A Z3-Graded Lie Superalgebra with Cubic Vacuum Triality

We construct a finite-dimensional Z3-graded Lie superalgebra of dimensions (12,4,3), featuring a grade-2 sector that obeys a cubic bracket relation with the fermionic sector. This induces an emergent triality symmetry cycling the three components. The full set of graded Jacobi identities is verified analytically in low dimensions and numerically in a faithful 19-dimensional matrix representation, with residuals ≤8×10−13 over 107 random tests. Explicit quadratic and cubic Casimir operators are computed, with proofs of centrality, and the adjoint representation is shown to be anomaly-free. The algebra provides a minimal, closed extension beyond conventional Z2 supersymmetry and may offer an algebraic laboratory for models with ternary symmetries.