From Exceptional Jordan Geometry to Clifford Algebra: An Immersive Bridge via the Fano Plane

From Exceptional Jordan Geometry to Clifford Algebra: An Immersive Bridge via the Fano Plane We construct a rigorous immersive bridge between the exceptional Jordan algebra h_3(O)^C and the associative Clifford algebra on a 14-dimensional complex vector space. The construction resolves the apparent incompatibility between the non-associative structure of Jordan algebras and the associative framework of Clifford-Dirac operators, a gap that has hindered the direct application of exceptional algebraic structures in quantum field theory and tensor network methods. The central technical contribution is a hybrid representation of Spin(10), obtained via restriction to Spin(7) subset Spin(8) subset Spin(10), which naturally yields a 14-dimensional operational subspace identified with the point-line incidence geometry of the Fano plane. We prove that the stabilizer algebra of a rank-one idempotent in h_3(O)^C is faithfully and differentiably realized as a subalgebra of the Clifford algebra on C^14, with injective differential and spectral invariants given by the minimal polynomial m_X(x) = x^4 - 13x^2 + 30. This result provides a direct algebraic link between the exceptional Jordan geometry underlying recent proposals for the Standard Model gauge group and its left-right extensions, and the associative Clifford-Dirac framework widely used in quantum field theory, gravity, and tensor network formulations. The bridge is verified by a fully reproducible numerical script, making the construction accessible for further theoretical and computational developments. The present work is concerned exclusively with the algebraic immersion. Potential applications include the realization of exceptional symmetries in tensor network models, the embedding of Jordan structures into noncommutative geometry, and the unification of gauge and gravitational sectors in a common algebraic framework.