Sectorial Green Functions in the Ternary Algebra C3 and Their Curved-Space Extension

Green functions associated with higher-order differential operators typically lead to special-function expressions in curved or bounded geometries, obscuring analytic transparency. In this work we develop the sectorial Green function for the cubic operator $(D^3+1)$ within the ternary algebra $C_3$ (defined by $j^3=-1$). The algebra admits three exponential carriers and divides the complex plane into six Stokes sectors, in each of which the Green kernel assumes a closed exponential--trigonometric form. We compute explicit responses to box and Gaussian sources, extend the construction to higher algebras $C_4$ and $C_5$, and interpret the resulting kernels as propagators for multi-carrier quantum systems. The central novelty is that, unlike the quadratic $(D^2+1)$ case where curved backgrounds induce Bessel or Airy functions, the $C_3$ Green function in curved space reduces exactly to its flat-space form upon reparametrization by the geodesic coordinate. Curvature merely deforms the coordinate without altering the analytic structure, ensuring closed-form transparency even in curved geometries.