Fixed-Point Existence and Convergence for the Hierarchical Compression Evolution Map

Provides the formal dynamical-systems foundation for Hierarchical Compression (HC). Defines the HC evolution map F, regularity conditions R1–R4, and proves fixed-point existence and local convergence. Defines Condition C (sub-Shannon compression) and connects it to HC fixed points via a stated compression interpretation. Together with the information-theoretic CER foundation and the bridge-assumption architecture, this note completes the formal mathematical layer of the HC framework. Includes extensions to normally hyperbolic invariant manifolds (NHIMs) and the LaSalle invariance principle. Terminology. The foundational publications developed the framework under the name Harmonic Coherence, reflecting its origin in physics domains where phase-locked resonance and coherent-structure formation are the operative bridge mechanisms. As the framework extended to number theory, complexity theory, and algebraic geometry—domains with no physical harmonic content—the name became a source of friction. This paper uses Hierarchical Compression (HC) for the domain-independent Layer 1 framework and reserves Harmonic Coherence (always spelled out, never abbreviated) for the Layer 2 physics-domain bridge instantiation. The abbreviation HC refers exclusively to the framework. Companion documents: CER Foundation | Hanners Theorem | Reconciliation | Bridge Synthesis