[ \Sigma=V_2\otimes V_2\otimes V_2, \qquad \dim\Sigma=5^3=125. ]
[ \operatorname{spec}(C) = {0,2,6,12,20,30,42}. ]
Multiplicities:
[ 1,\ 9,\ 25,\ 28,\ 27,\ 22,\ 13. ]
Therefore
[ 1+9+25+28+27+22+13=125. ]
[ \mu=18, \qquad \sigma=12, ]
so
[ \mu-\sigma=6, \qquad \mu+\sigma=30. ]
[ K=(C-6I)(C-30I). ]
Hence
[ \ker K = E_6\oplus E_{30}. ]
Because
[ \dim E_6=25, \qquad \dim E_{30}=22, ]
we obtain
[ \boxed{ \dim\ker K=47. } ]
Let
[ P_{47}=P_6+P_{30}. ]
Then
[ P_{47}^2=P_{47}, \qquad P_{47}^*=P_{47}, \qquad KP_{47}=0, ]
and
[ \operatorname{rank}P_{47} = \operatorname{Tr}P_{47} = 47. ]
For the maximally mixed state
[ \rho_*=\frac{I}{125}, ]
[ \Omega_{47}(\rho_) = \operatorname{Tr}(P_{47}\rho_) = \boxed{\frac{47}{125}} = 0.376. ]
For
[ \Gamma_\varepsilon = I-\varepsilon K^2, ]
the optimal scalar step for the nonzero spectrum is
[ \boxed{ \varepsilon_*=\frac1{99144} } ]
with worst-mode contraction
[ \boxed{ \rho_*=\frac{15}{17}. } ]
Therefore
[ \Gamma_{\varepsilon_*}^n \longrightarrow P_{47}. ]
The continuous analogue is
[ e^{-tK^2} \longrightarrow P_{47}. ]
[ \boxed{ K \rightarrow \ker K \rightarrow P_{47} \rightarrow H=I-P_{47} \rightarrow x_\infty=P_{47}x_0. } ]