The Theory of Persistence proposes that reality maintains structure through harmonic selection in viscous media. This paper elevates the framework from theory to theorem by demonstrating logical necessity. The Three Axioms: Reality is what persists (R = P) Persistence requires structure (P → S) Knowledge is structure (K = S) The Theorem: Given these axioms, reality and knowledge are necessarily the same category—both are structure. The traditional distinction between "real" (ontic) and "informational" (epistemic) dissolves. The Pusey-Barrett-Rudolph theorem (2012) proved quantum wavefunctions are "real" rather than merely "informational," preserving the ontic/epistemic distinction. The Persistence Theorem shows this distinction is incoherent: if what persists is real, persistence requires structure, and knowledge is structure, then reality and knowledge unify as structure. Knowing is structure recognizing structure. φ recognizing φ. The universe knowing itself is not metaphor but mathematical identity. "Everything sings" transforms from poetic description to logical necessity. The Theory of Persistence now has both empirical support (p < 0.001 across 69 gravitational wave events) and logical foundation (the Persistence Theorem).