A Closed, Observable, Composable, and Provably Safe Dynamical Kernel for SIS-10 over Apache Kafka

This paper introduces SIS-10 (Safety Integrity System–10), a minimal axiomatic kernel comprising ten formally independent axioms that collectively guarantee safety invariance in autonomous and AI-driven systems. Unlike probabilistic safety frameworks that quantify tolerable failure rates, SIS-10 defines safety as a logical invariant derived from first principles: no axiom is derivable from the remaining nine, each constrains a distinct behavioral dimension, and their conjunction is both necessary and sufficient for invariance, bounded recovery, and compositional safety. We formalize the system model with extended definitions for multi-step transitions and parallel composition, prove axiom independence constructively via finite-state countermodels, and establish nine theorems covering safety invariance, bounded recovery, timed progress, idempotent retry safety, compositionality preservation, non-absorbing unsafe states, observable degradation, fail-safe containment, and deployment safety. A Metatheorem on Kernel Tightness shows that each axiom is load-bearing for at least one theorem. We establish a novel LTL Encoding of all ten axioms and prove that SIS-10 compliance is decidable in PSPACE for finite-state systems. The kernel is instantiated on a class of event-driven transactional architectures with Apache Kafka as a concrete exemplar and operationalized through a formalized CI/CD safety gate and runtime monitoring framework. A five paradigm comparison positions SIS-10 relative to probabilistic SIL frameworks, formal verification, control-theoretic stability, process algebras, and emerging AI governance mandates, supported by a formal coverage table showing which axioms each paradigm addresses. SIS-10 does not replace existing safety standards; it defines the logical foundation they presuppose but have not formalized. Keywords: safety integrity, axiomatic kernel, formal methods, linear temporal logic, autonomous systems, CI/CD enforcement, functional safety, compositional verification, PSPACE decidability.