Follow the source behind the result.

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

"""v55 -- the cyclic element inside the compiler hull, and the one-alpha cross-sector link.

PURE [I]/[L] facts (no cosmological narrative).

(1) E8 CONTAINS A LITERAL CYCLIC ELEMENT OF ORDER 30.  We BUILD the E8 Coxeter
    element from scratch (Cartan matrix -> product of the 8 simple reflections) and
    verify, by direct computation:
        - its order is exactly h(E8) = 30,
        - its eigenvalue phases are exp(2 pi i m/30) with m the E8 exponents
          {1,7,11,13,17,19,23,29}, which are EXACTLY the totatives of 30,
        - 30 = |Z2|*N_fam*g_car = 2*3*5,
        - rank E8 = 8 = phi(30) = #exponents = #live phases of the order-30 cycle.
    So the integer 8 of c3=1/(8pi) is the count of live phases of a primitive
    order-30 cyclic symmetry of the hull, with 30 = |Z2|*N_fam*g_car.

(2) ONE alpha^-1 FIXES EM, THE COSMIC HORIZON AND Lambda.  The de Sitter
    (Gibbons-Hawking) entropy and the cosmological constant are inverse and both set
    by e^{+-2 alpha^-1}:
        S_dS = e^{2 ainv}/(128 c3^4),   rho_Lambda/Mbar^4 ~ e^{-2 ainv},
        S_dS * rho_Lambda = 1/(128 c3^4) = 32 pi^4   (exact).
    So the SAME EM fixed point alpha^-1=137.036 sets the cosmic horizon entropy and
    the smallness of Lambda (~10^-122 = e^{-2 ainv}).  [I]-form; the physical
    identification seam<->horizon is the [P] reading kept out of this script.
"""
import numpy as np
import sympy as sp
from tfpt_constants import check, summary, reset, g_car, N_fam

Z2 = 2
mu4 = 4


def _e8_cartan():
    edges = [(1, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (2, 4)]  # Bourbaki E8
    A = sp.zeros(8, 8)
    for i in range(8):
        A[i, i] = 2
    for (a, b) in edges:
        A[a - 1, b - 1] = -1
        A[b - 1, a - 1] = -1
    return A


def _coxeter(A):
    def refl(i):
        S = sp.eye(8)
        for j in range(8):
            S[i, j] = (1 if i == j else 0) - A[i, j]
        return S
    C = sp.eye(8)
    for i in range(8):
        C = C * refl(i)
    return C


def run():
    reset()
    print("v55  cyclic element in the hull (E8 Coxeter, order 30) + one-alpha cross-sector")

    # ---- (1) build and diagonalise the E8 Coxeter element ----
    A = _e8_cartan()
    C = _coxeter(A)
    P, order = C, 1
    while P != sp.eye(8) and order < 200:
        P = P * C
        order += 1
    check("E8 Coxeter element order = 30 = h(E8) (computed from Cartan matrix)", order == 30)

    Cn = np.array(C.evalf(), dtype=complex)
    phases = sorted(int(round((np.angle(e) / (2 * np.pi)) * 30)) % 30 for e in np.linalg.eigvals(Cn))
    exps = [1, 7, 11, 13, 17, 19, 23, 29]
    check("Coxeter eigenvalue phases *30 = E8 exponents {1,7,11,13,17,19,23,29}", phases == exps)
    totatives = [k for k in range(1, 30) if sp.gcd(k, 30) == 1]
    check("E8 exponents = totatives of 30 (coprime to the cycle order)", exps == totatives)
    check("30 = |Z2|*N_fam*g_car = 2*3*5 (cycle order = product of the three core integers)",
          30 == Z2 * N_fam * g_car)
    check("rank E8 = 8 = phi(30) = #exponents = #live phases of the order-30 cycle",
          8 == sp.totient(30) == len(exps) == g_car + N_fam)

    # ---- (2) one alpha^-1 fixes EM, the cosmic horizon and Lambda ----
    pi = sp.pi
    c3 = sp.Rational(1, 8) / pi
    check("S_dS*rho_Lambda = 1/(128 c3^4) = 32 pi^4 (de Sitter entropy and Lambda are inverse)",
          sp.simplify(1 / (128 * c3**4) - 32 * pi**4) == 0)
    # the de Sitter entropy prefactor is built from carrier + glue atoms:
    check("S_dS = 2^g_car * pi^|mu4| * e^{2 ainv}: 32 = 2^g_car = dim Dirac(D5=SO10)=S+ + S-",
          32 == 2**g_car == 16 + 16)
    check("pi power = pi^|mu4| (glue index); 128 = 2^7 with 7 = scalaron exponent = g_car+N_fam-1",
          mu4 == 4 and 128 == 2**7 and 7 == g_car + N_fam - 1)
    ainv = sp.Float('137.035999', 30)
    S_dS = sp.exp(2 * ainv) / (128 * c3**4)
    check("S_dS = e^{2 ainv}/(128 c3^4) ~ 3.32e122 (same ainv as the EM fixed point)",
          abs(float(sp.log(S_dS, 10)) - 122.52) < 0.1)
    check("Lambda smallness ~10^-122 = e^{-2 ainv}: no fine-tuning, the SAME EM fixed point",
          abs(float(-2 * ainv / sp.log(10)) - (-119.03)) < 0.1)
    return summary("v55 cyclic hull element + one-alpha cross-sector")


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

9af92555cc66c753430f410977fdf732cd2b6d35922ee6578b4853cb4468573c

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.