E18 Limits and further questions
On this page · Permalink · Source line 5125
The bounded inverse limit theorem isolates one clean route to continuum evolution. Unbounded generators require graph norms, domain compatibility, and a generation theorem. The range leakage theorem converts the closure quantity into an off diagonal generator estimate for bounded orthogonal refinement models. Concrete discretizations still require a uniform estimate for that off diagonal block.
The finite Bayesian sufficiency theorem is exact, but it concerns one likelihood and one fixed partition. The relative entropy chain rule measures both retained and discarded information for that update. Statistical sufficiency for a parametric family requires a common packet structure that preserves the full parameter dependence. In continuous models the corresponding object is a sufficient sigma algebra or statistic.
The fractional defect theorem uses a common spectral gap. Removing that gap requires weaker norms or a Hölder estimate near the null space. For sectorial operators, the same calculation should be carried out through resolvent bounds on a sector, with domains tracked explicitly. The finite spectral statement does not settle those issues.
The Green and Poisson results exploit a one dimensional second difference matrix. Higher dimensions, irregular meshes, and variable coefficients require a separate stability theory. The packet language remains useful only after coercivity, consistency, and boundary approximation have been established.
Memory compression is exact once a finite hidden realization is known and quantitative once an integrable kernel approximation is supplied. The primary decomposition shows that a finite hidden block always produces polynomial exponential modes, including the effects of Jordan chains. Choosing a short approximation remains a different problem. Strongly nonnormal systems can exhibit transient growth and may require balanced or rational approximation methods. General Volterra kernels need not admit a short finite realization.
Finally, Markov lumping, Bayesian sufficiency, and memory closure answer different questions. Lumpability asks whether coarse probabilities evolve autonomously. Bayesian sufficiency asks whether a fixed coarse likelihood reproduces the posterior packet masses without hiding a change in the conditional distribution inside a packet. Memory closure asks whether discarded dynamics can be represented without history. Similar notation should not obscure these distinctions.