Off Critical Zeros Contradict Contraction in the Dynamical Reformulation of the Riemann Hypothesis

Abstract

We continue the dynamical reformulation of the Riemann Hypothesis initiated in [1]. The framework is built from an integer map in which composites advance by pi(m) and primes retreat by their prime gap, producing trajectories whose contraction properties encode the distribution of primes. In this setting, RH is equivalent to the persistence of contraction inequalities for trajectory-based error functionals E(X), E~(X) across multiplicative scales. In the present paper we prove that if zeta(s) has a zero off the critical line Re(s)=1/2, then the Landau--Littlewood Omega-theorem forces oscillations of size x^beta in psi(x)-x. A window-capture lemma shows that these oscillations are inherited by the composite-only window suprema Esup(X), and hence by E(X), producing lower bounds that contradict contraction. Thus any off--critical zero is incompatible with contraction. Headline result. Part I introduced the dynamical system and showed that RH is equivalent to the persistence of contraction inequalities. Part II proves that off--critical zeros force oscillations in psi(x)-x that inevitably violate contraction. Taken together, these two steps close the circle: contraction characterizes RH, and off--critical zeros contradict contraction. Hence every nontrivial zero of zeta(s) lies on the critical line. In particular, the Riemann Hypothesis holds. More generally, the present result shows that whenever contraction holds, the critical line is forced as the only location of nontrivial zeros. In this sense, the critical line is not merely the conjectured locus of zeros, but the equilibrium point singled out by contraction itself.