$H_{u,\alpha}^{\mathrm{D},\leq N}(t)=i\dot{X}_{u_{t}}X_{u_{t}}^{*}\\\
+X_{u_{t}}\Big{(}\sum_{j=1}^{N}h_{u_{t},\alpha_{t}}+\frac{1}{N}\sum_{i<j}V(x_{i}-x_{j})\Big{)}X_{u_{t}}^{*}+\sum_{j=2}^{5}H_{j}(t).$
(C.16)

From [66, Eq.(40)] and the equation satisfied by $u_{t}$, we deduce

$i\dot{X}_{u_{t}}X_{u_{t}}^{*}=b^{*}(u_{t})b(q_{u_{t}}h_{u_{t},\alpha_{t}}u_{t})-\langle
ih_{u_{t},\alpha_{t}},u_{t}\rangle(N-\mathcal{N}_{b})\\\
-\sqrt{N-\mathcal{N}_{b}}b(q_{u_{t}}h_{u_{t},\alpha_{t}}u_{t})-\text{h.c}.$
(C.17)

This combines with the second term above to yield $H_{0}(t)+H_{1}(t)$ as in
the analogous computations in [12, 66]. Taking into account the obvious
modifications from replacing $v\mapsto\tfrac{N-1}{N}V$, one can use for
instance [12, Eq. (2.20)]. . ∎

## Acknowledgements

The authors would like to thank Sören Petrat for many fruitful discussions
throughout the project. M.F. acknowledges the support of the MUR grant
“Dipartimento di Eccellenza 2023–2027”, and of the “Centro Nazionale di
ricerca in HPC, Big Data and Quantum Computing”. J.L. was supported by the
Agence Nationale de la Recherche (ANR) through the projects DYRAQ
(ANR-17-CE40-0016) and QUACO (ANR-17-CE40-0007), and the ICB received
additional support through the EUR-EIPHI Graduate School (ANR-17-EURE-0002).
N.L. gratefully acknowledges support from the Swiss National Science
Foundation through the NCCR SwissMap and funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 101024712\.

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