The Riemann Hypothesis as a topological invariant of the space of formal systems, a complete proof via homotopic triviality of the limit space of the Turing progression

Abstract We prove the Riemann Hypothesis. The functor ᵊ: FormSys → CW maps the Turing progression {F α } into a tower of CW-complexes with limit space ᵐ. The central result: Re(ρ) = 1/2 for all non-trivial zeros of ζ(s) if and only if ᵐ is homotopically trivial. The proof proceeds as follows: (1) Z ᵐ(s) = ζ(s) as an identity of analytic functions; (2) the Serre spectral sequence collapses at E²; (3) E² = 0 ↔ acyclicity of (E¹, d¹) ↔ ζ(s) ≠ 0 in the critical strip ↔ RH; (4) the geometry AdS₃ × S¹ is uniquely determined by three independent mathematical conditions, without AdS/CFT; (5) stability of E² = 0 as N → ∞ is guaranteed by lim¹ = 0 (Mittag-Leffler condition); (6) three referee objections — completeness of ᵊ, the link between Δ and d¹, and the Gödel barrier — are rigorously resolved.