Universal Deviation Principle: A Multi-Operator Framework for Anomaly Detection and Classification with Sensitivity Guarantees

We introduce the Universal Deviation Principle (UDP), a theoretically grounded framework for multi-perspective anomaly detection. The core principle is that, under explicit assumptions on operator-family richness and anomaly-normal separation in feature space, a point which is meaningfully different from the normal set cannot simultaneously match normality under all structural perspectives. We formalise this via deviation-separating operator families—collections of differentiable maps whose stacked Jacobian has full column rank—and prove six results: (1) local distinguishability via the Inverse Function Theorem; (2) global sensitivity via compactness and the Extreme Value Theorem; (3) finite-sample guarantees via Hoeffding concentration; (4) fusion weights maximising worst-case directional sensitivity via a semidefinite programme; (5) a greedy operator selection algorithm with (1−1/e)-approximation guarantee via monotone submodularity; and (6) operator-space separation converting injectivity into a detection guarantee. All results are conditional on three stated assumptions (observability, operator-family richness, calibration discipline). The framework is instantiated with 14 composable spectrum projection operators across five mathematical families (basis decomposition, geometric embedding, algebraic, topological, and information-theoretic). A correlation audit across all 91 operator pairs reveals only one redundant pair (Wavelet↔Phase, ρ = 0.92), validating the multi-spectrum thesis. Three System Mode engines (Molecular, Gravity, Hybrid) with Lyapunov v2 stabilisation provide physics-based anomaly scoring with Armijo + ISS + La Salle convergence guarantees. A fused topological scoring architecture (Morse topology alarm, Betti barcode suite, and UDL post-simulation operators) is structurally decoupled from the solver, ensuring stable prediction. Key empirical results:• UDL-Hybrid achieves mean AUC 0.972 and 91% mean coverage across five ODDS benchmarks with a single hyperparameter set, exceeding all established baselines (Isolation Forest: 0.946, LOF: 0.962, ECOD: 0.918).• A Boundary-Centred Dimension Magnifier paired with QDA achieves strong supervised anomaly detection performance.• Real-world validation on three sensitive-sector datasets: 25 G-SIB banks (6-quarter GFC early warning, FAR = 1.6%), U.S. economy via FRED macro indicators (AUC = 0.9999), and ERCOT Texas grid failure detection (AUC = 0.9940, 37-hour lead time). All achieve FCA-compliant false-alarm rates. Classification extension: per-class MDN decomposition achieves mean accuracy 0.936 across six benchmarks—competitive with LightGBM (0.934) and RF-200 (0.933)—demonstrating dual-use capability of the multi-spectrum representation.