Analytic Bipartite Inverse Design of Doppler-Compensated Relativistic Lightsails: Projection Invariants under SO(2) and Scale-Invariant Riccati Formalism under SO(1,1)

Beamed-energy propulsion of ultra-light relativistic lightsails confronts two simultaneous electrodynamic instabilities: the continuous reflectivity degradation caused by the relativistic Doppler redshift along the propagation axis, and the transverse radiation-pressure gradient that displaces the sail from the optical axis. Current inverse-design methods address these challenges exclusively through computationally expensive stochastic algorithms with no physical insight into the structure of the solution space. We present a fully deterministic, closed-form analytical framework by decomposing the sail's effective dielectric tensor as ε(r,z)=ε⊥(r)⊗ε||(z) within the paraxial scalar approximation. The transverse sector is solved via the inverse Abel transform grounded in the universality theorem for $SO(2)$-invariant rotating curve families: a quadratic target phase Φ(y)=Φ₀(1-y²/R²) yields the exact closed-form graded-index profile n₁(r)=(2Φ₀/π R²)√R²-r² bounded within the Si/SiO₂ range [1.45, 3.48] at λ₀=1064~nm. The longitudinal sector is governed by the nonlinear Riccati equation for the local reflection coefficient, derived rigorously from the coupled-wave reduction of Maxwell's equations. A formal self-similarity argument under the Lorentz boost subgroup $SO(1,1)$ is used to prove that a log-periodic layer distribution Λⱼ=Λ₀eα j is a sufficient condition for broadband Doppler invariance across the target bandwidth [ω₀e^-ηₘₐₓ,ω₀]. A corrected thin-film Stefan-Boltzmann thermal analysis, using emissivity εₜₕ≈αIRd valid for sub-wavelength films, shows that the log-periodic design maintains equilibrium temperatures below the Si melting point of 1687 K at I₀=1~GW/m², whereas the conventional distributed Bragg reflector (DBR) exceeds this limit already at β≈0.07. This work provides a rigorous group-theoretic foundation that reduces the broadband inverse-design problem from a stochastic search to a direct algebraic verification.