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This note provides the formal dynamical-systems foundation for the Hierarchical Compression (HC) framework—referred to as Harmonic Coherence in the foundational publications. It defines the HC evolution map ℱ, the regularity conditions R1–R4, and proves fixed-point existence and convergence under those conditions. It also defines Condition C (sub-Shannon compression) and sketches how HC fixed points imply Condition C under the stated compression interpretation. Together with the information-theoretic foundation in the Contextual Entropy Reduction (CER) theorem and the bridge-assumption architecture in the Reconciliation document, this note completes the formal layer of the HC framework. Contents: HC configuration space, HC evolution map, regularity conditions R1–R4, four background assumptions (probability conservation, entropy reduction, coherence damping, forward invariance and precompactness), fixed-point existence theorem (compactness and continuity argument), convergence and normal hyperbolicity theorem (LaSalle invariance + NHIM persistence), degenerate-case corollary (dim M* = 0), empirical evidence remark (transformer NHIM data), Condition C definition and bridge theorem (proof sketch). Companion documents:• CER Theorem: 10.5281/zenodo.18668434• Reconciliation: 10.5281/zenodo.18671909• Hanners Theorem (master formalization): 10.5281/zenodo.15288890• Transformer Boundary Case (Paper A): 10.5281/zenodo.18974716• Bridge Synthesis (Paper C): 10.5281/zenodo.18977541