Follow the source behind the result.

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

"""v56 -- the unique attractor: a gapped boundary transport selects ONE fixed point.

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

(1) GAPPED TRANSFER => UNIQUE ATTRACTING FIXED POINT.  The boundary transport
    spectrum {1,(2/3)^6,(1/3)^6} has a spectral gap Delta = -log(2/3)^6 = 6 log(3/2)
    = 2.4328 > 0.  By Perron-Frobenius, a primitive operator with a gap has a UNIQUE
    dominant eigenvector, and iteration converges to it geometrically at the rate
    lambda2/lambda1 = (2/3)^6, independently of the initial state.  We demonstrate
    this directly: two very different start vectors iterate to the SAME fixed
    direction (cosine = 1), and the convergence rate is exactly (2/3)^6 -- the SAME
    lambda2 that governs the SM flavor gap and the horizon Page recovery (v54).
    This is the dynamical-systems content of "parameter-free via fixed point":
    the realized state is an attractor, not a tuning.

(2) THE COXETER ROTATION LIVES IN |mu4|=4 INVARIANT PLANES.  The E8 exponents pair
    as m + (30-m) = 30 -> {(1,29),(7,23),(11,19),(13,17)}, i.e. rank/2 = 4 = |mu4|
    conjugate eigenvalue pairs (4 invariant 2-planes of the order-30 Coxeter
    rotation).  Sum of exponents = 120 = |R+(E8)| (positive roots); #roots(E8) =
    rank*h = 8*30 = 240 = (live phases)*(cycle order).

(3) ENTROPY RESET = THE GAP.  Iterating the gapped operator converges to the rank-1
    projector onto the leading eigenvector (T^n/lambda1^n -> P_1, rank 1): the
    sub-leading microstate content is contracted away, only the rank-1 boundary law
    survives.  [I]-form; the seam<->cosmological "reset" identification 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, mu4 = 2, 4


def run():
    reset()
    print("v56  unique attractor: gapped boundary transport selects one fixed point")

    # ---- (1) gapped transfer => unique attractor at rate (2/3)^6 ----
    lam2 = (sp.Rational(2, 3))**6
    gap = -sp.log(lam2)
    check("spectral gap Delta = -log(2/3)^6 = 6 log(3/2) > 0",
          sp.simplify(gap - 6 * sp.log(sp.Rational(3, 2))) == 0 and float(gap) > 0)

    spec = np.array([1.0, (2 / 3)**6, (1 / 3)**6])
    V = np.array([[1, 1, 1], [0, 1, 2], [0, 0, 1]], float)  # invertible, non-diagonal
    T = V @ np.diag(spec) @ np.linalg.inv(V)
    evm = sorted(np.abs(np.linalg.eigvals(T)), reverse=True)
    check("constructed primitive operator has spectrum {1,(2/3)^6,(1/3)^6}",
          abs(evm[0] - 1) < 1e-9 and abs(evm[1] - (2 / 3)**6) < 1e-9)
    check("convergence rate lambda2/lambda1 = (2/3)^6 (geometric)",
          abs(evm[1] / evm[0] - (2 / 3)**6) < 1e-9)

    def it(v):
        v = v.astype(float)
        for _ in range(60):
            v = T @ v
            v = v / np.linalg.norm(v)
        return v
    a, b = it(np.array([1., 0., 0.])), it(np.array([0.1, 0.7, 0.2]))
    cos = abs(a @ b) / (np.linalg.norm(a) * np.linalg.norm(b))
    check("two different starts converge to the SAME fixed direction (cos=1) => unique attractor",
          abs(cos - 1.0) < 1e-9)

    # ---- (2) Coxeter rotation in |mu4|=4 invariant planes ----
    exps = [1, 7, 11, 13, 17, 19, 23, 29]
    pairs = [(m, 30 - m) for m in exps if m < 15]
    check("exponents pair m+(30-m)=30 => rank/2 = 4 = |mu4| invariant 2-planes",
          all(p[0] + p[1] == 30 for p in pairs) and len(pairs) == mu4 == 4)
    check("sum of exponents = 120 = |R+(E8)| (positive roots)", sum(exps) == 120)
    check("#roots(E8) = rank*h = 8*30 = 240 = (live phases)*(cycle order)", 8 * 30 == 240)

    # ---- (3) entropy reset = the gap: T^n / lambda1^n -> rank-1 projector ----
    Tn = np.linalg.matrix_power(T, 80)
    Tn = Tn / (1.0**80)
    rank = np.linalg.matrix_rank(np.round(Tn, 6))
    check("iterated operator -> rank-1 projector (only the boundary 'law' survives)", rank == 1)
    return summary("v56 unique attractor")


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

d0870475b4b56dc386935a8845a7494120da713a2f6a60488460faafe63f39fc

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.