The Limits of Empirical Science: Why Theorems Cannot Fix Their Own Axioms, and What Inter-Theoretic Incompatibility Proves

Physical theories are commonly treated as candidates for fundamentality once they achieve sufficient empirical confirmation within some domain. This paper argues that confirmation within a domain cannot establish fundamentality, because the empirical method operates exclusively on theorems — scale-relative deductive consequences — and a theorem, however well confirmed, never uniquely determines the axioms that entail it. This is not a rhetorical claim: it is a formal fact about deductive systems, illustrated within pure mathematics — axiomatically equivalent systems, the interchangeability of Euclid's Fifth Postulate and the triangle-angle-sum theorem, and the base-relativity of equivalences in reverse mathematics — before being applied to physical theory. From this formal fact follows a diagnostic: since true physical axioms, being axioms of one consistent reality, cannot contradict one another, any two mutually incompatible, well-confirmed physical theories cannot both rest on true axioms — at least one, generically both, rest on theorems mistaken for axioms. Quantum mechanics and general relativity are argued to be the paradigm instance. This reframes the search for a unified theory: literal unification of the Standard Model's and general relativity's founding postulates is not an unfinished technical project but a category error, since no operation exists for consistently merging two postulate sets neither of which is axiomatic. The paper engages directly with the strongest available objection — that general relativity is already treated as an effective field theory with a known finite domain of validity, making the incompatibility unremarkable — and argues that this framing is itself a tacit concession of the paper's central claim, not a refutation of it, since the standard remedy of pursuing a UV completion by the same effective-theory method cannot, by the underdetermination result, converge on true axioms regardless of how far it is extended. A remediation is developed from within the existing QGD/MPDT programme: axioms must be chosen independently of the empirical bottom-up method and certified instead by their derived, scale-bridging, testable consequences — a standard the programme's existing apparatus (the Constructibility Criterion, the Uniqueness Theorem) already supplies. Structural realism and the pessimistic meta-induction are engaged as independent corroborating lines of evidence, and the paper closes by applying its own diagnostic to itself.