Toward an Algebraic Reformulation of Calabi–Yau Geometry: Structures from E₆ Symmetry and the Exceptional Jordan Algebra

We present an algebraic reformulation of Calabi–Yau geometry grounded in the complexified exceptional Jordan algebra and its automorphism group . In this framework, core features of Calabi–Yau threefolds—including complex structure, Kähler form, SU(3) symmetry, and Ricci-flatness—arise not from differential geometry, but as intrinsic consequences of algebraic symmetries and trace identities. The complex structure is realized via scalar multiplication, the Kähler form through the Jordan trace and triality, and Ricci-flatness as a trace-free condition on commutator curvature in the derivation algebra. A non-vanishing holomorphic volume form is naturally provided by the cubic norm of , ensuring vanishing first Chern class.Rather than treating Calabi–Yau geometry as a set of imposed geometric conditions, this approach suggests it can be emergent from exceptional algebra. The abundance of SU(3)-compatible configurations within the algebra provides a constructive moduli space, offering insight into the multiplicity of Calabi–Yau structures. We further outline how this algebraic framework may connect to quantum gravity and unification, where both spacetime and curvature arise from derivation symmetries. Our aim is not to replace classical results, but to propose a new lens through which their deeper algebraic origin may be understood.