HOLOMORPHIC HOPF–BELTRAMI MAXWELL MODES ON THE ROUND THREE-SPHERE: Exact Construction, Conformal Transport, and Penrose Representation

This work develops an explicit global construction of a distinguished family of null electromagnetic modes on the Einstein cylinder ℝ × S³. The starting point is the Hopf geometry of the round three-sphere and a global orthonormal frame X₁, X₂, X₃. For every homogeneous holomorphic polynomial p(z₁,z₂) of degree m ≥ 0, the complex field F₀ = p(z₁,z₂)(X₂ + iX₃) is shown to satisfy simultaneously curl F₀ = ((m+2)/R)F₀, div F₀ = 0, and F₀ · F₀ = 0. It therefore generates an exact source-free Maxwell solution with ωₘ = c(m+2)/R. The associated electric and magnetic fields have equal magnitude, are mutually orthogonal, and their Poynting flux follows the Hopf direction away from the nodal set. The zeros of p lift naturally to Hopf fibres, giving the construction a direct geometric interpretation. A central structural result is that these solutions form a precise holomorphic chiral sector of the Maxwell spectrum, rather than the complete curl eigenspace. At degree m, the polynomial construction has complex dimension m+1, whereas the full positive curl eigenspace has dimension (m+1)(m+3). At the same time, a converse theorem proves that within the fixed complex-line ansatz f(X₂+iX₃), every smooth curl eigenfield is necessarily generated by a homogeneous holomorphic polynomial. The work also obtains closed expressions for the polynomial norm, cylinder energy, and magnetic helicity; on the unit sphere, H_B = N(p)/(m+2). Thus the spectral label, geometry, energy flow, nodal Hopf structure, and helicity are controlled within one exact framework. The construction is then transported from the Einstein cylinder to Minkowski spacetime by conformal compactification. In Bateman variables α and β, the resulting field takes the compact form F_M = p(α,β) ∇α × ∇β, producing smooth finite-energy null Maxwell fields. The same sector is connected explicitly to twistor theory through a homogeneous degree −4 Penrose representative, f_p(Z) = −4 p(C,W)/(A B^(m+3)), whose contour transform reproduces the Minkowski Maxwell spinor with full normalization. The conformal-generator weight of this representative is −i(m+2), matching the curl eigenvalue and the cylinder frequency. The result therefore establishes a direct, convention-explicit bridge between Hopf geometry, Beltrami eigenfields, exact Maxwell dynamics, Bateman variables, conformal transport, and the Penrose transform, while keeping the mathematical scope precise: the construction identifies an exact and exhaustive holomorphic sector within a fixed polarization, not a classification of all Maxwell fields on S³.