The De Bruijn Functional: A Unified Operator-Theoretic Metric for Safety Instrumented Systems (SIS, SIF, SIL)

This paper introduces the De Bruijn functional Λ(T, H) as a mathematically rigorous, operator-theoretic, and kernel-based metric suitable for Safety Instrumented Systems (SIS), Safety Instrumented Functions (SIF), and Safety Integrity Level (SIL) assurance. We show that Λ(T, H) provides a localized, positive, self-adjoint quadratic form capable of quantifying unsafe deviation, fault intensity, and risk-weighted deviation. Three plant-relevant definitions of Z(t) are developed: control error, diagnostic residual, and consequence-weighted deviation. It is a similar concept as I presented in my paper (Functional Safety is Not a Snapshot: Modeling SIL(t) via Mosaic Aging and Stochastic Hazard Rates. https://doi.org/10.5281/zenodo.18705930) We demonstrate that each yields a structurally robust safety metric with clear SIL claim patterns. This is futuristic approach to functional safety.