 Solution

Assume the quoted intensities represent relative emitted photon numbers, with efficiency corrections implicit, as required for population-flow bookkeeping. The prominent branch obeys $I_1=I_2+I_4$, and $I_2=I_3$ identifies a serial two-photon branch. The energies obey $E_2+E_3\simeq E_4$ and $E_1+E_4\simeq E_5$ within their uncertainties. The weak fifth line directly spans D to A. Reversing which end contains the line-1 step and interchanging the serial line-2/line-3 order produces the four possibilities.

Take A as zero. A gamma-ray energy-network fit to the five energy differences gives the following approximate excitation energies in keV:The line connections, in the order $1,2,3,4,5$, are respectively: I: $(B\to A,C\to B,D\to C,D\to B,D\to A)$; II: $(B\to A,D\to C,C\to B,D\to B,D\to A)$; III: $(D\to C,B\to A,C\to B,C\to A,D\to A)$; IV: $(D\to C,C\to B,B\to A,C\to A,D\to A)$.

For I and II, the B population entering through lines 2 or 3 and line 4 exits through line 1. For III and IV, line 1 feeds C and its intensity splits into the direct line 4 and serial lines 2/3. The small direct D-to-A branch bypasses these intermediate balances. The central measured energies do not obey every sum exactly; all four fitted arrangements have ${\chi }^2\simeq 0.81$ for two energy constraints. Assigning the weakest line's central energy as an exact excitation energy would discard the more precise cascade measurements.