Appendix: The Complex Structure is a Consequence, Not a Posit — How the Complex Plane Arises from Two Real Components

This appendix develops a point compressed in sections 4 and 7.4 of the related report: that without positing the imaginary unit i in the axioms, the complex structure arises as a consequence of two real components and a rotation. The axioms place only two independent real conserved quantities (the sum of squares g, and the area omega = delta1 delta2 = 1/2 of Axiom 4) and the rotation connecting them. Writing z = nu1 + i nu2 = nu e^{i theta}, the norm |z|^2 = nu1^2 + nu2^2 = g is the real part and the area omega is the imaginary (symplectic) part. The imaginary unit i is the sign of the 90-degree rotation J (J^2 = -1: a quarter-turn twice is a half-turn, i.e. multiplication by -1); two independent axes alone give only the real plane R^2, and adding the rotation J makes it the complex plane C. That the complex plane is two-dimensional reflects the limit of anonymity = 2. No physical identification is made; this is a formal observation, not a claim to explain why quantum theory is complex.