Transported Romberg Extrapolation for the Arcsine Schröder Linearizer: Polynomial Composition Lifts and a Flat Infinite-Depth Limit

This research investigates transported Richardson–Romberg extrapolation for the arcsine Schröder linearizer through finite polynomial-composition lifts. For odd integers m ≥ 3, the multiple-angle polynomial S_m(z) = sin(m arcsin z) provides the composition structure. The resulting extrapolation hierarchy admits exact finite-depth cancellation, explicit leading approximation defects, reciprocal scale and composition identities, and geometric Lagrange weights expressed through q-products. The infinite-depth analysis establishes a smooth limiting lacunary trigonometric series, a nonzero flat discrepancy at the fixed point, and nowhere-real-analytic behavior under the stated hypotheses. The work distinguishes its specific polynomial-composition realization and infinite-depth results from the classical foundations of Schröder linearization, Richardson–Romberg extrapolation, and lacunary-series theory.