A Time–Scale Operator Geometry for Safety Integrity Based on the De Bruijn Functional

This paper develops a mathematically rigorous operator theoretic framework for dynamicsafety assessment by introducing the De Bruijn functionalΛ(T,H) = ⟨ZT,AHZT⟩,ZT ∈L2([0,T];Rn),where AH : L2([0,T];Rn) → L2([0,T];Rn) is a positive self-adjoint integral operatorwith hazard-severity kernel H. The kernel H encodes time- and state-dependent weighting ofunsafe behavior and may be prescribed from engineering design data, derived from diagnosticmodels, or constructed from consequence severity mappings. Under the assumption H ∈L2 (a scale-dependent operator geometry on a Hilbert space), AH is bounded, translationcovariant, and compact, ensuring a discrete spectral representation. The functional providesa continuous measure of unsafe deviation and degradation accumulation over a finite horizon.Three realizations of the safety signal Z(t) are examined. (i) When Z(t) denotes closedloop deviation from a safe operating manifold, Λ(T,H) quantifies accumulated unsafe excursion. (ii) When Z(t) is a diagnostic residual, the functional characterizes latent fault growthand sensor–actuator inconsistency. (iii) When H is chosen as a consequence dependentseverity kernel, the framework distinguishes low impact deviations from high consequencehazardous states.The paper proves positivity, boundedness, and compact spectral structure of AH, andestablishes a uniform quadratic form boundcH ∥ZT∥2L2 ≤ Λ(T,H) ≤ CH∥ZT∥2L2,showing that all realizations of Λ(T,H) are norm equivalent on L2. These results providea mathematically closed foundation for interpreting Λ(T,H) as a stable, operator norm basedmeasure of dynamic safety integrity.The contribution is positioned as a complementary analytical layer to IEC 61508: thefunctional does not replace probabilistic SIL metrics such as PFDavg or PFH, but augmentsthem with areal time capable, operator theoretic measure that resolves transient degradationand evolving fault structure. This establishes a precise mathematical basis for SIL aligneddynamic safety assurance.