Physics-RAG: Invariant Latent Manifolds, Conformal Uncertainty Quantification, and Hierarchical Cascade Retrieval for Continuous Physical Systems

Standard Retrieval-Augmented Generation (RAG) and dense vector search frameworks rely on unconstrained text embeddings that frequently yield physically inconsistent retrievals when applied to continuous physical systems. In environments governed by non-linear conservation laws, linguistic proximity precipitates the Semantic-Physical Disconnect: models retrieve states that share superficial textual descriptions but differ fundamentally in dimensional scale, phase boundaries, or non-equilibrium thresholds. Furthermore, database querying encounters a dual failure mode: relational queries suffer from chronological blindness (defaulting to calm pre-storm onset hours rather than dynamic apexes), whereas unconstrained metric vector search suffers from cross-decadal manifold degeneracy (matching scale-invariant dimensionless numbers across disparate epochs and losing synoptic meteorological context). Moreover, standard surrogate models output brittle point predictions lacking distribution-free uncertainty guarantees, leaving downstream foundation models susceptible to unquantified uncertainty during acute physical shocks. To resolve these failure modes, we propose the Two-Tier Hybrid Cascade Physics-RAG architecture augmented with Conformalized Quantile Regression (CQR). Our framework projects heterogeneous multi-modal telemetries into a scale-invariant, 14-dimensional physical manifold derived from first-principles conservation laws and dimensional similitude. Retrieval is executed through a two-tier cascade: Tier 1 performs sub-15-millisecond zero-copy columnar relational pre-filtering via DuckDB to enforce spatial, temporal, and hard boundary constraints, while Tier 2 executes non-Euclidean manifold ranking governed by an asymmetric Directional Pareto objective that prioritizes physical disequilibrium excess over symmetric geometric proximity. To combat Markovian memorylessness, the architecture hydrates continuous parent sequences (±72 hours). Crucially, we integrate a distribution-free Conformal Prediction engine calibrated via multi-site stratified chronological splits, generating finite-sample valid prediction intervals that dynamically expand and contract with hydrograph volatility. We evaluate this architecture across an empirical proving ground of extreme coastal morphodynamics spanning 175,392 contiguous hourly hydrodynamic sensor records and N = 1,228 authentic in-situ topographic profile surveys across four contrasting global archetypes (USACE FRF Duck, NC; UNSW WRL Narrabeen, Australia; SIO Torrey Pines, CA; and Univ. Bordeaux Truc Vert, France). On a comprehensive N = 100 query benchmark evaluated against deterministic physical ground-truth predicates, the Two-Tier Hybrid Cascade achieves statistically significant performance gains over both standard external baselines (Okapi BM25: P@5 = 0.052, p = 8.23e-19; Google Dense Embeddings: P@5 = 0.366, p = 0.00019) and internal ablation variants (P@5 = 0.574, NDCG@5 = 0.597, MRR = 0.598, Regime Purity = 0.614). Out-of-sample CQR validation across N = 35,084 records achieves 91.2% empirical coverage for a 90.0% target with 314.5x dynamic interval adaptation (0.018 m to 5.662 m). To rigorously evaluate continuous SciML surrogate rollouts without observational survey intermittency, we formulate an Observing System Simulation Experiment (OSSE) synthetic nature run driven by authentic multi-year boundary forcing; we demonstrate that invariant manifold features dominate multi-year correlation (r = 0.574, reaching r = 0.839) while PINN soft-loss regularization stabilizes acute storm shocks (RMSE = 0.342 m), explicitly distinguishing synthetic continuous-operator verification from real-world in-situ field survey validation. Finally, we outline how these invariant-guided retrieval principles extend conceptually to other continuous cyber-physical systems. Code, benchmark query suites, and replication runners are openly accessible at:https://github.com/sam-tritto/unified-coastal-erosion-index