Let
[ E=\ker K ]
and let
[ P:\mathcal H\to E ]
be the orthogonal projector.
Define
[ H=I-P. ]
Then
[ P^2=P=P^*, ]
[ H^2=H, ]
and
[ PH=HP=0. ]
Every state decomposes uniquely as
[ x=Px+Hx. ]
Define the functional
[ F[x] = \frac12|Hx|^2. ]
Then
[ \nabla F=Hx. ]
The gradient flow
[ \dot x=-Hx ]
has exact solution
[ x(t) = e^{-tH}x_0 = Px_0+e^{-t}Hx_0. ]
Therefore
[ \boxed{ \lim_{t\to\infty}x(t)=Px_0. } ]
The invariant component is preserved exactly:
[ Px(t)=Px_0. ]
The transverse component decays exponentially:
[ Hx(t)=e^{-t}Hx_0. ]
Thus projection is not merely an endpoint operation. It is the asymptotic state of an exact contraction flow.