Exact Riemann Theta Function Evaluation at Ultra-High Genus via S(k,k) Block-Diagonal Factorization: Implementation and Benchmark

We report a computational study of Riemann theta function evaluation under S(k,k)-type block-diagonal constraints on the period matrix. The factorization identity underlying this method follows directly from the definition and is algebraically exact; no independent error analysis is required beyond floating-point rounding. This enables exact reference values at genus g > 20,000 within this structural class, a regime where existing methods — tensor-train approximations (Chimmalgi–Wahls 2023, g ~ 60, relative error ~0.01) and the FLINT library (Elkies–Kieffer 2025, certified precision for general period matrices) — do not provide exact benchmarks.

A central observation, which we do not consider obvious, is that approximation error under off-diagonal perturbation decreases as genus increases, with sharp transition-like convergence depending on the combination of Ω and g. This behavior — falling more than 15 orders of magnitude below CLT-based worst-case predictions — suggests a strong cancellation mechanism intrinsic to the lattice structure, whose theoretical explanation remains open. The minimum eigenvalue λ_min of Im(Ω) emerges as the dominant practical predictor of approximation accuracy.

Each base k yields a nearly disjoint series of exactly computable genera (2ⁿ, 3ⁿ, 5ⁿ, …), forming a structured benchmark dataset across a wide range of g. This provides exact reference values for validating approximation methods in high-dimensional theta function research, with natural applications in integrable systems and Hitchin systems where S(k,k) structure arises geometrically. The present method and FLINT are complementary: FLINT targets certified precision for general Ω; this method targets exact computation within specific structural classes at ultra-high genus.

A supporting implementation, including a browser-based demonstration up to g = 64 solitons, is publicly available.

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