= Solution
The qualitative explanation is <nuclear pairing>: $^{40}\mathrm K$ is odd-odd ($Z=19,N=21$), while both $^{40}\mathrm{Ar}$ and $^{40}\mathrm{Ca}$ are even-even. At $A=40$, moving from the odd-odd to either even-even branch lowers the pairing contribution to the atomic <mass> by
$$
\frac{(+11.2)-(-11.2)}{\sqrt{40}}=3.542\,\mathrm{MeV}.
$$
Two neighboring even-even <nuclear isobars> can therefore both be lower in mass than the intervening odd-odd member. The two possible directions are
$$
{}^{40}_{19}\mathrm K\to{}^{40}_{18}\mathrm{Ar}
\quad\text{by electron capture or positron emission},\qquad
{}^{40}_{19}\mathrm K\to{}^{40}_{20}\mathrm{Ca}
\quad\text{by beta-minus decay}.
$$
However, the particular coefficients printed in this paper do \b[not] quantitatively reproduce both allowed directions. Let $D=m_p+m_e-m_n\simeq-0.7823\,\mathrm{MeV}$. Using every supplied term, rather than pairing alone, gives
$$
\begin{aligned}
M(40,19)-M(40,18)
&=D+\frac{0.71(19^2-18^2)}{40^{1/3}}
+\frac{23.3(2^2-4^2)}{40}+\frac{22.4}{\sqrt{40}}
\simeq+3.451\,\mathrm{MeV},\\
M(40,19)-M(40,20)
&=-D+\frac{0.71(19^2-20^2)}{40^{1/3}}
+\frac{23.3(2^2-0^2)}{40}+\frac{22.4}{\sqrt{40}}
\simeq-1.443\,\mathrm{MeV}.
\end{aligned}
$$
Thus this parametrization would forbid the calcium beta-minus branch. The physical atomic mass differences are instead approximately $+1.505\,\mathrm{MeV}$ to argon and $+1.311\,\mathrm{MeV}$ to calcium, so both physical directions are allowed.<The KDK experiment by KDK collaboration (2021)>[Decay scheme and atomic Q values]{ref}
The valid interpretation is therefore \b[pairing explains how both directions can be possible, but the supplied smooth mass formula is insufficient to establish both for this isotope]. The <semi-empirical mass formula> has MeV-scale residual errors and omits <nuclear shell closure> corrections; $^{40}\mathrm{Ca}$ has the especially stable $Z=N=20$ shell closures. One must use measured <masses> or a model with those corrections for a quantitative proof, rather than changing the printed numbers silently.
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