GEOMETRY OF LIGHT ON THE ROUND THREE-SPHERE: Hopf–Jacobi Dynamics, Maxwell Intensity, Dual Tomography, and Nonlinear Spectral Certificates

We develop an algebraically parameterized geometry-of-light research framework on the fixed round three-sphere. Smooth null Beltrami fields built from homogeneous binary forms provide a common mathematical input for two distinct dynamical equations: source-free Maxwell evolution and exact pressure-balanced viscous flow. For the latter, we establish both an unforced multispectral toroidal Jacobi family and a symmetry-protected nonresonant Hopf–Killing family, distinguishing spectral degeneracy from vanishing projected nonlinear interaction. We derive energy, helicity, enstrophy, pressure-beat and Noether identities, a signed-curl spectral variance, and physical-time Fubini–Study/Fisher formulas for reducing viscous histories. A positive moment–Hankel hierarchy gives finite-rank spectral certificates with stated conditioning limits. In an externally controlled fixed- Hopf sector, projective forcing has a Berry-geometric cost and permits a smooth exchange of linked nodal fibres. Our new cross-domain reconstruction theorem shows that (m + 1)(2m + 1) calibrated equal-time intensity or pressure-deficit samples recover the degree-m binary form up to global phase and therefore recover its Hopf nodal divisor. A separate finite energy–helicity Prony procedure resolves occupied absolute curl shells and both chirality weights. An explicit counterexample proves that integrated energy–helicity histories alone cannot identify nodal topology, even without noise. The manuscript includes reproducible algebraic verification and sharply typed transfer gates from OpenAI Math. Polynomial optical knots, stellar representations and moment inversion have substantial prior art; the worldwide novelty of the joint construction remains unestablished. We make no claim of Maxwell–Navier–Stokes physical identity, universal PDE regularity, quantum-gravity closure or observed cosmological topology.