The Constructibility Criterion: A Unified Foundation for Physically Derivable Mathematics and Physically Derivable Theory

This paper establishes the Constructibility Criterion — a single formal requirement operating at a logical level prior to truth, provability, and falsifiability — from which both the foundations of Physically Derivable Set Theory (PDST) and the conditions of the Minimally Physically Derivable Theories (MPDT) metatheory follow as derived consequences. The Criterion consists of three jointly necessary conditions on any statement, whether mathematical or physical: domain constructibility (FC1 — every object quantified over must be producible by a finite physical process), predicate decidability (FC2 — every predicate applied must be decidable by a finite physical process), and operation termination (FC3 — every operation invoked must terminate within the universe's finite resource bound ∞̂). A statement that fails any condition is pre-formally void: it names no determinate subject matter and admits no verdict of truth, falsity, provability, unprovability, falsifiability, or unfalsifiability. It does not occupy a position in the logical space of verdicts — it falls outside that space entirely. The Criterion is derived from QGD's two foundational axioms (discrete preonic space; kinetic preonic matter), not postulated independently. The paper proves the Preonic Formulation Theorem (PFT), that a statement is preonic-well-formed if and only if it satisfies FC1–FC3 jointly, and derives five corollaries: C1 (Mathematical Voidness): Classical set-theoretic results requiring completed infinite objects — the Continuum Hypothesis, Cantor's theorem on ℙ(ℕ), Gödel incompleteness sentences, the Banach–Tarski decomposition — are pre-formally void, not false or unprovable. C2 (Physical Voidness): Every major current physical framework contains at least one pre-formally void axiom — the continuous spacetime manifold (FC1), the infinite-dimensional Hilbert space (FC1 + FC3), or the vacuum energy sum over infinite modes (FC1 + FC3). C3 (Voidness Propagation): Voidness propagates through every derivation depending on a void axiom; the three-step downstream pathology pattern (void import → inadmissible consequences → ad hoc suppressor) is its formal description; no within-theory suppressor can remove it. C4 (MPDT Derivability): The four MPDT conditions (C1–C4 of the Uniqueness Theorem) are FC1–FC3 applied to the four structural roles of a fundamental physical theory's axiom set. MPDT is not an independent metatheory — it is a derived application of the Criterion. C5 (PDST Maximality): PDST is the unique maximal preonic-well-formed mathematical system. A dedicated section (§5.9) establishes consistency, completeness, and decidability of PDST on purely logical grounds — via the truth-determinancy of classical two-valued logic over decidable predicates on finite domains — independently of any physical commitment, with the standard V_ω model within ZFC acknowledged as an independently available verification. The two domains of application — pure mathematics and physical theory — are unified under a single criterion because mathematics is a subset of physics: mathematical practice is a physical process, and the bounds on physical construction are the bounds on mathematical construction. Wigner's puzzle about the unreasonable effectiveness of mathematics in physics dissolves: the mathematics that is effective is precisely the mathematics that was never pre-formally void. The paper is P40 of the QGD/MPDT programme and is the capstone of its mathematical foundations sub-programme (companion papers: P12, P17, P27, P28, P36) and its metatheoretic sub-programme (P1, P18, P21, P22).