Fast Algorithm for Riemann Theta Functions via S(2,2) Decomposition

The Riemann theta function is a fundamental special function in mathematical physics, algebraic geometry, and complex analysis. However, its numerical evaluation is notoriously difficult as the computational complexity increases exponentially with the genus g (O(bᵍ)). Traditional direct summation methods typically reach their practical limit around g≈10. RLD_theta_engine.jl is an open-source Julia implementation that leverages known addition formulas for period matrices with an S(2,2) locus (block-diagonal structure). By implementing "Recursive Log-Decomposition," this engine breaks the exponential barrier and enables exact evaluations in ultra-high-dimensional regimes exceeding g=20,000 in seconds.