Transformers as a Boundary Case for Hierarchical Compression: From Contraction Failure to NHIM Evidence

Tests whether transformer dynamics satisfy the contraction criterion required by Hierarchical Compression (HC). Seven formulations—per-layer Jacobians, end-to-end composed Jacobians, autoregressive rollout, joint conversation dynamics, trajectory divergence, the block-only update map, and a decisive rescue test—all return negative results, placing the tested transformers outside the contraction-based applicability regime. A generalization to normally hyperbolic invariant manifolds (NHIMs) resolves the boundary case: deflated spectral analysis reveals a low-dimensional invariant manifold (k*=2 median; 68% of prompts with k*≤3) with normal-bundle contraction (ρN = 0.73 at kremove = 5), while the full tangent space remains expansive. The paper also identifies three bridge-failure classes (B1 admissibility, B2 structure preservation, B3 entropy attribution) and draws methodological lessons about basis-dependent entropy disagreements and product-bound looseness. 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 | Fixed-Point Theorem | Bridge Synthesis