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 and structural degradation, supporting early anomaly detection under explicit modeling and observation assumptions. A heuristic bank is further developed into a deterministic material diagnostics library, linking residual signatures to candidate physical defect classes under stated admissibility conditions. The resulting formulation provides a structural bridge between lattice dynamics, spectral graph theory, and deterministic inference, extending DSFB into predictive materials analysis. The framework admits explicit detectability conditions linking perturbation scale, observation structure, and residual evolution under admissible envelopes. These constructions also suggest a reproducible benchmarking pathway in which structured residual behaviour can be systematically evaluated across nominal, perturbed, and non-ideal regimes. This formulation admits operator-level characterizations of detectability in terms of perturbation scaling, observation-induced gain, and admissible residual envelopes under non-ideal conditions. Detectability is shown to be governed by structural separation relative to regime-dependent admissibility thresholds, yielding a noise-aware scaling law for detection time. This work builds on a broader line of deterministic structural inference frameworks in which residual dynamics are treated as structured observables within trust-calibrated, non-stochastic causal systems, extending these ideas to lattice-resolved solid-state regimes. A central result is a noise-aware detection-time scaling law that explicitly links perturbation magnitude, admissible envelope level, non-structural error, and observation sensitivity.