Deterministic Structural Inference in Solid-State Systems: A DSFB Engine for Crystal Lattices, Phonons, and Structural Forensics - Operator-Theoretic Detectability and Scaling Laws

This paper develops a deterministic structural inference framework for crystalline materials by embedding lattice dynamics within the DSFB paradigm. Crystal lattices are interpreted as frozen swarms whose dynamical matrix acts as a Laplacian operator governing phonon modes. Deviations between predicted and observed lattice responses generate structured residuals, whose drift and slew provide deterministic forensic signals of defects, strain fields, and incipient phase transitions. We formalize a DSFB engine for solid-state systems, introducing residual envelopes, spectral perturbation bounds, and finite-time detectability conditions. The framework formulates a direct mapping between phonon spectrum evolution…