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We introduce a transfinite fixed-point operator, denoted $\phi^\infty$, 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 $\phi$, and repeated application of $\phi$ yields an ordinal chain that stabilizes at $\phi^\infty$. 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 $\phi_P$ 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 $\phi^\infty$-fixed objects under their respective functors. This establishes $\phi^\infty$ 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 $\phi^\infty$ as a structural mechanism for completing Hilbert's program through categorical logic and ordinal convergence.