Definition: A minimal operational evolution limit

For a declared finite observable or payload interface, an operational evolution limit consists of:

1. a sequence of finite realizations indexed by $n$, with tick durations ${\delta }_n\downarrow 0$;

2. channels ${\mathcal{C}}_n(t)$, encoders ${\mathcal{E}}_n$, and decoders ${\mathcal{D}}_n$ on common comparison interfaces;

3. a limiting family ${\mathcal{T}}_t$ satisfying ${\mathcal{T}}_0=\mathrm{id}$, a stated composition law, and a stated continuity condition;

4. a convergence theorem in a declared topology, for example

   $$\underset{0\le t\le T}{\sup }{d}_{\diamond }({\mathcal{D}}_n{\mathcal{C}}_n(t){\mathcal{E}}_n,{\mathcal{T}}_t)⟶0$$

   for each finite $T$.

A spatial continuum theory additionally requires spatial realizations, local access maps, and their compatibility.