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A nuclide is beta-stable if it has no energetically allowed single beta-minus decay, beta-plus decay, or electron capture to a lower-mass neighboring nuclear isobar, with electron thresholds treated consistently. Beta stability need not imply stability against alpha decay, double beta decay, or nuclear fission.

Double beta decay changes the atomic number of a nucleus by two while preserving its mass number. The ordinary two-neutrino process can connect even-even nuclear isobars when the intermediate odd-odd nucleus is too heavy for a single beta decay. Neutrinoless variants would violate lepton number and require additional physics.

At fixed mass number $A$, the smooth semi-empirical mass formula makes the atomic mass approximately quadratic and convex in atomic number $Z$. Neighboring nuclear isobars undergo energetically allowed beta decay toward a discrete local minimum, subject to the electron-mass threshold for beta-plus decay. For odd $A$ there is one smooth branch; for even $A$, nuclear pairing splits the even-even and odd-odd branches. A local minimum against a single change of $Z$ need not be stable against double beta decay.

All three processes are weak charged current transformations between neighboring nuclear isobars. Neglecting neutrino masses, and writing $M_a(A,Z)$ for a neutral atomic mass, their Q values are:

These formulae use the paper's natural units. With energies in MeV, beta-plus decay requires an atomic mass-energy difference exceeding $2m_e{c}^2=1.022\mathrm{MeV}$. The extra electron-mass subtraction arises from using neutral atomic rather than bare nuclear masses. Electron capture can occur when the atomic mass-energy difference is positive but below this threshold. For a particular capture shell or excited daughter, the available energy must also cover the electron-hole and nuclear excitation energies, and the small nuclear recoil. Small atomic binding corrections are suppressed in the displayed threshold formulae.

The question's fixed-$A$ family consists of nuclear isobars, rather than isotopes: isotopes have fixed $Z$. The semi-empirical mass formula gives a convex nuclear mass parabola as a function of $Z$ at fixed $A$. For odd $A$, nuclear pairing does not split the family into two branches, so there is typically one beta-stable isobar, at the discrete minimum. Neighboring members can lower their mass by changing $Z$ toward it.

For even $A$, nuclear pairing lowers even-even nuclei and raises odd-odd nuclei, producing two offset nuclear mass parabolas. There are often two beta-stable even-even isobars, separated by an odd-odd member to which neither can energetically decay. The number is not an exact universal rule: some even-$A$ chains have one or more than two such minima against single beta decay, and a few light odd-odd nuclei are stable. Stability against single beta decay also does not guarantee stability against double beta decay.