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

Version 4.0 — Complete argument chain with one identified gap. We propose a new reformulation of the Riemann Hypothesis based on the iterative complex rewriting of n in the zeta function. Four steps are demonstrated rigorously: (1) the complex rewriting lemma n^(s_n) = n; (2) transformation of zeta(s) with phase factorisation w_n = e^(2pit) * n^(-sigma); (3) phases centre on 2pi*sigma — equal to pi iff sigma = 1/2; (4) Unification Theorem — fixed-point and maximum conditions both reduce to Im = 0 iff sigma = 1/2. The closed loop condition is stated: zeta(s) = 0 iff the phase vectors form a closed loop iff sigma = 1/2. The single remaining gap is precisely identified — showing that asymmetric phase distribution prevents loop closure. The community is invited to close this gap. ORCID: 0009-0007-4590-9874