Adjoint Projections on Computational Hierarchies: A Metric Framework

This paper introduces a formal framework for hierarchical computation using projection and collapse operators that form an adjunction between finite machines. A behavioral metric is defined to measure cross-level similarity and shown to satisfy metric properties. The framework includes proofs of metric completion, examples using binary linear codes, and an efficient level-assignment algorithm. Connections are drawn to finite cursor machines, descriptive complexity, and information theory, providing a compact and computable account of hierarchical processing.