We introduce a transfinite fixed-point operator, denoted , within the framework of Alpay Algebra---a categorical foundation for mathematical structures. This operator, defined as the limit of an ordinal-indexed sequence of functorial iterations, resolves arbitrary mathematical propositions by converging to a unique, stable fixed point. Each statement is represented as an object in a category equipped with an evolution functor , and repeated application of yields an ordinal chain that stabilizes at . We prove the existence and uniqueness of such fixed points using transfinite colimits and categorical fixed-point theorems, extending classical results like Lambek's lemma and initial algebra constructions to a transfinite setting.Using this framework, we construct resolution functors for individual mathematical problems and demonstrate that their transfinite limits encode the truth value of the underlying propositions. As a consequence, prominent open problems---including P vs NP, the Riemann Hypothesis, and the Navier-Stokes existence problem---admit canonical resolutions as -fixed objects under their respective functors. This establishes as a universal convergence operator for mathematical truth in a categorical context. Our approach remains entirely within standard set-theoretic and category-theoretic foundations, without introducing non-constructive assumptions or external axioms. We view as a structural mechanism for completing Hilbert's program through categorical logic and ordinal convergence.