Nuclear mass staggering explains missing stable technetium and promethium
Technetium (Tc) and promethium (Pm) are the only elements that lack stable isotopes in the range up to bismuth, a long-standing puzzle in physics and chemistry. We treat this absence as a problem of selecting the lowest-mass integer proton number Z in beta-stable isobaric chains with odd mass number A. Three-point mass parabolas identify two-unit jumps in the local minimum, whereas a five-point decomposition separates the smooth quadratic component from odd-Z/odd-N mass staggering, defined here as the mass shift of odd-Z isobars relative to neighboring odd-N isobars. With this decomposition, a fitted bulk-plus-shell mass model reproduces the smooth trend and shell-driven bending near magic numbers, including conventional shell-closure skips. This model, however, does not include odd-Z/odd-N staggering and cannot account for the Tc and Pm skips. The separated odd-Z/odd-N staggering remains positive across the Tc and Pm regions and is large enough for the lowest-mass integer-Z sequence to skip Tc and Pm. Shell-model occupation analysis suggests that this regional staggering reflects an orbital-dependent tensor-force monopole effect in the proton-neutron interaction. We identify tensor-force-driven odd-Z/odd-N mass staggering as the origin of the Tc and Pm skips in the odd-A sequence of lowest-mass isobars.