c67725020e386e49eec32810ab25bb483c76bae99964399c2fd

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.