Geometric Theoretical Framework for Dynamic Protein Mutation Detection Models: Defect Awareness and Pathogenicity Prediction

Traditional protein mutation detection and pathogenicity prediction pipelines rely on static single-conformation structural modeling, inherently ignoring conformational flexibility, dynamic ensemble evolution, and the underlying manifold geometry of protein dynamics. This induces systematic detection failures in flexible regions, allosteric sites, and metastable functional domains, yet lacks a rigorous mathematical characterization of such failure mechanisms. In this work, we establish a theorem-driven geometric-algebraic framework for dynamic protein mutation modeling. Starting from a dynamic conformational Riemannian manifold, we construct the latent representation space via representation-induced completion of operator-valued observations, rather than pre-assumed embedding structures. Within this setting, algebraic constraints are not imposed axiomatically but relaxed into learnable approximate Lie algebra regularization, enabling statistical verification of structural consistency. By integrating Levi-Civita connection, geodesic deviation, and heat kernel asymptotics, we introduce a Lipschitz-stable topological spectral defect (TSD,{delta} spec) index that quantifies the intrinsic inconsistency between static representations and dynamic geometric invariants, linking it to curvature-induced instability and Lie algebra deformation. Under a functorial compatibility principle, we design a dual-branch architecture for pathogenicity prediction and defect awareness, realized via local Lie algebra encoding and low-rank spectral approximation. On multi-source datasets (108 curated PDB structures, 1060 validated residues from ClinVar, DMS, MaveDB, and gnomAD), we establish three fundamental theorems and validate key findings: TSD effectively distinguishes pathogenic/functional variants (PTM: {micro} = 0.386, OMIM: {micro} = 0.443, Clin-Var: {micro} = 0.302) from neutral ones (gnomAD: {micro} = -0.660) with high significance (P = 6.67 x 10 -18) and strong classification performance (AUC=0.82-0.86), while correlating strongly with protein stability ({Delta}{Delta}G, Spearman=0.9794, P = 5.38 x 10 -28). TSD further reveals PTM sites as topological hubs and neutral variants as evolutionary topological redundancy, enabling a paradigm shift from sequence alignment to geometric dynamics and providing a physics-based biomarker for variants of uncertain significance (VUS). These results upgrade protein mutation modeling from empirical static prediction to provable dynamic mechanism analysis. The source code of this work is publicly available at https://github.com/Harmenlv/LieFold-AI/tree/main.