The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy — from Cardano's Cubics to Twistor Space, via the Sixth Dimension of GG-Theory

Abstract

Complex numbers were invented in the sixteenth century to pass through the impossible square roots of the cubic; by 1925 they had become the working substance of physics, and the puzzle of why has stood ever since — posed at full strength by Penrose, for quantum amplitudes and for the holomorphic geometry of twistor theory alike. This preprint assembles a calibrated answer. Physics contains two complex structures, both derived from globally shared real data. The gauge phase of charged matter is a real angle on the compact, non-spatial phase circle of the GG-6 bulk: charge quantization is its compactness, the Aharonov–Bohm phase its holonomy, the imaginary unit its rotation generator. The state-space i is the polar phase of the constitutional clock — the Hilbert transform, a real operator — selected uniquely by causality and energy positivity; the two coincide on charged sectors exactly under the Feynman–Stückelberg pairing, so antiparticles are the consistency condition between circle and clock. The same selection extends, conditionally, to interacting matter — and to holomorphy itself: positive frequency is Hardy-space holomorphy, forward-tube analyticity its field form, and the twistor division PT^+/PT^- is the frequency splitting, conformally packaged. The experimental exclusion of real quantum mechanics is read as direct support: one circle and one clock supply the single shared i that experiment demands. What remains open is named exactly; why C is beautiful mathematics is conceded to lie beyond physics. Part of the GG-Theory program, a series of preprints developing a six-dimensional geometric framework and its consequences for particle physics, cosmology, and open-system dynamics. The complete series is available at https://preprints.arisaka-gg.org/