Real Matrix Realization of the Complex Unit and the Lorentzian Companion K = Jη

This bridge note presents a real two-dimensional matrix realization of the imaginary unit within a unital real-algebra framework. Let V = ℝ² and let J in End_R(V) be the standard linear complex structure satisfying J² = −I. The two-dimensional unital commutative real subalgebra A_J = {xI + yJ : x,y in ℝ} ⊂ End_R(V) is shown to be isomorphic to ℂ through the map Φ(x+iy) = xI + yJ, with Φ(i) = J. The note then formulates Euler’s identity in real matrix form: Φ(e^(iθ)) = exp(θJ) = I cos θ + J sin θ. It then introduces a Lorentzian companion construction. With the Lorentzian metric η = diag(1,−1), the companion operator K = Jη satisfies K² = I, and its exponential generates the standard 1+1-dimensional Lorentz boost. In a null basis, the boost diagonalizes as diag(e^δ, e^(−δ)). The document is intended as a compact bridge note for Arc Geometry applications. Its formal mathematical core is separated from the arc-geometric interpretive layer, where projection, time-torque, and holographic-track language are treated as research-development vocabulary rather than as additional assumptions in the proofs.