  The Broadhurst-Kataev conjecture, that the ``discrepancy'' in the connection
with the $\pi^0 \to \gamma\gamma$ anomaly equals the beta function
$\beta(\bar{\alpha})$ times a power series in the effective coupling
$\bar{\alpha}$, is proven to all orders of perturbative quantum chromodynamics.
The use of nested short-distance expansions is justified via Weinberg's
power-counting theorem.
