Range Control as the Foundational Unification of AI Alignment and Safety: Why Probabilistic Frameworks Necessarily Separate Them

This paper advances three substantive claims on the foundational architecture of AI alignment and safety. First, the mainstream distinction between alignment (goal fidelity) and safety (system resilience) is a byproduct of probabilistic frameworks, not a foundational necessity. Probabilistic optimization tools cannot provide hard guarantees and therefore require separate safety mechanisms as backup; the separation reflects tool limitations rather than categorical distinction. Second, in the framework of Structural Determination Theory (SDT), where modal-structural admissible ranges A_v are foundational, alignment and safety are achieved through a single design choice: constraining A_v as a subset of A_intended. Alignment is positively specified, safety is negatively achieved (harmful outputs excluded as logical consequence); the two are logically ordered—alignment-prior, safety-by-consequence. Third, this range-based foundation yields catastrophe prevention (via minimax sufficiency and the dual-layer paradigm of the SDT corpus) and probabilistic optimization efficiency as joint substantive benefits. The paper articulates four mathematical theorems (UA-A1 through UA-A4) formalizing the foundational unification, the probabilistic necessity of separation, the search space reduction, and the modal-structural inevitability of specification gaming. AI safety as a problem domain has a substantive modal-structural footprint—binary catastrophe, worst-case adversarial robustness, structural invariance across deployment—that probabilistic foundations cannot capture without loss. The paper situates Constitutional AI training, Constitutional Classifiers, RLHF, Defense in Depth, and constrained reinforcement learning within this framework, identifying their structural gap and providing foundational justification for their existing complementary deployment. The argument extends a modal-structural categorical-priority methodology to AI safety: such foundations categorically reorganize what entity-theoretic frameworks treat as separate categories.