The DeBruijn Functional: Enhanced Metric for Safety Intellegence Systems.

This paper introduces the DeBruijn functional Λ(T,H) as a mathematically rigorous operatortheoretic framework for the analysis and assurance of Safety Instrumented Systems (SIS), SafetyInstrumented Functions (SIF), and Safety Integrity Level (SIL) performance. The proposedfunctional defines a localized, positive, self-adjoint quadratic form capable of quantifying unsafedeviation, fault intensity, degradation accumulation, and consequence-weighted hazard exposurewithin a unified analytical structure. Three physically meaningful realizations of the safety signal Z(t) are developed. The firstinterprets Z(t) as closed-loop control deviation from a prescribed safe operating manifold, yielding a direct measure of accumulated unsafe process excursion. The second interprets Z(t) asa diagnostic residual generated from model–plant disagreement, thereby capturing latent faultgrowth, sensor degradation, actuator inconsistency, and structural anomalies. The third extends the framework to risk-weighted safety analysis by embedding spatially and operationallydependent hazard severity kernels into the functional, allowing the metric to distinguish betweenbenign deviations and deviations with catastrophic consequence potential. Under suitable kernel regularity conditions, the resulting DeBruijn functional admits acompact spectral representation and possesses strong stability, localization, and monotonicity properties relevant to SIL verification. The framework naturally unifies time-domain safetymonitoring, frequency-domain degradation characterization, and consequence-sensitive hazardquantification within a single mathematically closed formalism. Unlike conventional SIL metricsbased solely on probabilistic averages such as PFDavg and PFH, the proposed approach provides a continuous analytical measure of dynamic safety integrity capable of resolving localizedunsafe behavior, transient degradation phenomena, and evolving fault structures. The resultingtheory establishes a forward-looking foundation for mathematically rigorous functional safetyassessment, real-time safety intelligence, and next-generation operator-theoretic SIL assurancemethodologies.