Gödel's Theorem Does Not Obstruct Physical Closure: A Cost-Theoretic Resolution via Recognition Science

Gödel's incompleteness theorems establish that no consistent formal system containing arithmetic can prove all arithmetic truths. This is sometimes cited as an obstruction to closed physical theories. We show this objection is misapplied. Recognition Science (RS) defines truth not as Tarskian satisfaction but as stabilization under cost minimization, where the cost functional J(x) = 1/2 (x + x⁻¹)⁻¹ is the unique function satisfying the d'Alembert composition law with appropriate normalization and calibration. Under this definition, Gödel sentences—which query their own provability—translate to configurations that query their own stabilization status. We prove that such self-referential stabilization queries cannot have fixed points under J-iteration: they neither stabilize nor diverge cleanly, and hence fall outside the RS ontology entirely. The Gödel phenomenon is thereby reclassified: these are not true but unprovable statements, but rather non-configurations—syntactically well-formed strings that do not correspond to elements of the physical ontology. Closure in RS means a unique J-minimizer exists, not that all arithmetic truths are provable. Gödel's theorem, correctly understood, constrains formal proof systems, not cost-theoretic physics.