shown in Fig. \[fig:dispersion\](c,d) of the main text, we again fit the energies of the $s$-shell emission and the radiative Auger emission into both $p$-shells simultaneously. The coupling term $\Delta_{pd}$ is included as a fit parameter. The exciton $g$-factor is measured independently and not fitted.

For the GaAs QD shown in Fig. \[fig:mechanismFull\](c), we again fit the energies of the $s$-shell emission and the radiative Auger emission into both $p$-shells simultaneously. The exciton $g$-factor is fixed to a value typical for GaAs QDs.

When observable, all phonon-related features are described using the fit results described above. A constant phonon energy is used as a single fit parameter.

  --------------------------------------- ------------ ----------------- ----------- ----------------------- ------------------ --------------------- --------------------- ---------------------
                                           $E_0$ (eV)   $m_e^*$ ($m_0$)   $g_h-g_e$   $\hbar\omega_0$ (meV)   $\Delta_p$ (meV)   $\Delta_{pd}$ (meV)   $\Delta_{LO}$ (meV)   $\Delta_{TO}$ (meV)
  InGaAs, Fig. 2(b) (main text)              1.3214         0.0712          1.505             17.67                 1.26                1.12                  36.1                    –
  GaAs, Fig. 2(d) (main text)                1.5925         0.0757          1.135             13.84                 1.90                0.25                  36.3                  33.5
  InGaAs, Fig. \[fig:mechanismFull\](b)      1.3152         0.0762          1.968             20.98                 1.08                  –                     –                     –
  GaAs, Fig. \[fig:mechanismFull\](c)        1.5757         0.0737           1.1              17.59                 2.61                1.37                  36.5                    –
  --------------------------------------- ------------ ----------------- ----------- ----------------------- ------------------ --------------------- --------------------- ---------------------

Radiative Auger Process: Theory {#sec:theory}
===============================

To find the multi-particle eigenstates, $\{\Psi_{\textbf{n}}\}$, 