Canonical identity

$\mathcal S_S(t)=e^{-tK_S^2}$

$\mathcal S_S(t+s)=\mathcal S_S(t)\mathcal S_S(s)$

$\lim_{t\to\infty}\mathcal S_S(t)=P_S$

If $\gamma_S=\min(\operatorname{spec}(K_S^2)\setminus\{0\})$, then

$\left\|\mathcal S_S(t)-P_S\right\|_2=e^{-\gamma_S t}$

Evidence class

E0 finite-dimensional spectral theorem; E1 numerical realizations for selected kernels.

Civic role

Continuous-time convergence authority for representation-selected invariant subspaces.

Transit credential

Spectral Terminal → Continuous Flow Station → Projector Depot → Machine Certificates.

Boundary

The theorem requires Hermitian finite-dimensional $K_S$. Open-system physical implementation is a separate destination-specific question.