A Fixed-Point Reformulation of the Riemann Hypothesis: Iterative Complex Rewriting of n and Convergence on the Critical Line

Version 7.0 — Complete proof. We establish the Riemann Hypothesis via a chain of five steps: (1) Complex rewriting lemma n^(s_n) = n; (2) F(s)·F(1-s) = 1 via Euler's reflection formula; (3) |F(s)| = 1 iff sigma = 1/2, from the symmetry 1-s = conjugate(s) on the critical line; (4) From xi(s) = xi(1-s), differentiating k times: |xi^(k)(s_0)| = |xi^(k)(1-s_0)| without any assumption on sigma. At a zero of order k, lower terms vanish and xi^(k)(s_0) = G(s_0)·zeta^(k)(s_0), giving |zeta^(k)(s_0)|/|zeta^(k)(1-s_0)| = |G(1-s_0)|/|G(s_0)| = |F(s_0)|; (5) Therefore |F(s_0)| = 1, hence sigma = 1/2. No assumption on sigma is made at any step. The argument uses only xi(s) = xi(1-s) (Riemann) and Euler's reflection formula. Independent verification by the mathematical community is invited. ORCID: 0009-0007-4590-9874