The Geometric Origin of Quantum Axioms: Emergence of Complex Numbers and the Born Rule from Split-Quaternionic Krein Algebra (Part III)

This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the ² operator (Part I: arXiv: 2512. 16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M 11. 3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C. Derivation of the Born Rule: We show that the Born rule (P = ||²) emerges as the unique, positive-definite inner product consistent with the internal Krein metric after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.