The Homogeneous First-Order Roots of the Klein-Gordon Operator

This updated version classifies first-order factorisations of the Klein--Gordon operator acting on real multivector fields. The factorisation problem requires a field space closed under left Clifford-vector differentiation and right Clifford multiplication, which fixes the natural intrinsic setting as the real spacetime algebra \Cl(1,3). Within the scalar-mass ansatz \nabla\Psi=-\mu\Psi K, with K^2=-1 and K restricted to a single definite Clifford grade, four primitive branches arise: a spacelike vector, a timelike trivector, a spacelike bivector, and the pseudoscalar. After choosing a timelike unit vector, the even and odd parts of the multivector field form a natural Clifford doublet. The timelike trivector and spacelike bivector branches reduce to Dirac equations for even Clifford spinors, though in different bases of the doublet. In both cases the propagation is governed by a fixed right-acting spacelike bivector B satisfying B^2=-1, giving the usual Dirac complex structure geometrically inside the real spacetime algebra. The spacelike vector branch is also algebraically consistent and admits Klein-Gordon mass-shell oscillatory modes, but its effective imaginary structure is momentum-dependent and two-sided rather than a universal fixed right-acting bivector. The pseudoscalar branch reduces, after internal recombination, to the same vector-root type. This update clarifies the distinction between Dirac-type first-order propagation and structurally admissible massive first-order root structures that are not governed by a universal Dirac spin-plane phase. It also sharpens the role of the real spacetime algebra as the intrinsic field space forced by the factorisation problem.