Geometric Rigidity of Cauchy-Barycentric Coordinates and the Horizontal Confinement of Riemann Zetas

This paper introduces a unied geometric framework for the Riemann Hypothesis (RH) and prime number prediction, based on the Geometric Rigidity of Cauchybarycentric coordinates. We demonstrate that the non-trivial zeros of the Riemann zeta function ζ(s) correspond uniquely to states of perfect rotational equilibrium within a dynamic control polygon P n governed by the Cauchy kernel. Through the analysis of the Cauchy-Gram Matrix, we prove that any displacement δ from the critical line Re(s) = 1/2 induces a breakdown of circulant symmetry, generating a strictly positive Geometric Torque T (δ). This torque acts as a topological restoring force, enforcing the horizontal confinement of zeros on the symmetry axis. Beyond connement, we reveal that the system's stability reaches its global maximum when the polygon's cardinality n is a prime numbera conguration that minimizes spectral entropy and maximizes the Rigidity Gradient ∆(n). This fundamental property allows us to derive a general formula for the prime characteristic function ⊮ P (n) and an iterative successor operator to predict p k+1 from p k . By transmuting prime arithmetic into a dynamics of crystalline resonance, this work provides a deterministic perspective on the distribution of primes and the underlying architecture of the Zeta landscape.