From Structural Signal to Observable Prediction: Bridge Assumptions B1-B3 in Hierarchical Compression

This paper argues that the Hierarchical Compression (HC) framework—referred to as Harmonic Coherence in the foundational publications—should be read as a layered framework rather than a monolithic theory. It formalizes three bridge assumptions (B1–B3) that connect the abstract formal core to domain-level observable predictions, and uses a 45-row cross-domain audit to diagnose where the current corpus succeeds, where it fails, and what kind of failure dominates. The central finding is that the main bottleneck is not missing normalization but bridge incompleteness: wrong predictor choice, hidden-variable dependence, and entropy/observable mismatch. Two empirical benchmark domains anchor this diagnosis: (1) transformer language models, where a six-formulation admissibility study reveals boundary-case dynamics; and (2) gravitational-wave ringdown, where a non-circular numerical-relativity benchmark identifies dimensionless surface gravity κM as the leading Kerr coordinate for organizing quasinormal-mode damping. Three-layer architecture: • Layer 1 (Theorem layer): HC evolution map, regularity conditions R1–R4, fixed-point existence, local convergence, Condition C • Layer 2 (Bridge layer): Assumptions B1 (admissible context map), B2 (entropy–observable identification), B3 (constitutive law) • Layer 3 (Application layer): Domain-specific predictions tested against data Companion documents: • Fixed-Point Theorem Note: 10.5281/zenodo.18978490 • Paper A (Transformers): 10.5281/zenodo.18974716 • Paper B (Kerr Coordinates): 10.5281/zenodo.18976467 • CER Theorem: 10.5281/zenodo.18668434 • Reconciliation: 10.5281/zenodo.18671909 • Bridge Note: 10.5281/zenodo.18670126