Ramanujan, Landen, and Baker N-Series Refinement: Higher-Degree Modular Gap Laws and a Two-Axis Comparison

This preprint compares three distinct refinement mechanisms arising in elliptic perimeter approximation, modular transformations, and constructive geometry. Ramanujan's second ellipse-perimeter approximation acts on the eccentricity parameter, while geometric N-series extrapolation accelerates polygonal approximation by cancelling successive even powers of the resolution parameter. Using classical theta-function expansions, the paper derives the fixed-degree modular-gap refinement [δ(q^m)/δ(q)^m=4¹⁻ᵐ(1+4mq^2+O(q^4)), m≥2.] The analysis also connects the descending Landen contraction of the normalized arithmetic–geometric mean gap with the asymptotic correction coefficient (α_n/δ_n^2→1/8), consistent with the companion coefficient-ratio limit (αₙ₊₁/α_n^2→1/2). A separate comparison with a constructive cubic angle-trisection cascade distinguishes modular contraction, Richardson-type extrapolation, and nonlinear geometric error suppression. No equivalence between these operators is claimed. The repository contains the revised LaTeX manuscript, compiled PDF, numerical verification program, SHA-256 integrity manifest, and successful automated verification at 400-digit precision. Associated source repository: https://github.com/CWBaker360/ramanujan-landen-nseries-refinement Status: Revised preprint, October 21, 2026. Not independently peer-reviewed.