Mathematical Coherence in Spectral Dynamics: A Minimal Structural Framework

We present a minimal structural framework organizing spectral dynamics, structural observables, and detectability constraints into a coherent system. The framework is built on three layers: (1) exact spectral identities (r′ (t) = −Varₜ (λ) ), (2) observable structural indicators (σ₃ (H), Δ (t) ), (3) decision constraints (detectability thresholds and abstention). No global coercivity is claimed. No global equivalence between spectral, structural, and decision quantities is assumed. Each layer operates under explicit validity conditions. The role of this document is organizational: it connects independently valid components without extending their scope beyond proven or observable regimes. Error is treated as a structural component of the system, not as a failure. This framework does not impose conclusions. It defines when conclusions are justified.