Variational Foundation of the Split-Quaternionic Vacuum: Deriving the Algebraic Structure and Negative Norms from First Principles (Part IX)

This paper constitutes Part IX of the research series on the Higher-Derivative Lee-Wick (HDLW) theory and the Split-Quaternionic Algebra (PQA). While previous works in this series established the phenomenological success of the PQA framework in unifying particle physics and gravity, this study addresses the fundamental question: "Why does nature select this specific algebraic structure?" Moving beyond phenomenological justification, we propose that the Split-Quaternionic geometry and the associated Real-Time Negative-Norm Principle (RTNNP) are dynamical consequences of a unified variational principle. By modeling the internal basis elements as dynamical matrix variables, we introduce an "algebraic tension" action analogous to matrix models.