Continuous Extension and Lyapunov Stability in the Gilbreath Difference Triangle

Building on the exact signed-path expansion of the Gilbreath difference triangle, we map the discrete prime gap sequence into a continuous piecewise-linear dynamical system. We prove that the discrete path coefficients derived in prior work are exact evaluations of the system's Jacobian matrix. By resolving non-differentiable hinges using the Clarke subdifferential, we establish that the zero-sum coefficient law is a direct consequence of gauge invariance under uniform translation. Finally, we demonstrate that the Gilbreath Conjecture is mathematically equivalent to proving that the upper Lyapunov exponent of this specific subgradient product is strictly negative, reducing the conjecture to a problem of asymptotic stability and ergodic perturbation extinction.