Higher-order dual extensions of dual-type octonions with explicit inverses and block matrix representations

A third-order nilpotent augmentation of the dual-type octonion framework was obtained by replacing the first-order dual layer with a higher-order infinitesimal structure. Algebraic closure and coefficient-wise product expansions were established, enabling all subsequent derivations to be performed in a finite form. A comprehensive invertibility condition was established by limiting the problem to the leading component, and a closed-form inverse was constructed directly up to the second order, producing a computation-ready formula for reciprocal evaluation. The structure of zero divisors and nilpotent elements was characterized, showing how degeneracy and non-invertibility spread via higher-order perturbation layers. The nilpotent constraint was used to establish finite exponential and power expansions, resulting in identities of the Euler and De Moivre types that were obtained for the third-order scenario and explicit formulations for repeated products. A block triangular left-multiplication matrix representation was constructed, from which the determinant and spectral consequences were obtained with the inversion formulas. The resulting framework provides an algebraic tool for explicit computation within dual-type octonionic models and enables higher-order perturbation encoding.