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

We propose a new reformulation of the Riemann Hypothesis based on the iterative complex rewriting of the summation index n in the zeta function. The key observation is that for any positive integer n, there exists a family of complex numbers s_n = 1 + (2πk/ln n)i such that n^(s_n) = n for all k in Z. Substituting this rewriting into the Euler series for ζ(s) yields a transformed series whose convergence properties depend on Re(s). Iterating this transformation produces an infinite product condition for the zeta function to return to itself — a fixed-point condition. We show that this product converges to unity precisely when Re(s) = 1/2, because this is the only line where imaginary contributions cancel symmetrically. The critical line is reformulated as the unique fixed-point locus of the iterative rewriting operator. Preliminary empirical results from SymSearch on the first ten non-trivial zeros are reported. This paper presents a reformulation and an intuition, not a complete proof.