Why the exceptional criterion is joint

The two conditions

are simultaneous conditions on the entire angular coefficient. They are stronger than cancellation at $\delta =0$, or at any other chosen angular slice.

When they hold, every Bessel order is an integer:

The finite-character heat projection then proves the exact finite-image covariance, not merely cancellation of a first diffractive asymptotic coefficient. When they do not hold, analyticity and closedness of the wavefront set prove the full ${\mathcal{D}}_{\mathrm{ap}}^{+}$ lower bound. These two different arguments are what make the exceptional classification exhaustive.