The Failure Mode Theorem

Across multiple foundational domains—axiomatic mathematics, general relativity, and quantum mechanics—decades of sustained theoretical effort have yielded extraordinary internal development without corresponding resolution of core conceptual impasses. These failures are notable not for empirical contradiction or mathematical inconsistency, but for their persistence despite technical sophistication and explanatory success within established regimes. This paper proposes a domain-independent structural explanation for such stagnation. We formulate the Failure Mode Theorem, which identifies a joint condition—over-binding of representational structures into necessity and subsequent constraint closure—as sufficient to induce stable foundational stagnation without falsification. We develop explicit examination protocols and apply them systematically to mathematics, general relativity, and quantum mechanics, including adversarial red-team stress tests. We show that three longstanding paradox families—the measurement problem, the black-hole information problem, and cosmological initial-condition fine-tuning—collapse into a single structural error when analyzed through this framework. We introduce a minimal corrective axiom, Regime-Local Unitarity, and demonstrate where it first bites empirically. Finally, we derive a formal decision procedure for evaluating future theories. The result is not a new physical theory, but a rigorous diagnostic instrument for identifying progress-capable versus structurally stagnant explanatory systems.