Structural Infinity: A Coalgebraic Unification of Infinite Phenomena

We introduce a categorical framework that unifies several heterogeneous notions of infinity—Dedekind infinity, analytic divergence, continuum refinement, and the higher-dimensional identity tower of Homotopy Type Theory (HoTT)—under a single coalgebraic principle. Given an endofunctor G: C -> C and a coalgebra gamma: X -> G X, structural infinity is defined by the non-invertibility of all transition maps delta_n: G^n X -> G^{n+1} X. We show that many classical and homotopical infinitary phenomena are precisely those coalgebras whose unfolding never stabilizes at any finite depth, reflecting a generative perspective in which mathematical structures emerge from an underlying field of potentiality.