This paper provides a analytical proof of the Riemann Hypothesis using a differential interaction operator Φ(s,δ) on the Hilbert space l2(N). By mapping the Dirichlet η-function to a trace-class operator representing the interaction between states shifted by ±δ from the critical line, we derive a Phase-Torque J(δ,t) governed by a hyperbolic sine bias. We establish a Product Criterion showing that the operator trace vanishes if and only if a zero exists at either 1/2 + δ + it or 1/2 − δ + it. Finally, we establish the convergence criteria for this operator and demonstrate that the Diophantine independence of prime logarithms, amplified by the hyperbolic lever, prevents the trace from vanishing off the critical line.