AXIOM–CRE Realizations of su(n) and so(n): Complex-Structure Centralizers, Clifford Bivectors, and the Octonionic n = 3 Bridge

This record contains the preprint “AXIOM–CRE Realizations of su(n) and so(n): Complex-Structure Centralizers, Clifford Bivectors, and the Octonionic n = 3 Bridge” together with its companion exact-arithmetic verification script. The work organizes two standard realizations of compact classical Lie algebras within a unified AXIOM–CRE framework. The signed-XOR Clifford sector realizes so(n) through normalized bivectors, while the Conjugation–Real Embedding (CRE) realizes su(n) as the J-centralizing, J-trace-free subalgebra of so(2n). The natural inclusion so(n) → su(n) provides a commutative bridge between these realizations. For n = 3, the construction is shown to be compatible with the octonionic route: fixing a unit imaginary octonion yields the classical SU(3) stabilizer inside G₂, whose real action agrees, in an adapted basis, with the CRE realization. The framework also intertwines SU(n) connections, curvature, gauge transformations, and the Yang–Mills trace pairing. The accompanying script, verify_axiom_su_so.py, provides independent exact-arithmetic checks of representative algebraic identities and finite-dimensional constructions appearing in the paper, including the CRE constraints, Clifford bivector relations, signed-XOR matrix reconstruction, and the octonionic derivation/stabilizer computation. It is provided as a reproducibility aid and is not a substitute for the mathematical proofs in the paper. The classical Lie-theoretic embeddings themselves are not claimed as new. The contribution is their exact organization, compatibility, and executable realization within the AXIOM, signed-XOR, ALH, and CRE framework.