The exact 125-site Rubik carrier now carries all 15 declared outer and inner slice permutations. Canonical finite identities:
$$ P^TP=I,\qquad P^4=I,\qquad P\mathbf1=\mathbf1, $$
$$ L=L_{C_5}\oplus L_{C_5}\oplus L_{C_5},\qquad \lambda(k)=\sum_{j=1}^3\left(2-2\cos{2\pi k_j\over5}\right), $$
$$ \Delta E_P(\rho)=\rho^T(P^TLP-L)\rho. $$
For $\dot\rho=-L\rho$ and $E_{1/2}=\frac12\rho^TL\rho$,
$$ \dot E_{1/2}=-\|L\rho\|_2^2\le0, \qquad (I-\varepsilon L)^n\to P_0={\mathbf1\mathbf1^T\over125}. $$
Typed boundary: $\operatorname{rank}P_0=1$ while $\operatorname{rank}P_{47}=47$. Therefore the shared 125-dimensional carrier does not identify the two kernels, and the C₅³ flow does not derive $47/125$.
Exact Python proof and corrected theorem
For $a,b>0$ and $|\kappa|<1$, the positive fixed point exists uniquely and satisfies
$$ \rho_^2-\sigma_^2=a-b, \qquad (1-\kappa^2)p_^2-\kappa(a+b)p_-ab=0, \qquad p_=\rho_\sigma_*. $$
Its Jacobian has spectrum $\{0,\tau\}$ with
$$ \tau={\kappa(a+b)\over2p_*}+\kappa^2. $$
The exact local-stability interval is
$$ -\kappa_c(a,b)<\kappa<1, $$
where $\kappa_c=1$ if $a=b$ and otherwise $\kappa_c=\sqrt{z_c}$, with $z_c$ the positive root of
$$ (a-b)^2z_c^2+(3a^2-2ab+3b^2)z_c-4ab=0. $$
Superseded wording: |τ|<1 for all |κ|<1 when $a\ne b$; ρ²−σ²=a−b as an orbitwise conserved quantity; and a global positive-quadrant self-map for every negative $\kappa$.