= Solution
For a single-photon transition, <angular momentum addition> requires $|J_i-J_f|\le L\le J_i+J_f$, $L\ge1$, 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 $0\to0$; 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, \b[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.
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