NULL BELTRAMI GEOMETRY ON THE ROUND THREE-SPHERE: Hopf Classification, Bateman Links, Twistor Cohomology, and Structural Obstructions

This work develops a rigorous geometric framework for a class of null Beltrami and electromagnetic fields whose structure links spectral geometry, topology, Maxwell theory, twistor cohomology, and gravitational deformation theory. Its central result is a global classification of smooth nonzero null curl eigenfields on the round three-sphere S³: every positive-eigenvalue solution is, up to an orientation-preserving round isometry, of the form F = p(z₁,z₂)(X₂ + iX₃), λ = m + 2, where p is a homogeneous holomorphic polynomial of degree m. The significance of this theorem is not merely classificatory. It shows that a seemingly large nonlinear family of fields is governed globally by Hopf geometry and finite-dimensional polynomial data, even across its nodal set. For the associated Bateman electromagnetic fields, the compactified spatial maps are proved to be diffeomorphisms, the nodal sets become exact linked circles, and their evolution is controlled by ambient isotopy. Under sufficiently small transverse C¹ deformations this link structure is conditionally stable. These results provide a mathematically precise foundation for studying topologically structured electromagnetic fields, optical vortices, and knotted field configurations, where linking and nodal topology can act as robust degrees of freedom. They are therefore relevant at a foundational level to topological photonics, structured light and optical information, while avoiding the stronger claim that the present mathematics by itself already constitutes a communication or quantum-engineering protocol. A second perspective comes from plasma and field theory. Beltrami equations such as curl B = λB describe force-free magnetic configurations and appear naturally in magnetohydrodynamics, relaxed plasma states, and astrophysical magnetic structures. The classification on S³ and the resolution of the nodal-set removability problem provide an unusually rigid model in which global topology, spectral data and field-line geometry can be studied simultaneously. The work also constructs a precise bridge from the Maxwell sector to twistor geometry: an explicit Penrose representative embeds the fixed-Hopf family into H¹(𝒰,𝒪(−4)), while multiplication by a nonzero sextic gives a surjection to the polynomial Hamiltonian sector H¹(𝒰,𝒪(2))ₘ, with dim ker = 6(m + 1) for m ≥ 4. Selected representatives can then be integrated to nonlinear complex Ricci-flat half-flat geometries. This does not establish a universal electromagnetic-to-gravitational “double copy”; rather, it identifies a concrete cohomological mechanism through which spin-1 field data and spin-2 geometric deformations can be compared within one exact mathematical language. That distinction is important for future work on twistor descriptions of gauge–gravity relations and on the geometric structures underlying classical and quantum field theories. Equally important are the paper’s structural obstruction theorems, because they identify routes that a more complete theory cannot take. A Poincaré-equivariant linear operation cannot convert the massless Hopf sector into a massive irreducible state; holomorphic ambitwistor data alone cannot choose retarded rather than advanced propagation; the pure Weyl square of a null Maxwell spinor is necessarily Petrov type N and cannot reproduce the expanding vacuum Kerr–Schild II/D sector; and there is no fibre-degree-preserving zeroth-order universal morphism 𝒪(−4) → 𝒪(−6) implementing Maxwell-to-graviton spin raising. Instead of forcing a formal analogy beyond its domain of validity, the analysis isolates the full twisting Einstein–Maxwell CR system as the appropriate Lorentzian backreaction problem and separates the low spherical harmonics carrying energy–momentum from the invertible radiative sector through ℒ = ΔS²(ΔS² + 2). In this sense, the broader value of the work is methodological as well as mathematical: it simultaneously establishes new constructive bridges and proves where those bridges must end. Like other foundational developments in geometry and mathematical physics, its technological relevance is therefore primarily long-range: by reducing complicated knotted field configurations to controlled Hopf, polynomial, cohomological and CR structures, it supplies analytic machinery that may ultimately inform topological control of electromagnetic and plasma fields, robust field-based information structures, and more rigorous approaches to the relation between gauge fields and gravitation, while clearly separating such research directions from claims not yet implied by the mathematics.