Emergent Photons, Speed of Light Invariance, and Spin-1 Polarization via Two-Stage Symplectic-to-Polar Projections and Complexified Misner Space

The internal structure of the photon, the fundamental origin of the invariant speed of light (c), and the geometric mechanism behind spin-1 circular polarization remain open questions in contemporary quantum field theory. In this paper, we derive a deterministic geometric model of massless elementary quanta directly from a 6-dimensional symplectic phase-space manifold (MH⁶) through a two-stage dimensional reduction. First, a Cartesian metric-induced imaginary projection (ΠI) collapses the three momentum degrees of freedom into a unified imaginary axis (iI), mapping TqMH⁶ onto an abstract 4D complex-product tangent space (TqXC³⁺¹). Second, applying Euler's polar transformation (ΦEuler: zʲ = Tj exp(±iφj)) in conjunction with a canonical radial projection (ΠR) conceals the complex structure behind real hyper-polar coordinates: three tangential angular bases (∂φi) spanning observable 3D space, and a shared real radial axis (∂T) representing chronological proper time. By examining the near-origin limit (r → 0) of a deformed Hamiltonian metric on the constituent Riemann surfaces (XCMi¹), we establish an exact metric equivalence with the vacuum Misner spacetime subjected to a complex angular rotation (z = iφ). Solving the null-geodesic condition (ds² = 0) at the singularity (T → 0) yields T = exp(±iφ), demonstrating that primordial motion is strictly tangential (|v| = |∓∂φ| = c = 1) with zero radial proper-time flow (vT = 0). Finally, treating the primordial complex singularity as an infinite quantized source of complex energy and modeling a complex-product singularity quantum as a 4D sheaf of three intersecting complex planes reproduces the complex Maxwell electromagnetic field vector (F = E/c + iB) and naturally accounts for left- and right-handed photon spin states (|L⟩, |R⟩). Author's Note: This article presents a formalized mathematical synthesis and extension of Chapters 3 and 4 from the author's monograph: Theory of Everything: Complex Singularity Origin of Matter and Antimatter Universes (BookEdit, 2026, ISBN 978-83-68750-17-1).