CM Optical Papers III: Dispersion as a Non-linear Temporal Response and the Deterministic Derivation of the Abbe

This paper provides the first deterministic derivation of optical dispersion and the Abbe invariant from the principles of Cognitional Mechanics (CM). Moving beyond the empirical fits of the 19th-century Sellmeier equations, we redefine the refractive index not as a material constant, but as a measure of computational latency within the spacetime manifold Mn(C). The core of our argument establishes that physical frequency represents the density of unitary information updates per manifold clock cycle. By utilizing the spatial packing invariant (n = 4/3) established in the first paper of this series, we demonstrate that dispersion is a non-linear temporal response caused by the computational saturation of the manifold. The highlight of this work is the quantitative derivation of the Abbe number (V), reinterpreted as a "Stability Coefficient" of the manifold's temporal update process. We validate our model against experimental data for distilled water, achieving a refractive index prediction accuracy of better than 0.03% across the visible spectrum and predicting the Abbe number with less than 2.0% error from first principles. This work bridges the gap between static spatial refraction and dynamic temporal response, completing the core axis of the CM optical framework and setting the stage for the final study on information decoherence (scattering).