Curvature–Modularity Correspondence for Affine (1,1)-Curves

Classical period pairings between geometric curvature and automorphic masses typically rely on spectral regularization, analytic continuation, or height-theoretic machinery. This work isolates a regularization-free correspondence at the level of absolutely convergent periods for a family of affine (1, 1)-curves in C2. Pulling back the ambient Euclidean metric yields a rational curvature density with exact O(|𝑡 |−6) decay, guaranteeing absolute convergence of the linear period ∫R 𝐾(𝑡) 𝑑𝑡 without boundary counterterms or meromorphic continuation. Through an affine coordinate lift to the Poincaré upper half-plane and the Maass operator identity 𝜉2𝐸∗2 = 3/𝜋, this geometric period is structurally aligned with the hyperbolic mass 𝐼𝐸 =∫F 𝐸∗2 𝑑𝜇. Their ratio defines a finite, scaling-invariant period constant CMCC, which quantifies the coupling strength between affine differential geometry and weight-2 automorphic analysis. The pairing is enforced by three compatible principles: affine scaling invariance, exact polynomial decay, and the weight-stripping differential action of 𝜉2. The result is strictly global, invariant under Aff+(R), and provides a computable template for extending regularization-free period architectures to higher-degree rational curves or completed Eisenstein series of weight 𝑘 ≥ 4 under matching decay conditions.