Refinement, weighted transfer, and integral separation

Abstract

A complete family of idempotents gives exact coordinates, but a single family does not record scale or integral visibility. We study towers in which every coarse idempotent is the sum of its fine descendants. The coordinate changes are invertible and coherent, and every operator block is the sum of its refined blocks. Residue selectors give a model over arbitrary rings. When roots of unity are available they agree with character averages, and coprime moduli factor through the Chinese remainder theorem. In a complete filtered algebra, a diagonal operator plus positive weight transfers has an inverse obtained by summing admissible weighted paths. The formula truncates exactly at finite precision, while finite transfer graphs have rational path series. For a rational spectral decomposition over a discrete valuation ring, resultants give denominator bounds. The exact denominator is the largest Smith exponent of the lattice gluing module. This yields a necessary and sufficient precision delay, monotonicity under refinement, and a full congruence solving formula. We close with exact formulas for the propagation of a coarsening defect through powers, resolvents, holomorphic functions, exponentials, and multilevel towers.