Solution
= Solution
At $k=6$, the two zeros coalesce and
$$
\Delta H=3\varepsilon\pi a^2(a^2-2)^2+O(\varepsilon^2).
$$
The first-order drift has a double zero and does not change sign there. The energy balance calculation is therefore degenerate: it neither proves persistence of a periodic orbit nor determines its stability near $k=6$. Higher-order terms are needed.