The Theory of Axiomatic Necessity (TNA) is frequently interpreted as a metaphysical or philosophical framework; however, this paper establishes it rigorously as a genuine ontological calculus. By defining three primitive domains—Unlimited Structural Possibility (), the Descriptive Domain (), and the Instantiated Domain (^*) —and three structural operators—Projection (), Coherence Restriction (C), and Instantiation (I) —TNA provides a formal syntax for computing the structural limits of any sufficiently rich system. The calculus is governed by the Structural Non-Derivability Corollary: ^* which mathematically isolates the Locus of Non-Derivability exclusively at the Instantiation operator (I). This proves that while and C are computable formal operations, I is structurally non-computable from within the descriptive domain. To demonstrate this calculus in action, consider the domain of Artificial Intelligence: a Large Language Model can perfectly execute (generating candidate structures) and C (filtering for syntactic and logical coherence within), but it structurally halts at I, as it cannot derive its own semantic instantiation (^*) from its weights and algorithms. By remaining semantically neutral, this ontological calculus unifies seemingly disparate limits—from Gödelian incompleteness and the Knowledge Argument to cosmological boundaries—into a single, invariant structural grammar, demonstrating that the Failure of Local Closure is not a metaphysical mystery, but a formal, computable boundary of reality.
Claudio Bresciano (Wed,) studied this question.