The leading constant for rational approximation of fractional inverse powers

For a fixed positive finite interval and a fixed exponent between zero and one, we prove the Braess--Hackbusch conjecture for the leading constant in best absolute uniform rational approximation of a fractional inverse power. The approximants have numerator degree at most $n-1$ and denominator degree at most $n$, with no restriction on poles outside the interval. The proof combines strong asymptotics for Jacobi-weight multipoint Padé interpolants with a weighted finite-Blaschke construction on a slit disk. Its alternating errors give the matching lower bound directly for the full real rational class.