Structural Determination Theory: Mathematical Certification Matrix

This paper renders Structural Determination Theory (SDT) as a formal deductive system in classical multi-sorted first-order logic, with ZFC as the metatheory. The central formal device is a two-tier explication relation (object-level and meta-level) that makes "conceptual explication" (the inherent unfolding of a single axiom together with the meaning of the terms it uses) a precisely defined formal relation, distinct from derivation from an axiom-set. This distinction is load-bearing: a formalization that conflated the two would render SDT, by its own formal output, a multi-premise theory rather than the single-axiom theory it claims to be. The paper formalizes the single axiom and its three-level semantic unfolding, formalizes the self-refutation argument as a four-branch reductio at the meta-level, gives formal definitions of all eleven core concepts (E, P, G, R, C, A, L, D, τ, T, G_D) each accompanied by a conservativity lemma, and certifies the four core arguments (R non-retraction, G constancy, time emergence, spacetime trichotomy) as strict object-level explications. Three metamathematical results are established: the minimal closure properties of the explication relation, the conservativity total theorem (no definition introduces new ontological commitment), and the relative consistency of the full system in ZF via a constructive model. The graph-theoretic content invoked here (the determination dependency graph G_D) is restated to the level the formal derivations require; its full development is canonical in the companion paper GD v6.0.