Universal emergence of rational corrections in Borel-based strong-coupling approximants

The corrected–Borel scheme provides a hierarchical framework for extracting strong–coupling information from short perturbative expansions. The method begins with an irrational core determined by the first two coefficients and the known strong–coupling exponent, and refines this core through a sequence of rational Borel–root corrections generated from successive residuals. Each correction depends only on a new pair of perturbative coefficients, ensuring transparency, stability, and coefficient locality even for short or divergent series. The scheme accommodates both extrapolation, where the strong–coupling amplitude is unknown, and interpolation, where the amplitude is prescribed. A recursive extension allows synthetic coefficients generated by the approximant itself to continue the hierarchy beyond the available data. Applications to representative test functions demonstrate that the corrected–Borel hierarchy yields accurate and systematically improvable estimates of strong–coupling amplitudes using minimal perturbative input.