  For a linearly driven quantum two-level system, or qubit, sets of
stroboscropic points along the cycloidal-like trajectory in the rotating frame
can be approximated using the exact rotating wave approximation introduced in
arXiv:1807.02858. That work introduces an effective Hamiltonian series
$\mathcal H_{\text{eff}}$ generating smoothed qubit trajectories; this series
has been obtained using a combination of a Magnus expansion and a Taylor
series, a Magnus-Taylor expansion. Since, however, this Hamiltonian series is
not guaranteed to converge for arbitrary pulse shapes, the same work
hypothesizes an axiomatic definition of the effective Hamiltonian. The first
two of the proposed axioms define $\mathcal H_{\text{eff}}$ to (i) be analytic
and (ii) generate a stroboscopic time evolution. In this work we probe a third
axiom---motivated by the smoothed trajectories mentioned above---namely, (iii)
a variational principle stating that the integral of the Hamiltonian's positive
eigenvalue taken over the full pulse duration is minimized by this $\mathcal
H_{\text{eff}}$. We numerically refute the validity of this third axiom via a
variational minimization of the said integral.
