The Mathematics of Transactional Homeostasis: Formal Foundations and Bayesian Risk Regulation for Stewardship Intelligence

AbstractThis paper presents the formal mathematical foundations for the Transactional HomeostasisFramework and Stewardship Intelligence architecture described in our companion paper. Wedevelop three interconnected formal systems: (1) a transactional calculus modeling reciprocalsignaling dynamics and temporal mismatch—presented as interpretation-agnostic formalisminspired by but not dependent on the Transactional Interpretation of quantum mechanics; (2) ahomeostatic attractor model describing how bonded systems maintain dynamic coherencethrough CDR (Condensation-Decoherence-Recoherence) cycles, with operational definitions forcomputational systems; and (3) a Bayesian risk regulation framework for the five-step safetypipeline.We demonstrate that the pipeline achieves substantial risk reduction, with illustrative parameterssuggesting approximately 12-19× improvement from baseline depending on correlationassumptions. We explicitly acknowledge that parameter values require empirical calibration; themathematical machinery is the contribution, with specific numbers serving as placeholders forfuture empirical work. We show that treating safety as continuous homeostaticregulation—rather than static alignment—enables auditable, adaptive risk management. TheBayesian framework models our epistemic state about system safety, complementing rather thanreplacing the structural safety constraints of the SHAI architecture. All models remainprovisional and open to productive failure and empirical refutation.Keywords: Transactional calculus, homeostatic attractors, Bayesian risk regulation, CDR cycle,dynamic coherence, safety pipeline, formal verification, Shared Humanistic AI