Harmonic Coherence: Comprehensive Mathematical Reconciliation and Unified Theoretical Summary

Presents a unified framework in which quantum dynamics, semiclassical gravity coupling, and entropy-constrained evolution are expressed in a common mathematical language under the Hierarchical Compression (HC) program. Introduces a coherence tensor C_μν, formalizes an entropy-descent closure, and states explicit falsifiable test channels in precision clocks, vacuum systems, and gravitational-wave residual analysis. Core claims are in falsifiable form with derivation sketches, parameterized models, and reproducibility-oriented appendices. Speculative extrapolations lacking bridge coordinates or measurable observables are explicitly excluded.

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: Hanners Theorem | Fixed-Point Theorem | Bridge Note | Bridge Synthesis | GW Kerr Benchmark | Transformer Boundary Case