KM Space Theory, Volume II: Coordinate Reduction, Topology, Algebraization, and Classical Return

Abstract

We develop KM space theory as a layered framework for coordinate reduction, observable topology, homological invariants, algebraization, derived comparison, and return to classical geometry. KM dimension is defined through minimal structure-preserving equation presentations rather than classical intrinsic or ambient dimension. We construct normalized coordinate cores, KM topology, homology and cohomology, fiberwise and stable invariants, KM schemes, cycle groups, and derived comparison mechanisms. Exact obstruction theories are established for descent, algebraic \(K\)-lifting, cycle realization, categorical reflection, and coherent coordinate return. Explicit relative solutions are obtained for finite objects, nilpotent thickenings, principal affine descent, and square-zero extensions. Throughout, internal KM statements are separated from classical transfer results. In particular, KM cycles do not automatically determine Chow cycles, and universal algebraic return remains equivalent to the unresolved rational Hodge conjecture. The resulting framework identifies precisely which reconstruction data are available, which defects are computable, and which universal obligations remain open. **Keywords** KM spaces; coordinate reduction; ambient dimension; observable topology; algebraic cycles; Chow groups; algebraic \(K\)-theory; derived categories; descent theory; obstruction theory; classical return; Hodge conjecture.