KM Space Theory, Volume V: Dynamical Screening, Invariant Structures, and Asymptotic Reconstruction
This volume develops a dynamical form of KM space theory based on admissible presentations. A dynamic presentation consists of a source evolution, a host evolution, an admissible presentation map, and the boundedness, topological, or measurable structures required to compare them. The theory also records admissible compressions, host costs, realization spectra, reflective cores, and the dynamic axioms DK1–DK12. Classical dynamical results and general comparison theorems are retained with their original hypotheses; a theorem is specifically KM only when its proof depends on the presentation structure. For metric phase spaces \(X\) and \(K\), semiflows \(S\) and \(T\), and a continuous map \(\kappa:X\to K\), the orbit comparison is between \(\kappa S(t)x\) and \(T(t)\kappa x\). At this stage, \(\kappa\) is a typed comparison map. It becomes a KM dynamic screen only when induced by an admissible compression, a core morphism, or the unit of a specified dynamic reflector. Vanishing orbit defect gives a semiconjugacy, but does not by itself provide an inverse, a section, or reconstruction of the dynamics hidden in the fibers. The analytic input comes from *KM Space Theory, Volume IV*. Closed generators, semigroup realizations, spectral and Fredholm data, energy forms, and solution spaces are used together with their domains and realization choices. Attractors, invariant measures, entropy, nonlinear stability, asymptotic compactness, statistical stability, and scattering require additional dynamical hypotheses. The theory distinguishes exact semiconjugacy, finite-time approximation, orbit shadowing, asymptotic and statistical comparison, measurable factor maps, conjugacy, extensions, and reverse trajectory lifting. Long-time invariants are organized into typed dynamic profiles that retain presentation costs, structural and presentation redundancy, orbit and attractor defects, invariant-measure and entropy loss, scattering data, and reconstruction groupoids. Reverse problems are formulated through lift spaces, kernel actions, core groupoids, convex fibers of invariant probabilities, conditional measures, relative entropy, and asymptotic-state fibers. Existence, uniqueness, naturality, categorical canonicality, and strict canonical selection are treated as distinct properties. In particular, an attained minimal presentation need not be canonical, a screened attractor need not reconstruct a source attractor, and zero orbit defect need not imply KM equivalence. Finite-state, linear, symbolic, gradient, reaction–diffusion, Markov, skew-product, bifurcation, and scattering models illustrate the framework. The resulting theory provides a dependency-controlled interface for studying long-time dynamics before and after KM screening while sharply separating classical dynamics, typed comparison, and genuinely KM-axiomatic conclusions. Keywords KM space theory; dynamical screening; admissible presentation; dynamic compression; screening reflector; rigid core; dynamic realization spectrum; redundancy profile; semiflow; attractor; invariant measure; entropy; transfer operator; random dynamical system; scattering; kernel dynamics; core groupoid; asymptotic reconstruction.