 Solution

For a single-photon transition, angular momentum addition requires $|J_i-J_f|\le L\le J_i+J_f$, $L\ge 1$, together with the electric/magnetic parity rules above. Thus a known final state and a measured multipole restrict both the spin difference and parity of the emitting level. For example, an E1 line reverses parity and allows $\Delta J=0,1$ except $0\to 0$; a mixed M1+E2 line preserves parity and must satisfy both angular triangles. An E2 line preserves parity and allows spin differences up to two.

Line energies identify connected levels, while calibrated intensities and coincidences test cascade flow and branching. If a proposed assignment permits a fast low-order transition to a lower state, an experimentally stringent absence of that line argues against the assignment. Nevertheless, an allowed transition can have a small or vanishing nuclear matrix element. Pure E2 radiation alone does not prove that M1 is forbidden by angular momentum. Detection sensitivity, efficiency and overlap must be considered, and in real nuclei internal conversion can remove photons; this question explicitly excludes that competing channel. Lifetimes or reduced-strength estimates make a missing-line argument substantially stronger than a bare selection-rule statement.