Non-Finite H: Infinite Histories, Limits, and Colimits

This paper removes the finiteness assumption. In 2026a, we proved no unital map T → E for H = S^1. In 2026b, we generalized to the triple H^n(T(H)) = (dim H^n, dim C^n, 1) for all finite H, all n. In 2026c, we defined the Bridge Conjecture. In 2026d, we proved the eraser must be CPTP decoherence. In 2026e, we proved all four follow from one Unifier Axiom: [x,y] = 1, and no unital φ: T(H) → E preserves [x,y]. In 2026f, we categorified H: histories as objects, T as functor, erasure as functor, No-Go as natural. In 2026g, we proved the Unifier survives in characteristic p, via the Frobenius center k[x^p,y^p]. Here we ask: does the Unifier survive when H is infinite? A real spacetime M is not finite. A real history is not a finite simplicial set. We define H = lim→ H_i as a colimit of finite H_i, and T(H) = lim← T(H_i) as a limit. We prove the No-Go survives the limit: if it holds for each finite H_i, it holds for H. We prove the triple extends to infinite H as (dim H^n, dim C^n, 1) where dimensions may be infinite, but c = 1 remains 1. We prove erasure in the infinite case must still be CPTP, and must be continuous in the limit topology. This does not change the experiment. It removes the last toy restriction: finiteness. Tseglin 2026h = Ch. 143. Cites Tseglin 2026a DOI 10.5281/zenodo.23977165, 2026b DOI 10.5281/zenodo.23980498, 2026c DOI 10.5281/zenodo.23980738, 2026d DOI 10.5281/zenodo.23981860, 2026e DOI 10.5281/zenodo.23982063, 2026f DOI 10.5281/zenodo.23982359, 2026g DOI 10.5281/zenodo.23985216.