Proof. For clarity, we organize the proof so that each type of error is estimated once and then reused throughout the argument. The proof is divided into four steps. Step 1 reduces the variational principle to the projected space. Step 2 controls the localization and entropy errors arising from the non-factorized trial state. Step 3 establishes the finite-dimensional semiclassical bound. Step 4 rewrites this projected estimate in the normalized Gaussian form stated in the proposition.

Step 1: Reduction to a finite-dimensional estimate. We start by defining the interacting Gibbs state on \({\mathfrak{F}}(P{\mathfrak{H}})\): \[\begin{align} \label{interacting32Gibbs32in32trial} \Gamma^{f_\eta}_{\tau,P}:=\frac{e^{-\mathbb{H}_{\tau,P}}f_\eta\left({\mathcal{N}}_{P}/\tau\right)}{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}\Big(e^{-\mathbb{H}_{\tau,P}}f_\eta\left({\mathcal{N}}_{P}/\tau\right)\Big)}, \end{align}\tag{48}\] with \({\mathcal{N}}_{P}= {\mathord{{\rm d}}}\Gamma(P)\) and \[\begin{align} \label{regularized32Hamiltonian} \mathbb{H}_{\tau,P}:=\mathbb{H}_{\tau,0,P}-\mathbb{W}_{\tau,P}:=\frac{1}{\tau}{\mathord{{\rm d}}}\Gamma(PhP)-\frac{1}{\tau^3} \mathbb{W}_{P}. \end{align}\tag{49}\] Here, \(\mathbb{W}_P\) is the second quantization of the operator \(W_P^\varepsilon:=P^{\otimes 3} W^\varepsilon P^{\otimes 3}\). With the same conventions, we define the free Gibbs state on \({\mathfrak{F}}(P{\mathfrak{H}})\) as \[\begin{align} \label{non-interacting32Gibbs32in32trial} \Gamma^{f_\eta}_{\tau,0,P}:=\frac{e^{-\mathbb{H}_{\tau,0,P}}f_\eta\left({\mathcal{N}}_{P}/\tau \right)}{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \Big(e^{-\mathbb{H}_{\tau,0,P}}f_\eta\left({\mathcal{N}}_{P}/\tau\right)\Big)},\qquad \Gamma_{\tau,0,P}:=\frac{e^{-\mathbb{H}_{\tau,0,P}}}{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \Big(e^{-\mathbb{H}_{\tau,0,P}}\Big)} . \end{align}\tag{50}\] Similarly, we define \(\Gamma^{f_\eta}_{\tau,0,Q}\) and \(\Gamma_{\tau,0,Q}\) by replacing \(P\) with \(Q=\mathbb{1}-P\) in the above definitions. Now we explain the choice of the trial state, which is a key difference from the previous works [3], [55], [56]. Following the strategy in those papers, one would naturally consider the trial state \[\Gamma_{\mathrm{ts}}= {\mathcal{U}}^* ( \Gamma^{f_\eta}_{\tau,P}\otimes \Gamma^{f_\eta}_{\tau,0,Q} ) {\mathcal{U}}.\] However, in the present setting, the truncated free Gibbs state \(\Gamma^{f_\eta}_{\tau,0}\) does not factorize. After applying the variational principle (see 52 below), the term \(\mathrm {Tr}\big[\Gamma_{\mathrm{ts}}(\log f_\eta({\mathcal{N}}/\tau) -\log (f_\eta({\mathcal{N}}_P/\tau) \otimes f_\eta({\mathcal{N}}_Q/\tau) ) )\big]\) appears in the discrepancy between the relative entropy \({\mathcal{H}}(\Gamma_{\mathrm{ts}}, \Gamma^{f_\eta}_{\tau,0})\) and its \(P\)-localized counterpart. This term is difficult to control directly. We therefore introduce the following trial state on \({\mathfrak{F}}({\mathfrak{H}})\), with a cutoff given by the full number operator: \[\begin{align} \label{def:trial:state} \Gamma_{\mathrm{ts}}:=\frac{e^{-\mathbb{H}_{\tau,0}+{\mathcal{U}}^* ( \mathbb{W}_{\tau,P} \otimes \mathbb{1}_{{\mathfrak{F}}(Q{\mathfrak{H}})} ) {\mathcal{U}}}f_\eta\left({\mathcal{N}}/\tau\right)}{\mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})}\Big( e^{-\mathbb{H}_{\tau,0}+ {\mathcal{U}}^* ( \mathbb{W}_{\tau,P} \otimes \mathbb{1}_{{\mathfrak{F}}(Q{\mathfrak{H}})} ) {\mathcal{U}}}f_\eta\left({\mathcal{N}}/\tau\right) \Big)}, \end{align}\tag{51}\] where the associated partition function is denoted by \({\mathcal{Z}}_{\mathrm{ts}}\). Using the variational principle 34 , it follows that \[\begin{align} \label{upper32bound32to32finite32dimension} - &\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{{\mathcal{Z}}^{f_\eta}_{\tau,0}} \leqslant{\mathcal{H}}\left( \Gamma_{\mathrm{ts}} , \Gamma^{f_\eta}_{\tau,0} \right) - \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \mathbb{W}_\tau \Gamma_{\mathrm{ts}} \right) \\&= {\mathcal{H}}\left(\Gamma^{f_\eta}_{\tau,P}, \Gamma^{f_\eta}_{\tau,0,P} \right) - \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( \mathbb{W}_{\tau,P} \Gamma^{f_\eta}_{\tau,P} \right) - \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \mathbb{W}_{\tau} \Gamma_{\mathrm{ts}} \right) + \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( \mathbb{W}_{\tau,P} \Gamma^{f_\eta}_{\tau,P} \right) \nonumber \\ & \qquad \qquad \qquad \qquad \quad +{\mathcal{H}}\left( \Gamma_{\mathrm{ts}}, \Gamma^{f_\eta}_{\tau,0} \right)- {\mathcal{H}}\left( \Gamma^{f_\eta}_{\tau,P}, \Gamma^{f_\eta}_{\tau,0,P} \right) \nonumber \\ & = -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau,P}}{{\mathcal{Z}}^{f_\eta}_{\tau,0,P}} - \left[ \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \mathbb{W}_{\tau} \Gamma_{\mathrm{ts}} \right) -\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( \mathbb{W}_{\tau,P} \Gamma^{f_\eta}_{\tau,P} \right) \right] +\left[{\mathcal{H}}\left( \Gamma_{\mathrm{ts}}, \Gamma^{f_\eta}_{\tau,0} \right)- {\mathcal{H}}\left(\Gamma^{f_\eta}_{\tau,P}, \Gamma^{f_\eta}_{\tau,0,P} \right) \right], \nonumber \end{align}\tag{52}\] where we denote \({\mathcal{Z}}^{f_\eta}_{\tau,P}=\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( e^{-\mathbb{H}_{\tau,P}}f_\eta\left({\mathcal{N}}_{P}/\tau\right) \right)\) and \({\mathcal{Z}}^{f_\eta}_{\tau,0,P}=\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( e^{-\mathbb{H}_{\tau,0,P}}f_\eta\left({\mathcal{N}}_{P}/\tau\right) \right)\). Although the trial state \(\Gamma_{\mathrm{ts}}\) does not factorize, we next show that the error terms in 52 are sufficiently small.

Step 2: Error estimates for the trial state. First, using the characterization of second quantization 23 , the unitary equivalence \({\mathcal{U}}\) and 41 , we have \[\label{rewrite:first} \begin{align} \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left({\mathcal{U}}^*\big( \mathbb{W}_{\tau,P} \otimes \mathbb{1}_{{\mathfrak{F}}(Q{\mathfrak{H}})} \big) {\mathcal{U}}\Gamma_{\mathrm{ts}} \right) &= \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}}) \otimes {\mathfrak{F}}(Q{\mathfrak{H}}) } \left(\big( \mathbb{W}_{\tau,P} \otimes \mathbb{1}_{{\mathfrak{F}}(Q{\mathfrak{H}})} \big) {\mathcal{U}}\Gamma_{\mathrm{ts}}{\mathcal{U}}^* \right) \\ &= \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}}) } \left( \mathbb{W}_{\tau,P} \big( \Gamma_{\mathrm{ts}}\big)_P \right) =\frac{1}{\tau^3} \mathrm {Tr}\big(W_P^\varepsilon\big((\Gamma_{\mathrm{ts}}\big)_P\big)^{(3)} \big) . \end{align}\tag{53}\] Then, we use 53 to rewrite the second term on the right-hand side of 52 as \[\begin{align}\label{error:terms95in95upper95bound} \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} & \left( \mathbb{W}_{\tau} \Gamma_{\mathrm{ts}} \right) -\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( \mathbb{W}_{\tau,P} \Gamma^{f_\eta}_{\tau,P} \right) \\ &= \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \left(\mathbb{W}_{\tau} -{\mathcal{U}}^*\big( \mathbb{W}_{\tau,P} \otimes \mathbb{1}_{{\mathfrak{F}}(Q{\mathfrak{H}})} \big) {\mathcal{U}}\right) \Gamma_{\mathrm{ts}} \right) \\ &\qquad \qquad + \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( {\mathcal{U}}^* \big( \mathbb{W}_{\tau,P} \otimes \mathbb{1}_{{\mathfrak{F}}(Q{\mathfrak{H}})}\big) {\mathcal{U}}\Gamma_{\mathrm{ts}} \right) -\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left(\mathbb{W}_{\tau,P} \Gamma^{f_\eta}_{\tau,P} \right) \\&= \frac{1}{\tau^3} \mathrm {Tr}\left( \big(W^\varepsilon- W^\varepsilon_P\big)\Gamma_{\mathrm{ts}}^{(3)} \right) +\left[ \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( {\mathcal{U}}^* \big( \mathbb{W}_{\tau,P} \otimes \mathbb{1}_{{\mathfrak{F}}(Q{\mathfrak{H}})}\big) {\mathcal{U}}\Gamma_{\mathrm{ts}} \right) -\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left(\mathbb{W}_{\tau,P} \Gamma^{f_\eta}_{\tau,P} \right) \right]. \end{align}\tag{54}\] The next lemma isolates the interaction part of the upper-bound error. Its content is that once the high-mode mass of the trial state is small, replacing \(W^\varepsilon\) by \(W^\varepsilon_P\) costs only a negligible amount. We now control the first term on the right-hand side.

Next, we control the second term on the right-hand side of 54 as well as the difference of the relative entropies in the last term of 52 . The purpose is to recover, from the full quantum trial state, the entropy term that naturally appears in the finite-dimensional classical variational problem. The difficulty lies precisely in the lack of factorization induced by the cutoff: after localization, the \(P\)- and \(Q\)-modes are still coupled through \(f_\eta({\mathcal{N}}/\tau)\), hence [56] is not applicable. We therefore rewrite the entropy difference into interaction contributions and logarithms of truncated partition functions, and then bound these pieces separately using the localization estimates already obtained, together with a uniform positive lower bound on the truncated free partition function.

Combining Lemmas 3 and 4, we obtain that, for all sufficiently large \(\tau\) and \(\Lambda_e\), and all \(\eta\in \big(0,\frac{1}{2}{\mathcal{K}}^2\big)\), \(\varepsilon\in(0,1)\) satisfying \(\eta^{-1}e^{M\varepsilon^{-2}} \leqslant\Lambda_e^{\frac{1}{8}} \leqslant\tau^{\frac{1}{32}}\), \[\label{all:error95in95upper95bound} \begin{align} &\left| \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( \mathbb{W}_{\tau,P} \Gamma^{f_\eta}_{\tau,P} \right) - \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \mathbb{W}_{\tau} \Gamma_{\mathrm{ts}} \right) +{\mathcal{H}}\left( \Gamma_{\mathrm{ts}}, \Gamma^{f_\eta}_{\tau,0} \right)- {\mathcal{H}}\left(\Gamma^{f_\eta}_{\tau,P}, \Gamma^{f_\eta}_{\tau,0,P} \right)\right| \leqslant O \big( \Lambda_e^{-\frac{1}{8}} \big). \end{align}\tag{85}\]

Step 3: Semi-classical analysis in projected space. Let \(J:= \mathrm{dim}(P{\mathfrak{H}})= \mathrm {Tr}(P) \leqslant\sqrt{\Lambda_e}\). We now prove the following projected semiclassical estimate. For all sufficiently large \(\tau\) and \(\Lambda_e\), and all \(\eta\in \big(0,\frac{1}{2}{\mathcal{K}}^2\big)\), \(\varepsilon\in(0,1)\) satisfying \(\eta^{-1}e^{M\varepsilon^{-2}}\leqslant\Lambda_e^{\frac{1}{8}}\leqslant\tau^{\frac{1}{32}}\), we have \[\begin{align} \label{upper32bound:end1} \frac{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}\left(e^{-\mathbb{H}_{\tau,P}}f_\eta(\mathcal{N}_P/\tau) \right) }{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}(e^{-\mathbb{H}_{\tau,0,P}})} \geqslant\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0,P}(u) - C\Lambda_e^{-\frac{1}{2}}. \end{align}\tag{86}\] where \(C>0\) depends only on \(w\) and \({\mathcal{K}}\). The proof proceeds naturally in two steps: first, we perform a coherent-state decomposition and separate the principal term from the cutoff errors; second, we bound the principal term from below and remove the cutoff error.

Step 3a: Coherent-state decomposition and separation of principal and error terms. Since \([\mathbb{H}_{\tau,P}, {\mathcal{N}}_P]= 0\), the coherent-state trace can be decomposed into particle-number sectors. More precisely, by the resolution of the identity in terms of coherent states 43 , we obtain \[\begin{align} \label{conherent-state:identity} \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}&\left(e^{-\mathbb{H}_{\tau,P}}f_\eta(\mathcal{N}_P/\tau) \right) =\pi^{-J} \int_{P{\mathfrak{H}}} \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( e^{-\mathbb{H}_{\tau,P}} f_\eta(\mathcal{N}_P/\tau) \big|\xi(u) \big\rangle \big \langle \xi(u)\big| \right) {\mathord{{\rm d}}}u \nonumber \\ &=\frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} \left\langle \xi(\sqrt{\tau}u), e^{-\mathbb{H}_{\tau,P}} f_\eta(\mathcal{N}_P/\tau) \xi(\sqrt{\tau}u) \right\rangle {\mathord{{\rm d}}}u \\ &\geqslant\frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\}} \left\langle \xi(\sqrt{\tau}u), e^{-\mathbb{H}_{\tau,P}} f_\eta(\mathcal{N}_P/\tau) \xi(\sqrt{\tau}u) \right\rangle {\mathord{{\rm d}}}u \nonumber \\ & =\frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\}} e^{-\tau \|Pu\|_{L^2}^2} \sum_{n=0}^\infty \frac{(\tau\|Pu\|_{L^2}^2 )^n}{n!} f_\eta \left(\frac{n}{\tau}\right) \left\langle \frac{(Pu)^{\otimes n} }{\|Pu\|_{L^2}^n} ,e^{-\mathbb{H}_{\tau,P}^{(n)}} \frac{(Pu)^{\otimes n} }{\|Pu\|_{L^2}^n}\right\rangle {\mathord{{\rm d}}}u .\nonumber \end{align}\tag{87}\] Here, the \(n\)-body Hamiltonian is defined by \(\mathbb{H}_{\tau,P}^{(n)}= \frac{1}{\tau}\sum_{j=1}^n (PhP)_{j}- \frac{1}{\tau^3}\sum_{1 \leqslant i_1 < i_2< i_3 \leqslant n} (W_P^\varepsilon)_{i_1,i_2,i_3}\), \(n\geqslant 3\), whereas for \(n=1,2\), only the kinetic part remains. In the formulas above we first assume \(\|Pu\|_{L^2}>0\). The resulting estimate extends to the case \(\|Pu\|_{L^2}=0\) since the corresponding coherent-state Poisson distribution has mean zero and only the sector \(n=0\) contributes. By the Peierls-Bogoliubov inequality \(\langle x,e^A x \rangle \geqslant e^{\langle x,Ax \rangle}\) for normalized vector \(x\), we obtain \[\begin{align} \left\langle \frac{(Pu)^{\otimes n} }{\|Pu\|_{L^2}^n} ,e^{-\mathbb{H}_{\tau,P}^{(n)}} \frac{(Pu)^{\otimes n} }{\|Pu\|_{L^2}^n}\right\rangle \geqslant\mathrm{exp} \left( - \left\langle \frac{(Pu)^{\otimes n} }{\|Pu\|_{L^2}^n} ,\mathbb{H}_{\tau,P}^{(n)} \frac{(Pu)^{\otimes n} }{\|Pu\|_{L^2}^n}\right\rangle \right) =: e^{-{\mathcal{E}}_{n,\tau} (Pu)}. \end{align}\] For \(\|Pu\|_{L^2}>0\), this exponent is explicitly \[\begin{align} {\mathcal{E}}_{n,\tau} (Pu) = \begin{cases} \frac{n}{\tau\|Pu\|_{L^2}^2 }\langle u,PhPu \rangle - \frac{n(n-1)(n-2)}{\tau^3\|Pu\|_{L^2}^6 } {\mathcal{W}}^\varepsilon_P,& n\geqslant 3, \\ \frac{n}{\tau\|Pu\|_{L^2}^2 }\langle u,PhPu \rangle, & n=1,2, \\ 0, & n=0. \end{cases} \end{align}\] Consequently, inserting this expression into 87 yields \[\begin{align} \label{tr:lower95two95part} \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( e^{-\mathbb{H}_{\tau,P}}f_\eta(\mathcal{N}_P/\tau) \right) & \geqslant\frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\}} e^{-\tau \|Pu\|_{L^2}^2} \sum_{n=0}^\infty \frac{(\tau\|Pu\|_{L^2}^2 )^n}{n!} f_\eta \left(\frac{n}{\tau}\right) e^{-{\mathcal{E}}_{n,\tau} (Pu)} {\mathord{{\rm d}}}u \nonumber \\ & = \frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\}} \mathbf{E} \Big( f_\eta \Big( \frac{\mathbf{X}}{\tau}\Big) e^{-{\mathcal{E}}_{\mathbf{X},\tau } (Pu) } \Big) {\mathord{{\rm d}}}u, \end{align}\tag{88}\] where, for each fixed \(u\), \(\mathbf{X}\) denotes a Poisson random variable with mean \(\tau \|Pu\|_{L^2}^2\). Setting \({\mathcal{E}}(Pu):= \langle u,PhPu \rangle- {\mathcal{W}}^\varepsilon_P\), we claim that, \[\begin{align} \label{assertion:diff:Hamiltonian} \mathbb{1}_{\{n/\tau \leqslant{\mathcal{K}}^2,\, \|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } \left| {\mathcal{E}}_{n,\tau} (Pu) - {\mathcal{E}}(Pu) \right | \leqslant C_{{\mathcal{K}},w} (\Lambda_e +\varepsilon^{-2}) \left(\Big| \frac{n}{\tau } - \|Pu\|_{L^2}^2 \Big| +\tau^{-1}\right). \end{align}\tag{89}\] Indeed, since \(PhP\leqslant\Lambda_e P\), the kinetic contribution is bounded, for all \(n\geqslant 0\), by \[\begin{align} \left| \frac{n}{\tau\|Pu\|_{L^2}^2 }\langle u,PhPu \rangle - \langle u,PhPu \rangle \right| \leqslant\left| 1- \frac{n}{\tau\|Pu\|_{L^2}^2 } \right | \langle u,PhPu \rangle \leqslant\Lambda_e \left| \frac{n}{\tau } - \|Pu\|_{L^2}^2 \right |. \end{align}\] For the interaction contribution, using \({\mathcal{W}}^\varepsilon_P \leqslant\varepsilon^{-2} \|w\|_{L^\infty}^2 \|Pu\|_{L^2}^6\), we obtain, for \(n\geqslant 3\), \[\begin{align} & \mathbb{1}_{\{n/\tau \leqslant{\mathcal{K}}^2,\, \|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } \Big| \frac{n(n-1)(n-2)}{\tau^3\|Pu\|_{L^2}^6 } {\mathcal{W}}^\varepsilon_P - {\mathcal{W}}^\varepsilon_P \Big| \\ &\qquad \leqslant\mathbb{1}_{\{n/\tau \leqslant{\mathcal{K}}^2,\, \|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } \Big| \frac{n(n-1)(n-2) }{\tau^3 } - \|Pu\|_{L^2}^6 \Big| \varepsilon^{-2} \|w\|_{L^\infty}^2 \\ & \qquad \leqslant\mathbb{1}_{\{n/\tau \leqslant{\mathcal{K}}^2,\, \|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } \left( \Big| \frac{n^3 }{\tau^3 } - \|Pu\|_{L^2}^6 \Big| + \frac{3 n^2 }{\tau^3 } + \frac{2n }{\tau^3 } \right) \varepsilon^{-2} \|w\|_{L^\infty}^2 \\ &\qquad \leqslant C_{{\mathcal{K}},w} \left(\Big| \frac{n}{\tau } - \|Pu\|_{L^2}^2 \Big| +\tau^{-1}\right) \varepsilon^{-2} . \end{align}\] For \(n=0,1,2\), the interaction term is absent from \({\mathcal{E}}_{n,\tau}\). In this case the same bound follows from \[\begin{align} & \mathbb{1}_{\{n/\tau \leqslant{\mathcal{K}}^2,\, \|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } {\mathcal{W}}^\varepsilon_P \leqslant\varepsilon^{-2} \|w\|_{L^\infty}^2{\mathcal{K}}^4 \|Pu\|_{L^2}^2 \\ &\qquad \leqslant\varepsilon^{-2} \|w\|_{L^\infty}^2{\mathcal{K}}^4 \Big|\frac{n}{\tau}-\|Pu\|_{L^2}^2\Big| + 2 \varepsilon^{-2} \|w\|_{L^\infty}^2{\mathcal{K}}^4 \frac{1}{\tau} \leqslant C_{{\mathcal{K}},w} \left(\Big| \frac{n}{\tau } - \|Pu\|_{L^2}^2 \Big| +\tau^{-1}\right) \varepsilon^{-2} . \end{align}\] This proves the claim 89 . Turning back to 88 , we use \(e^{-x} \geqslant 1-x\) together with 89 to obtain \[\begin{align} \label{transform:numbercut:to:masscut} & \mathbb{1}_{\{ \|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } f_\eta \Big( \frac{\mathbf{X}}{\tau}\Big) e^{-{\mathcal{E}}_{\mathbf{X},\tau } (Pu) } \geqslant\mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } f_\eta \Big(\frac{\mathbf{X}}{\tau}\Big) e^{-{\mathcal{E}}(Pu)} e^{-|{\mathcal{E}}_{\mathbf{X},\tau}(Pu)-{\mathcal{E}}(Pu) |} \nonumber \\ & \quad \geqslant\mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } f_\eta \Big(\frac{\mathbf{X}}{\tau}\Big) e^{-{\mathcal{E}}(Pu)} - \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } f_\eta \Big(\frac{\mathbf{X}}{\tau}\Big) e^{-{\mathcal{E}}(Pu)} |{\mathcal{E}}_{\mathbf{X},\tau}(Pu)-{\mathcal{E}}(Pu) | \nonumber \\ & \quad \geqslant\mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } \Big( f_\eta \big(\|Pu\|_{L^2}^2\big) -\|f_\eta' \|_{L^\infty} \Big| \frac{ \mathbf{X}}{\tau}- \|Pu\|_{L^2}^2 \Big|\Big) e^{-{\mathcal{E}}(Pu)} \nonumber \\ & \qquad\qquad \qquad-C_{{\mathcal{K}},w} (\Lambda_e +\varepsilon^{-2}) \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } \left( \Big| \frac{ \mathbf{X}}{\tau } - \|Pu\|_{L^2}^2 \Big| +\tau^{-1}\right) e^{-{\mathcal{E}}(Pu)} . \end{align}\tag{90}\] Since \(\mathbf{X}\) is a Poisson random variable with mean \(\tau\|Pu\|_{L^2}^2\), we have \[\begin{align} \label{expectation:Poisson} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } \mathbf{E} \Big| \frac{ \mathbf{X}}{\tau}- \|Pu\|_{L^2}^2 \Big| \leqslant\tau^{-1} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } \left( \mathbf{E} \Big| \mathbf{X}- \tau \|Pu\|_{L^2}^2 \Big|^2 \right)^{1/2} \leqslant{\mathcal{K}}\tau^{-\frac{1}{2}}. \end{align}\tag{91}\] Taking expectation on both sides of 90 and inserting the resulting bound into 88 , we obtain \[\begin{align} \label{lower:bound3} \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}& \left( e^{-\mathbb{H}_{\tau,P}}f_\eta(\mathcal{N}_P/\tau) \right) \geqslant\frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} f_\eta \big(\|Pu\|_{L^2}^2\big) e^{-{\mathcal{E}}(Pu)} {\mathord{{\rm d}}}u \\ &- \frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} \eta^{-1} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } \mathbf{E} \Big| \frac{ \mathbf{X}}{\tau}- \|Pu\|_{L^2}^2 \Big| e^{-{\mathcal{E}}(Pu)} {\mathord{{\rm d}}}u \nonumber \\ &- C_{{\mathcal{K}},w} (\Lambda_e +\varepsilon^{-2}) \frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} }\left( \mathbf{E} \Big| \frac{ \mathbf{X}}{\tau } - \|Pu\|_{L^2}^2 \Big| +\tau^{-1}\right) e^{-{\mathcal{E}}(Pu)} {\mathord{{\rm d}}}u \nonumber \\ &\geqslant\frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} f_\eta \big(\|Pu\|_{L^2}^2\big) e^{-{\mathcal{E}}(Pu)} {\mathord{{\rm d}}}u- C_{{\mathcal{K}},w} \frac{ \Lambda_e}{\sqrt{\tau}} \frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } e^{-{\mathcal{E}}(Pu)} {\mathord{{\rm d}}}u,\nonumber \end{align}\tag{92}\] where we used 91 and \(\eta^{-1}\varepsilon^{-2}\leqslant\Lambda_e\) in the last inequality.

Step 3b: Lower bound on the principal term and removal of the cutoff errors. From [3], the free partition function is given by \(\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}(e^{-\mathbb{H}_{\tau,0,P}})=\prod_{j=1}^J \frac{1}{1-e^{-\lambda_j/\tau}}\). As already observed in [56], Bernoulli’s inequality, together with \(\Lambda_e\leqslant\tau^{\frac{1}{4}}\) yields \[\begin{align} \label{bernoulli:inequality} \prod_{j=1}^J \frac{\tau}{\lambda_j} (1-e^{-\lambda_j/\tau}) \geqslant\prod_{j=1}^J \left( 1- \frac{\lambda_j}{2\tau}\right) \geqslant 1-\sum_{j=1}^J \frac{\lambda_j}{2\tau} \geqslant 1-\frac{\Lambda_e^{3/2}}{\tau} \geqslant 1- \Lambda_e^{-1}. \end{align}\tag{93}\] Applying 93 to the first term on the right-hand side of 92 , we obtain \[\begin{align} \label{first95hard95term1} &\frac{1}{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}(e^{-\mathbb{H}_{\tau,0,P}})}\frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) e^{-{\mathcal{E}}(Pu)} {\mathord{{\rm d}}}u \\ & \qquad = \prod_{j=1}^J \frac{\tau}{\lambda_j} (1-e^{-\lambda_j/\tau}) \int_{P{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) e^{ {\mathcal{W}}_P^\varepsilon} \prod_{j=1}^J \frac{\lambda_j}{\pi} e^{-\langle u,PhPu\rangle} {\mathord{{\rm d}}}u \nonumber \\& \qquad \geqslant\left( 1-\Lambda_e^{-1} \right) \int_{P{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{0,P}(u) \geqslant\int_{P{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{0,P}(u) - C\Lambda_e^{-\frac{1}{2}} , \nonumber \end{align}\tag{94}\] where the last inequality follows from the pointwise bound on the support of the cutoff, \[\begin{align} \Lambda_e^{-1} \int_{P{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{0,P}(u) \leqslant\Lambda_e^{-1} e^{\varepsilon^{-2}\|w\|_{L^\infty}^2 {\mathcal{K}}^6} \leqslant\Lambda_e^{-\frac{1}{2}}. \end{align}\] For the second term on the right-hand side of 92 , the same argument gives \[\begin{align} \label{second95easy95term} &C_{{\mathcal{K}},w} \frac{ \Lambda_e}{\sqrt{\tau}} \frac{1}{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}(e^{-\mathbb{H}_{\tau,0,P}})} \frac{\tau^J}{\pi^{J}} \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } e^{-{\mathcal{E}}(Pu)} {\mathord{{\rm d}}}u \\ & =C_{{\mathcal{K}},w} \frac{ \Lambda_e}{\sqrt{\tau}} \prod_{j=1}^J \frac{\tau}{\lambda_j} (1-e^{-\lambda_j/\tau}) \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{0,P}(u) \leqslant C_{{\mathcal{K}},w} \frac{ e^{\varepsilon^{-2}\|w\|_{L^\infty}^2 {\mathcal{K}}^6} \Lambda_e }{\sqrt{\tau}} \leqslant C \Lambda_e^{-\frac{1}{2}}, \nonumber \end{align}\tag{95}\] where in the last inequality we used \(\Lambda_e \leqslant\tau^{\frac{1}{4}}\). Combining 92 , 94 and 95 , we obtain 86 .

Step 4: Conclusion of ?? . Going back to 52 , we first use 86 to derive \[\label{finite95analysis1} \begin{align} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau,P}}{{\mathcal{Z}}^{f_\eta}_{\tau,0,P}} &= -\log \frac{ \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}\left (e^{-\mathbb{H}_{\tau,P}}f_\eta(\mathcal{N}_P/\tau)\right) }{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}( e^{-\mathbb{H}_{\tau,0,P}})} + \log \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}(\Gamma_{\tau,0,P} f_\eta({\mathcal{N}}_P/\tau)) \\ &\leqslant-\log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0,P}(u) - \Lambda_e^{-\frac{1}{2}} \right) +\log \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}\big(\Gamma_{\tau,0,P} f_\eta({\mathcal{N}}_P/\tau)\big) \\ &= -\log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0,P}(u) - \Lambda_e^{-\frac{1}{2}} \right) \\ &\qquad \qquad + \log \left(\int_{P {\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0,P}(u) \right) + \log \left( \frac{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}\big(\Gamma_{\tau,0,P} f_\eta({\mathcal{N}}_P/\tau)\big)}{\int_{{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0}(u) }\right). \end{align}\tag{96}\] To control the right-hand side, we first provide a lower bound for \(\int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0,P}(u)\). Using 183 , we obtain that for all sufficiently large \(\Lambda_e>0\) satisfying \(\eta^{-1}\leqslant\Lambda_e^{\frac{1}{8}}\), \[\label{lower:bound:for:integral:f40Pu41} \begin{align} \int_{P {\mathfrak{H}}} & e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0,P}(u) \geqslant\int_{{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0}(u) \geqslant\int_{{\mathfrak{H}}} f_\eta\left( \|u\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0}(u) -C\Lambda_e^{-\frac{1}{8}} \\ & \geqslant\int_{{\mathfrak{H}}}\mathbb{1}_{\{ \|u\|_{L^2}^2 \leqslant\frac{{\mathcal{K}}^2}{2}\}} {\mathord{{\rm d}}}\mu_{0}(u) -C\Lambda_e^{-\frac{1}{8}} \geqslant\frac{1}{2} \int_{{\mathfrak{H}}}\mathbb{1}_{\{ \|u\|_{L^2}^2 \leqslant\frac{{\mathcal{K}}^2}{2}\}} {\mathord{{\rm d}}}\mu_{0}(u) >0. \end{align}\tag{97}\] By this bound and the fact \(\log(1+t)=O(|t|)\) for small \(|t|\), we deduce that for sufficiently large \(\tau>0\) and \(\Lambda_e>0\) satisfying \(\eta^{-1}\leqslant\Lambda_e^{\frac{1}{8}}\), \[\begin{align} \label{finite95analysis2} - &\log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0,P}(u) - \Lambda_e^{-\frac{1}{2}} \right) + \log \left(\int_{P {\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0,P}(u) \right)\nonumber \\ &= -\log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu_{0,P}^{f_\eta}(u) \right) -\log \left( 1- \frac{\Lambda_e^{-\frac{1}{2}}}{ \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0,P}(u) } \right) \\& \leqslant-\log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu_{0,P}^{f_\eta}(u) \right) + C\Lambda_e^{-\frac{1}{2}}.\nonumber \end{align}\tag{98}\] For the third term on the right-hand side of 96 , using 80 , 97 , 178 and \(\log(1+t)=O(|t|)\) for \(|t|\) small again, we derive for sufficiently large \(\tau>0\) and \(\Lambda_e>0\) satisfying \(\eta^{-1}\leqslant\Lambda_e^{\frac{1}{8}}\leqslant\tau^{\frac{1}{32}}\), \[\begin{align} \label{finite95analysis3} \log &\left( \frac{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}(\Gamma_{\tau,0,P} f_\eta({\mathcal{N}}_P/\tau))}{\int_{{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0}(u) }\right) = \log \left(1+ \frac{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}(\Gamma_{\tau,0,P} f_\eta({\mathcal{N}}_P/\tau))-\int_{{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0}(u) }{\int_{{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0}(u) } \right) \nonumber \\ &\leqslant C\left|\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}\big(\Gamma_{\tau,0,P} f_\eta({\mathcal{N}}_P/\tau)\big)-\int_{{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0}(u) \right| \leqslant C \tau^{-\frac{1}{4}} \leqslant C\Lambda_e^{-\frac{1}{8}}. \end{align}\tag{99}\] Finally, collecting all error terms and substituting 85 , 96 , 98 and 99 into 52 , we obtain \[\begin{align} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{{\mathcal{Z}}^{f_\eta}_{\tau,0}} \leqslant- \log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) \right) + O\left( \Lambda_e^{-\frac{1}{8}}\right). \end{align}\] This concludes the proof of Proposition 9. ◻

Proof. Let \(J:=\mathrm {Tr}[P]\) and define \(W^\varepsilon_P=P^{\otimes 3} W^\varepsilon P^{\otimes 3}\). We denote by \((\Gamma^{f_\eta}_{\tau})_{P}^{(3)}= P^{\otimes 3} (\Gamma_{\tau}^{f_\eta})^{(3)} P^{\otimes 3}\) the \(P\)-localization of \((\Gamma_{\tau}^{f_\eta})^{(3)}\), and by \(\mu_{P,\tau}^{\tau^{-1}}\) and \(\mu_{P,0}^{\tau^{-1}}\) the corresponding lower symbols of \((\Gamma^{f_\eta}_\tau)_P\) and \((\Gamma^{f_\eta}_{\tau,0})_P\), respectively.

Step 1: Localization of the interaction and control of the error term. By applying the variational principle 34 , 36 and the Berezin-Lieb type inequality 47 , we obtain that \[\label{lower:variational} \begin{align} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} &={\mathcal{H}}(\Gamma^{f_\eta}_{\tau},\Gamma^{f_\eta}_{\tau,0}) - \frac{1}{ \tau^3} \mathrm{Tr}\big( W^\varepsilon(\Gamma_{\tau}^{f_\eta})^{(3)} \big) \\ &\geqslant{\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu_{P,0}^{\tau^{-1}})- \frac{1}{\tau^3} \mathrm {Tr}\left(W^\varepsilon_{P} (\Gamma_{\tau}^{f_\eta})^{(3)} \right) - \frac{1}{\tau^3}\mathrm {Tr}\left( \left(W^\varepsilon-W^\varepsilon_{P}\right) (\Gamma_{\tau}^{f_\eta})^{(3)} \right). \end{align}\tag{100}\] We begin with the localized interaction terms in 100 . By 45 , we derive \[\begin{align} -& \frac{1}{\tau^3} \mathrm {Tr}\left(W^\varepsilon_{P} (\Gamma_{\tau}^{f_\eta})^{(3)} \right)= - \frac{1}{\tau^3} \mathrm {Tr}\left(W^\varepsilon_{P} P^{\otimes 3}(\Gamma_{\tau}^{f_\eta})^{(3)} P^{\otimes 3} \right) \\ &= - \frac{1}{\tau^3} \mathrm {Tr}\left(W^\varepsilon_P \big((\Gamma_{\tau}^{f_\eta})_P\big)^{(3)} \right) \nonumber =- \frac{1}{6} \int_{P {\mathfrak{H}}} \langle u^{\otimes 3}, W^\varepsilon_{P} u^{\otimes 3}\rangle {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u) \\&\qquad \qquad \qquad \qquad \qquad \qquad + \frac{1}{ \tau^3} \sum_{\ell=0}^{2}{3\choose \ell}^2 \frac{(3-\ell +J-1)!}{(J-1)!} \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \Big( W^\varepsilon_P \big((\Gamma^{f_\eta}_{\tau})_{P}^{(\ell)} \otimes_s \mathbb{1}_{\otimes_s^{3-\ell} P{\mathfrak{H}}}\big) \Big) . \end{align}\] Since \(W^\varepsilon_P\geqslant 0\) and \((\Gamma^{f_\eta}_{\tau})_{P}^{(\ell)} \otimes_s \mathbb{1}_{\otimes_s^{3-\ell} P{\mathfrak{H}}}\geqslant 0\), each trace term in the sum is nonnegative. Therefore, \[\begin{align} \label{lower32bound32interaction32energy1} - \frac{1}{\tau^3} \mathrm {Tr}\left(W^\varepsilon_{P} (\Gamma_{\tau}^{f_\eta})^{(3)} \right)\geqslant-\frac{1}{6} \int_{P{\mathfrak{H}}} \langle u^{\otimes 3}, W^\varepsilon_P u^{\otimes 3}\rangle {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u)= - \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P(u) {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u). \end{align}\tag{101}\]

We next estimate the localization errors on the second line of 100 . As in Lemma 3, these terms are reduced to the high-frequency one-body mass. More precisely, repeating the computations in 56 –61 , we obtain \[\begin{align} \left| \frac{1}{\tau^3}\mathrm {Tr}\left( \left(W^\varepsilon-W^\varepsilon_{P}\right) (\Gamma_{\tau}^{f_\eta})^{(3)}\right)\right| & \leqslant C \varepsilon^{-2}\|w\|^2_{L^\infty} {\mathcal{K}}^3 \left( \frac{1}{\tau} \left| \mathrm {Tr}\left( Q \big( (\Gamma_{\tau}^{f_\eta})^{(1)} - \Gamma_{\tau,0}^{(1)} \big) \right) \right|+ \frac{1}{\tau} \mathrm {Tr}\left( Q\Gamma_{\tau,0}^{(1)} \right) \right)^{\frac{1}{2}} \\& \quad + C\varepsilon^{-2}\|w\|^2_{L^\infty} \left( \frac{1}{\tau} \left| \mathrm {Tr}\left( Q \big((\Gamma_{\tau}^{f_\eta})^{(1)} - \Gamma_{\tau,0}^{(1)} \big) \right) \right|+ \frac{1}{\tau} \mathrm {Tr}\left( Q\Gamma_{\tau,0}^{(1)} \right) \right). \end{align}\] To bound the right-hand side, it is enough to control the relative entropy \({\mathcal{H}}(\Gamma^{f_\eta}_{\tau}, \Gamma_{\tau,0})\), thanks to 63 . Repeating the argument in 64 –68 , we obtain for sufficiently large \(\tau>0\) \[\begin{align} {\mathcal{H}}\big( \Gamma^{f_\eta}_{\tau}, \Gamma_{\tau,0} \big) & \leqslant-\mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \mathbb{W}_{\tau} \Gamma^{f_\eta}_{\tau,0}\right)-\log\left( \frac{{\mathcal{Z}}^{f_\eta}_{\tau,0}}{{\mathcal{Z}}_{\tau,0}}\right) + \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \mathbb{W}_{\tau}\Gamma^{f_\eta}_{\tau} \right) \\& \leqslant-\log\left[ \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \Gamma_{\tau,0} f_\eta \left(\frac{{\mathcal{N}}}{\tau}\right) \right) \right]+2\|w\|_{L^\infty}^2\varepsilon^{-2}{\mathcal{K}}^6 \\ & \leqslant\log \left( 1/ C({\mathcal{K}})\right)+2\|w\|_{L^\infty}^2\varepsilon^{-2}{\mathcal{K}}^6. \end{align}\] The remaining estimates are identical to those in 69 . Hence, for all sufficiently large \(\tau>0\), \(\Lambda_e>0\), \(\eta\in \big(0,\frac{1}{2}{\mathcal{K}}^2\big)\) and \(\varepsilon\in(0,1)\) satisfying \(\varepsilon^{-3}\leqslant\Lambda_e^{\frac{1}{8}}\), we conclude that \[\begin{align} \label{errors:localization95interaction2} \left| \frac{1}{\tau^3} \mathrm {Tr}\left( \left(W^\varepsilon-W^\varepsilon_{P}\right) (\Gamma_{\tau}^{f_\eta})^{(3)}\right) \right| \leqslant C\Lambda_e^{-\frac{1}{8}}. \end{align}\tag{102}\] Combining 100 –102 and invoking the classical variational principle 36 once more, we obtain that for all sufficiently large \(\tau>0\), \(\Lambda_e>0\), \(\eta\in \big(0,\frac{1}{2}{\mathcal{K}}^2\big)\) and \(\varepsilon\in(0,1)\) satisfying \(\varepsilon^{-3}\leqslant\Lambda_e^{\frac{1}{8}}\), and any \(R>0\), \[\begin{align} \label{variation} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} &\geqslant{\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu_{P,0}^{\tau^{-1}})- \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u)-C\Lambda_e^{-\frac{1}{8}} \nonumber \\ & = {\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu_{P,0}^{\tau^{-1}})- \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P \mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant R \}} {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u)- \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P \mathbb{1}_{\{\|u\|_{L^2}^2 > R \}} {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u)-C\Lambda_e^{-\frac{1}{8}} \nonumber \\ & \geqslant-\log \left( \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u) \right)- \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P \mathbb{1}_{\{\|u\|_{L^2}^2 > R \}} {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u) -C\Lambda_e^{-\frac{1}{8}} . \end{align}\tag{103}\] Step 2: Interacting lower-symbol tail estimate. To control the second term on the right-hand side of 103 , we first prove a general tail estimate for interacting lower symbols.

Applying Lemma 5 to the Gibbs state \(\Gamma_\tau^{f_\eta}\), with \(P=\mathbb{1}(h\leqslant\Lambda_e)\) and \(R={\mathcal{K}}^2+1\), we conclude that for all sufficiently large \(\tau>0\) satisfying \(\varepsilon^{-3}\leqslant\Lambda_e^{\frac{1}{8}} \leqslant\tau^{\frac{1}{32}}\), \[\begin{align} \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P \mathbb{1}_{\{\|u\|_{L^2}^2 > {\mathcal{K}}^2+1 \}} {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u) & \leqslant\varepsilon^{-2} \|w\|^2_{L^\infty} \int_{\{\|Pu\|_{L^2}^2>{\mathcal K}^2+1\}}\|Pu\|_{L^2}^{6}{\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u) \\&\leqslant C({\mathcal{K}},w) \varepsilon^{-2} e^{-c({\mathcal{K}})\tau} \leqslant C\Lambda_e^{-\frac{1}{8}}. \end{align}\] Inserting this bound into 103 gives \[\begin{align} \label{lower:holder3} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} \geqslant-\log \left( \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2+1 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u) \right) -C\Lambda_e^{-\frac{1}{8}} . \end{align}\tag{104}\]

Step 3: Quantitative \(L^1\)-comparison between \(\mu_{P,0}^{\tau^{-1}}\) and \(\mu^{f_\eta}_{0,P}\). To prove ?? , it remains to compare \(\int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2+1 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u)\) with \(\int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}\). Since \(\mu_{P,0}^{\tau^{-1}}\) involves a particle number cutoff, whereas \(\mu^{f_\eta}_{0,P}\) incorporates a mass cutoff, a more refined comparison is needed. Our first goal is therefore to establish a quantitative \(L^1\)-estimate between them.

Step 4: Exponential moment estimate and conclusion of the lower bound. With the \(L^1\)-comparison in hand, we control the right-hand side of 104 by \[\label{lower:holder1} \begin{align} \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2+ 1\}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u) & \leqslant\int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) + \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2+ 1 \}} } | {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}- {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}| \\ &\leqslant\int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u)+ e^{\varepsilon^{-2} \|w\|_{L^\infty}^2({\mathcal{K}}^2+ 1)^3} \|\mu_{P,0}^{\tau^{-1}}-\mu^{f_\eta}_{0,P}\|_{L^1(P{\mathfrak{H}})} \\ &\leqslant\int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) + e^{M \varepsilon^{-2}}\Lambda_e^{-\frac{1}{4}}\leqslant\int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) + \Lambda_e^{-\frac{1}{8}}. \end{align}\tag{114}\] Using the fact that \(\log(1+t)=O(|t|)\) for small \(|t|\), together with 104 , we obtain \[\begin{align} \label{conclusion:lower32bound} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} \nonumber &\geqslant-\log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2+1 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u) \right)-C\Lambda_e^{-\frac{1}{8}} \\ &\geqslant-\log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) \right)-\log \left(1+\frac{C \Lambda_e^{-\frac{1}{8}} }{\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u)} \right)-C\Lambda_e^{-\frac{1}{8}} \\ & \geqslant-\log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u)\right)- O\left(\Lambda_e^{-\frac{1}{8}} \right), \nonumber \end{align}\tag{115}\] where we used \(\int e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) \geqslant 1\) in the last inequality. This concludes the proof of Proposition 12. ◻

Proof. Our starting point is the convergence of the quantum free energy established in the previous subsections. Combining Proposition 9 and Proposition 12, we obtain that, for sufficiently large \(\Lambda_e>0\), \(\eta\in \big(0,\frac{1}{2}{\mathcal{K}}^2\big)\) and \(\varepsilon\in(0,1)\) satisfying \(\Lambda_e^{\frac{1}{8}} \geqslant\eta^{-1}e^{M\varepsilon^{-2}}\), \[\begin{align} \label{P:to:infty2} \left| -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} + \log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (u)\right) \right| \leqslant O\left( \Lambda_e^{-\frac{1}{8}} \right), \end{align}\tag{116}\] where we have chosen \(\tau \to \infty\) sufficiently fast such that \(\Lambda_e \leqslant\tau^{\frac{1}{4}}\). Similarly, for any given \(\overline{\Lambda}_e\geqslant\Lambda_e\), we also obtain the same bound with \(P\) replaced by \(\overline{P}=\mathbb{1}(h\leqslant\overline{\Lambda}_e)\), provided that \(\tau\) is sufficiently large such that \(\eta^{-1}e^{M\varepsilon^{-2}}\leqslant\Lambda_e^{\frac{1}{8}} \leqslant\overline{\Lambda}_e^{\frac{1}{8}} \leqslant\tau^{\frac{1}{32}}\). More precisely, we may choose the same sufficiently large \(\tau\) such that \[\begin{align} \label{P:to:infty3} \left| -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} + \log \left( \int_{\overline{P}{\mathfrak{H}}} e^{{\mathcal{W}}_{\overline{P}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,\overline{P}} (u)\right) \right| \leqslant O\left( \overline{\Lambda}_e^{-\frac{1}{8}} \right) \leqslant O\left( \Lambda_e^{-\frac{1}{8}} \right), \end{align}\tag{117}\] and 116 hold simultaneously. Thus by the triangle inequality, we find that for all \(\overline{\Lambda}_e \geqslant\Lambda_e\) \[\begin{align} \label{P:to:infty4} \left| \log \left( \int_{\overline{P}{\mathfrak{H}}} e^{{\mathcal{W}}_{\overline{P}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,\overline{P}} (u)\right) - \log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (u)\right) \right| \leqslant O\left( \Lambda_e^{-\frac{1}{8}} \right). \end{align}\tag{118}\] Recalling the definitions of \({\mathcal{W}}^\varepsilon_P\) and \({\mathcal{W}}^\varepsilon\) in ?? and 19 , it follows that, for any given \(\varepsilon>0\) \[\begin{align} \left|{\mathcal{W}}_P^\varepsilon(u) -{\mathcal{W}}^\varepsilon(u) \right| &\leqslant\frac{1}{3!} \left| \big\langle u^{\otimes 3}, W^\varepsilon u^{\otimes 3}\big\rangle - \big\langle P^{\otimes 3} u^{\otimes 3}, W^\varepsilon P^{\otimes 3} u^{\otimes 3}\big\rangle \right| \\& \leqslant\frac{1}{3!} \|w^\varepsilon\|_{L^\infty}^2\left( \|Pu\|_{L^2}^4+ \|Pu\|_{L^2}^2 \|u\|_{L^2}^2+ \|u\|_{L^2}^4 \right) \big( \|Pu\|_{L^2}+ \|u\|_{L^2}\big) \|u-Pu\|_{L^2} \\ &\leqslant\|w^\varepsilon\|_{L^\infty}^2 \|u\|_{L^2}^5\|u-Pu\|_{L^2} . \end{align}\] Since \(\|w^\varepsilon\|_{L^\infty}<\infty\) and \(u\in L^2\) \(\mu_0\)-almost surely, it follows that \({\mathcal{W}}_P^\varepsilon(u)\to {\mathcal{W}}^\varepsilon(u)\) as \(\Lambda_e \to \infty\), \(\mu_0\)-almost surely. By the continuity of the exponential function and \(f_\eta\), we obtain \[\begin{align} \label{P:to:infty5} e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta(\|Pu\|_{L^2}^2) \xrightarrow{\Lambda_e \to \infty} e^{{\mathcal{W}}^\varepsilon(u) } f_\eta(\|u\|_{L^2}^2), \end{align}\tag{119}\] \(\mu_0\)-almost surely. Furthermore, for any given \(\varepsilon>0\), we have \[\begin{align} \label{P:to:infty6} \sup_{\Lambda_e \geqslant 1} e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta(\|Pu\|_{L^2}^2) \leqslant e^{\varepsilon^{-2}\|w\|_{L^\infty}^2 {\mathcal{K}}^6} \|f_\eta\|_{L^\infty} <\infty. \end{align}\tag{120}\] Combining 119 with 120 , then the dominated convergence theorem implies that \[\begin{align} \label{P:to:infty7} \lim_{\Lambda_e \to \infty} \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (u) = \lim_{\Lambda_e \to \infty} \frac{ \int_{{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } f_\eta(\|Pu\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0 (u)}{\int_{{\mathfrak{H}}} f_\eta(\|Pu\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0 (u) } = \int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0} (u). \end{align}\tag{121}\] Incorporating 121 with 118 and letting \(\overline{\Lambda}_e \to \infty\), we obtain ?? . ◻

Proof of ?? . The conclusion follows from a combination of Propositions 9, 12 and 14. We first verify that, under the assumptions of Proposition 8, the parameter constraints required in the above propositions can indeed be satisfied. The condition \(1>\varepsilon\geqslant M \left(\log \tau \right)^{-\frac{1}{2}}\) implies that \(e^{64M\varepsilon^{-2}} \leqslant\tau\). Consequently, for \(\eta \in \big[\tau^{-\frac{1}{64}}, \frac{1}{2} {\mathcal{K}}^2\big)\), we may choose \(\Lambda_e>0\) sufficiently large such that \[\begin{align} \label{relation:para} \eta^{-1}e^{M\varepsilon^{-2}} \leqslant\Lambda_e^{\frac{1}{8}}\leqslant\tau^{\frac{1}{32}} . \end{align}\tag{122}\] Therefore, combining ?? and 116 , it follows that \[\begin{align} & \left| -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} + \log \left( \int_{ {\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_0 (u)\right) \right| \leqslant\left| -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} + \log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (u)\right) \right| \\ &\qquad \qquad \qquad \qquad + \left| \log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (u)\right) -\log \left( \int_{ {\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_0 (u)\right) \right| \leqslant C \Lambda_e^{-\frac{1}{8}} . \end{align}\] By the elementary inequality \(|e^x-e^y| \leqslant e^y(e^{|x-y|}-1)\), \(x,y \in \mathbb{R}\), we obtain \[\begin{align} \left|\frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} -\int_{ {\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_0 (u) \right| & \leqslant\int_{ {\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_0 (u) \big(e^{C\Lambda_e^{-\frac{1}{8}}}-1\big) \leqslant Ce^{\|w\|_{L^\infty}^2{\mathcal{K}}^6\varepsilon^{-2}} \Lambda_e^{-\frac{1}{8}} \\ &\leqslant Ce^{\frac{1}{2} M\varepsilon^{-2}} \Lambda_e^{-\frac{1}{8}} \leqslant C \Lambda_e^{-\frac{1}{16}}, \end{align}\] where we used 122 in the last inequality. Letting \(\tau \to \infty\) and \(\Lambda_e \to \infty\) completes the proof. ◻

Proof of ?? . The proof consists of three steps. We first combine the quantitative de Finetti estimate with the \(L^1\)-control of the lower symbols to obtain trace-norm convergence after localization to \(P{\mathfrak{H}}\). We then remove the projection on the quantum side using the one-body tail estimate from Lemma 3. On the classical side, we compare the two partition functions directly via the pointwise convergence \({\mathcal{W}}_P^\varepsilon(u)\to {\mathcal{W}}^\varepsilon(u)\). Finally, we combine the localized convergence with these approximation estimates to obtain the full trace-norm statement. Throughout the proof, we assume that the parameters satisfy 122 , so that Propositions 9, 12 and 14 apply directly.

Step 1: convergence after localization. Let \(P=\mathbb{1}(h\leqslant\Lambda_e)\) and \(J=\mathrm {Tr}(P)\). Let \({\mathcal{W}}_P^\varepsilon\) and \(\mu^{f_\eta}_{0,P}\) be defined as in ?? and ?? , respectively. We introduce the probability measure \(\mu^{\varepsilon,f_\eta}_P\) on \(P{\mathfrak{H}}\) by \[\begin{align} {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u):=\frac{e^{{\mathcal{W}}_P^\varepsilon(u)} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u)}{\int e^{{\mathcal{W}}_P^\varepsilon(v)} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(v)}. \end{align}\] We also define \(\big((\Gamma_{\tau}^{f_\eta})_P\big)^{(k)}=P^{\otimes k}(\Gamma^{f_\eta}_{\tau} \big)^{(k)} P^{\otimes k}\). In this step, we prove that, for all sufficiently large \(\tau\) and \(\Lambda_e\) satisfying 122 , \[\begin{align} \label{density32matrix32convergence} \left\| \frac{k!}{\tau^k}\big((\Gamma_{\tau}^{f_\eta})_P\big)^{(k)} - \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant O\big(\Lambda_e^{-\frac{1}{32}}\big). \end{align}\tag{123}\] To this end, let \(\mu_{P,\tau}^{\tau^{-1}}\) denote the lower symbol associated with \(\Gamma^{f_\eta}_\tau\) on \(P{\mathfrak{H}}\). By the quantitative de Finetti estimate 46 , we have for every \(k\geqslant 1\), \[\begin{align} \label{de32finetti32error32trace32norm32localized} & \left\| \frac{k!}{\tau^k}\big((\Gamma_{\tau}^{f_\eta})_P\big)^{(k)} -\int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\tau^{-1}}_{P,\tau}(u) \right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \nonumber \\ &\quad \leqslant\frac{1}{\tau^k}\sum_{\ell=0}^{k-1}{k\choose \ell}^2\frac{(k-\ell+J-1)!}{(J-1)!}\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}\big({\mathcal{N}}_P^\ell (\Gamma^{f_\eta}_{\tau})_P\big) \\ &\quad \lesssim_{k} \sum_{\ell=0}^{k-1} \left(\frac{J}{\tau}\right)^{k-\ell} \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})}\left(\left(\frac{{\mathcal{N}}}{\tau}\right)^\ell \Gamma^{f_\eta}_{\tau}\right) \lesssim_{k} \sum_{\ell=0}^{k-1} \left(\frac{J}{\tau}\right)^{k-\ell} {\mathcal{K}}^{2\ell} \leqslant C_{k,{\mathcal{K}}}\Lambda_e^{-1}, \nonumber \end{align}\tag{124}\] where in the inequality we used \(J=\mathrm {Tr}(P)\leqslant\sqrt{\Lambda_e} \leqslant\tau^{\frac{1}{8}}\).

It therefore remains to prove that, for all sufficiently large \(\tau\) and \(\Lambda_e\) satisfying 122 , \[\begin{align} \label{density32matrix32convergence3} \left\| \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\tau^{-1}}_{P,\tau}(u)- \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant O \left( \Lambda_e^{-\frac{1}{32}}\right). \end{align}\tag{125}\] Since \(|u^{\otimes k}\rangle\langle u^{\otimes k}|\) is a positive rank-one operator with trace \(\|u\|_{L^2}^{2k}\), we apply Cauchy–Schwarz inequality to obtain \[\begin{align} \label{weighted32tv32operator32bound} & \left\| \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\tau^{-1}}_{P,\tau}(u)- \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \nonumber \\ &\quad \leqslant\int_{P{\mathfrak{H}}}\|u\|_{L^2}^{2k}\,\big|{\mathord{{\rm d}}}\mu^{\tau^{-1}}_{P,\tau}-{\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P\big| \\ &\quad \leqslant\left(\int_{P{\mathfrak{H}}}\|u\|_{L^2}^{4k}\,{\mathord{{\rm d}}}\mu^{\tau^{-1}}_{P,\tau}(u) +\int_{P{\mathfrak{H}}}\|u\|_{L^2}^{4k}\,{\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u)\right)^{1/2} \|\mu^{\tau^{-1}}_{P,\tau}-\mu^{\varepsilon,f_\eta}_P\|_{L^1(P{\mathfrak{H}})}^{1/2}. \nonumber \end{align}\tag{126}\] We next bound the moments appearing above. By the definition of \(\mu^{\varepsilon,f_\eta}_P\), \[\begin{align} \label{density32matrix32convergence35} \int_{P{\mathfrak{H}}}\|u\|_{L^2}^{4k}\,{\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u) =\frac{\int_{{\mathfrak{H}}}\|Pu\|_{L^2}^{4k} e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2)\,{\mathord{{\rm d}}}\mu_0(u)}{\int_{{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2)\,{\mathord{{\rm d}}}\mu_0(u)} \leqslant{\mathcal{K}}^{4k}. \end{align}\tag{127}\] For the moment associated with the lower symbol, applying 124 with \(2k\) in place of \(k\), we obtain for all sufficiently large \(\tau\) and \(\Lambda_e\) satisfying 122 , \[\begin{align} \label{density32matrix32convergence36} \int_{P{\mathfrak{H}}} &\|u\|_{L^2}^{4k} {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u) =\mathrm {Tr}\left(\int_{P{\mathfrak{H}}}|u^{\otimes 2k}\rangle\langle u^{\otimes 2k}|\,{\mathord{{\rm d}}}\mu^{\tau^{-1}}_{P,\tau}(u)\right) \nonumber \\ &\leqslant\frac{(2k)!}{\tau^{2k}} \mathrm {Tr}\left( \big((\Gamma_{\tau}^{f_\eta})_P\big)^{(2k)} \right) + \left\| \frac{(2k)!}{\tau^{2k}} \big((\Gamma_{\tau}^{f_\eta})_P\big)^{(2k)} -\int_{P {\mathfrak{H}}} |u^{\otimes {2k}}\rangle \langle u^{\otimes {2k}} | {\mathord{{\rm d}}}\mu^{\tau^{-1}}_{P,\tau}(u) \right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(2k)})} \\ &\leqslant\mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \left( \frac{{\mathcal{N}}}{\tau}\right)^{2k} \Gamma_{\tau}^{f_\eta} \right) + C_{k,{\mathcal{K}}} \leqslant{\mathcal{K}}^{4k} +C_{k,{\mathcal{K}}} . \nonumber \end{align}\tag{128}\] It thus suffices to derive an \(L^1\)-comparison between \(\mu_{P,\tau}^{\tau^{-1}}\) and \(\mu^{\varepsilon,f_\eta}_P\). We assert that, for sufficiently large \(\tau\) and \(\Lambda_e\) satisfying 122 , \[\begin{align} \label{density32matrix32convergence30} \| \mu_{P,\tau}^{\tau^{-1}}- \mu^{\varepsilon,f_\eta}_P \|_{L^1(P{\mathfrak{H}})} \leqslant C \Lambda_e^{-\frac{1}{16}} . \end{align}\tag{129}\] Using the variational principle 34 , the Berezin-Lieb inequality 47 , and invoking again 103 with 115 , we obtain that for \(R={\mathcal{K}}^2+1\), \[\begin{align} \label{density32matrix32convergence31} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} & \geqslant{\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu_{P,0}^{\tau^{-1}})- \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P \mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant R \}} {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u)-C\Lambda_e^{-\frac{1}{8}} \nonumber \\ & = {\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu') -\log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) \mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u)\right)-C\Lambda_e^{-\frac{1}{8}} \\ & \geqslant{\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu')-\log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u)} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u)\right)- C \Lambda_e^{-\frac{1}{8}} , \nonumber \end{align}\tag{130}\] where we used 115 in the last inequality and defined \[\begin{align} {\mathord{{\rm d}}}\mu'(u):=\frac{e^{{\mathcal{W}}_P^\varepsilon(u)\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u)}{\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(v)\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v)}. \end{align}\] By combining 130 with the free energy upper bound ?? , and using Pinsker’s inequality (see, e.g., [69]), we obtain, for sufficiently large \(\tau>0\) and \(\Lambda_e>0\), \[\begin{align} \label{density32matrix32convergence32} \| \mu_{P,\tau}^{\tau^{-1}}- \mu' \|_{L^1(P{\mathfrak{H}})} \leqslant\sqrt{2} {\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu')^{\frac{1}{2}} \leqslant C \Lambda_e^{-\frac{1}{16}} . \end{align}\tag{131}\] Next, we compare \(\mu'\) with \(\mu^{\varepsilon,f_\eta}_P\): \[\begin{align} \label{density32matrix32convergence33} \| \mu' - \mu^{\varepsilon,f_\eta}_P \|_{L^1(P{\mathfrak{H}})} & \leqslant\frac{1}{\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v)} \left\| e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}-e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} \right\|_{L^1(P{\mathfrak{H}})} \nonumber \\ & +\left| \frac{1}{\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v)} - \frac{1}{\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(v)} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (v)} \right| \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(v)\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (v) \nonumber \\ & +\frac{1}{\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(v)} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (v)} \int_{P{\mathfrak{H}}} \big| e^{{\mathcal{W}}_P^\varepsilon(v)\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant R \}} } - e^{{\mathcal{W}}_P^\varepsilon(v) } \big| {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (v) . \end{align}\tag{132}\] The last term on the right-hand side vanishes because \(\mu^{f_\eta}_{0,P}\) is supported on \(\{\|v\|_{L^2}^2 \leqslant{\mathcal{K}}^2 \}\) and \(R>{\mathcal K}^2\). By \(\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v)\geqslant 1\) and the same calculations as in 114 , the first term is controlled by \[\begin{align} \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant R \}} } \big| {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}} - {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} \big| \leqslant e^{\varepsilon^{-2} \|w\|_{L^\infty}^2R^3} \|\mu_{P,0}^{\tau^{-1}}-\mu^{f_\eta}_{0,P}\|_{L^1(P{\mathfrak{H}})} \leqslant\Lambda_e^{-\frac{1}{8}}. \end{align}\] For the same reason, the second term on the right-hand side of 132 is bounded by \[\begin{align} & \left| \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant R \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v)- \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(v) \right| \\ &\leqslant\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant R \}} } \big| {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}} - {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} \big|+ \int_{P{\mathfrak{H}}} \big| e^{{\mathcal{W}}_P^\varepsilon(v)\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant R \}} } - e^{{\mathcal{W}}_P^\varepsilon(v) } \big| {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (v) \leqslant\Lambda_e^{-\frac{1}{8}}. \end{align}\] Collecting the above bounds together with 131 and 132 , the bound 129 follows. Inserting 127 –129 into 126 , we obtain 125 .

Step 2: removal of the projection on the quantum side. We claim that, for all sufficiently large \(\tau\) and \(\Lambda_e\) satisfying 122 , one has \[\begin{align} \label{quantum32tail32claim32trace32norm} \left\| \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} - \frac{k!}{\tau^k}\big((\Gamma_{\tau}^{f_\eta})_P\big)^{(k)} \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant O\big(\Lambda_e^{-\frac{1}{16}}\big). \end{align}\tag{133}\] Indeed, since \(\big((\Gamma_{\tau}^{f_\eta})_P\big)^{(k)}=P^{\otimes k}( \Gamma^{f_\eta}_{\tau} \big)^{(k)} P^{\otimes k}\), the same argument as in 56 –59 yields \[\begin{align} & \left\| \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} - \frac{k!}{\tau^k}\big((\Gamma_{\tau}^{f_\eta})_P\big)^{(k)} \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant\mathrm {Tr}\left( ( \mathbb{1}^{\otimes k}-P^{\otimes k}) \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} \right) \\ &\qquad \qquad \qquad \qquad\qquad \qquad \qquad\qquad \qquad + 2\sqrt{ \mathrm {Tr}\left( ( \mathbb{1}^{\otimes k}-P^{\otimes k}) \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} \right) \mathrm {Tr}\left( \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} \right)} \\ &\qquad \qquad \leqslant\mathrm {Tr}\left( ( \mathbb{1}^{\otimes k}-P^{\otimes k}) \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} \right)+ 2{\mathcal{K}}^k \sqrt{ \mathrm {Tr}\left( ( \mathbb{1}^{\otimes k}-P^{\otimes k}) \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} \right) }. \end{align}\] It therefore remains to estimate the first trace term. To this end, we use the elementary inequality \[\begin{align} \mathbb{1}^{\otimes k}-P^{\otimes k}\leqslant\sum_{j=1}^k \mathbb{1}^{\otimes(j-1)}\otimes Q\otimes \mathbb{1}^{\otimes(k-j)}, \end{align}\] and then argue exactly as in the proof of 60 , with the \(3\)-body density matrix replaced by the \(k\)-body one. This yields \[\begin{align} \label{first32quantum32tail32estimate} \mathrm {Tr}& \left( ( \mathbb{1}^{\otimes k}-P^{\otimes k}) \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} \right) \leqslant\frac{k!}{\tau^k} \mathrm {Tr}\left[\left(\sum_{j=1}^k \mathbb{1}^{\otimes(j-1)}\otimes Q\otimes \mathbb{1}^{\otimes(k-j)}\right)(\Gamma^{f_\eta}_{\tau})^{(k)}\right] \nonumber \\& \lesssim_k \frac{1}{\tau} \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( {\mathcal{U}}^* \big(\mathbb{1}_{{\mathfrak{F}}(P{\mathfrak{H}})} \otimes {\mathcal{N}}_Q \big) {\mathcal{U}}\left( \frac{{\mathcal{N}}}{\tau} \right)^{k-1} \Gamma^{f_\eta}_{\tau} \right) \\ & \lesssim_k \frac{1}{\tau} {\mathcal{K}}^{2k-2} \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( {\mathcal{U}}^* \big(\mathbb{1}_{{\mathfrak{F}}(P{\mathfrak{H}})} \otimes {\mathcal{N}}_Q \big) {\mathcal{U}}\Gamma^{f_\eta}_{\tau} \right) = \frac{ C_k }{\tau} {\mathcal{K}}^{2k-2} \mathrm {Tr}\left( Q (\Gamma^{f_\eta}_{\tau})^{(1)} \right) \nonumber \\ &\lesssim_{k,{\mathcal{K}}} \frac{1}{\tau} \left| \mathrm {Tr}\left( Q \big((\Gamma^{f_\eta}_{\tau})^{(1)}- \Gamma_{\tau,0}^{(1)} \big) \right) \right|+ \frac{1}{\tau} \mathrm {Tr}\left( Q\Gamma_{\tau,0}^{(1)} \right) \leqslant C \Lambda_e^{-\frac{1}{8}}, \nonumber \end{align}\tag{134}\] where the last inequality follows from the same one-body tail estimates as those used in 62 –69 . This proves 133 .

Step 3: quantitative comparison between the finite-dimensional Hartree operator and the full Hartree operator. Following the strategy used in the proof of Proposition 14, we claim that, for all sufficiently large \(\tau\) and \(\Lambda_e\) satisfying 122 , \[\begin{align} \label{overline:mu:P:to:infty} \left\| \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u) - \int_{{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant O\big(\Lambda_e^{-\frac{1}{32}}\big). \end{align}\tag{135}\] Starting from 123 and 133 , it follows that \[\begin{align} \label{P:to:infty:new1} \left\| \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} - \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant O\big(\Lambda_e^{-\frac{1}{32}}\big) , \end{align}\tag{136}\] provided that \(\tau \to \infty\) sufficiently fast satisfying \(\Lambda_e\leqslant\tau^{\frac{1}{4}}\). For any given \(\overline{\Lambda}_e\geqslant\Lambda_e\) and \(\overline{P}=\mathbb{1}(h\leqslant\overline{\Lambda}_e)\), by the argument as in 116 –118 , we may choose the same sufficiently large \(\tau\) such that \[\begin{align} \left\| \frac{k!}{\tau^k} (\Gamma_{\tau}^{f_\eta})^{(k)} - \int_{\overline{P}{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_{\overline{P}}(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant O \left( \overline{\Lambda}_e^{-\frac{1}{32}}\right), \end{align}\] and 136 hold simultaneously. Thus by the triangle inequality, we find that for all \(\overline{\Lambda}_e\geqslant\Lambda_e\to \infty\), \[\begin{align} \label{P:to:infty:new} \left\| \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u)- \int_{\overline{P}{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_{\overline{P}}(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant O \left( \Lambda_e^{-\frac{1}{32}}\right). \end{align}\tag{137}\] Thus \(\int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u)\) is a Cauchy family in \(\mathfrak{S}^1({\mathfrak{H}}^{(k)})\). Then, by the definitions of \(\mu^{\varepsilon,f_\eta}_P\) and \(\mu^{\varepsilon,f_\eta}\), we obtain \[\begin{align} \label{P:to:infty:newS1} &\left\| \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u)- \int_{{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \\ &\leqslant\left| \frac{1}{\int_{{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0(u)} - \frac{1}{\int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0(u)} \right| \int_{{\mathfrak{H}}} \|Pu\|_{L^2}^{2k} e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0(u) \nonumber \\ & \quad +\frac{ \left\| \int_{{\mathfrak{H}}} \big( |(Pu)^{\otimes k}\rangle \langle (Pu)^{\otimes k} | e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2) - |u^{\otimes k}\rangle \langle u^{\otimes k} | e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2)\big){\mathord{{\rm d}}}\mu_0(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} }{\int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0(u)} \nonumber \\ & \leqslant\int_{{\mathfrak{H}}} \left| e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2)- e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2) \right| {\mathord{{\rm d}}}\mu_0(u) \frac{\int_{{\mathfrak{H}}} \|Pu\|_{L^2}^{2k} {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u)}{\int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0(u)} \nonumber \\ & \quad +\frac{\int_{{\mathfrak{H}}} \big\| |(Pu)^{\otimes k}\rangle \langle (Pu)^{\otimes k} | e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2) - |u^{\otimes k}\rangle \langle u^{\otimes k} | e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2) \big\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} {\mathord{{\rm d}}}\mu_0(u)}{\int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0(u)} . \nonumber \end{align}\tag{138}\] To estimate these two terms, we first observe that for any \(\varepsilon\in (0,1)\), \(\eta\in(0,\frac{1}{2} {\mathcal{K}}^2)\) and \(\Lambda_e>0\), \[\begin{align} \int e^{{\mathcal{W}}^\varepsilon} f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_{0}(u) &\geqslant\int f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_{0}(u) \geqslant\int \mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2/2\}} {\mathord{{\rm d}}}\mu_{0}(u)>0, \\ \int_{{\mathfrak{H}}} \|Pu\|_{L^2}^{2k} {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u)& \leqslant{\mathcal{K}}^{2k}. \end{align}\] Moreover, from 119 and 120 , it holds that \(\mu_0\)-almost surely, \[\begin{align} & \left\| |(Pu)^{\otimes k}\rangle \langle (Pu)^{\otimes k} | e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2) - |u^{\otimes k}\rangle \langle u^{\otimes k} | e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \\ &\quad \leqslant 2 \|u\|_{L^2}^{k} \|(Pu)^{\otimes k} -u^{\otimes k}\|_{L^2({\mathfrak{H}}^{(k)})} e^{\varepsilon^{-2}\|w\|_{L^\infty}^2{\mathcal{K}}^6}\|f_\eta\|_{L^\infty} \\ & \qquad + \|u\|_{L^2}^{2k} \big| e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2) -e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2) \big| \xrightarrow{\Lambda_e \to \infty} 0, \end{align}\] and \[\begin{align} &\left\| |(Pu)^{\otimes k}\rangle \langle (Pu)^{\otimes k} | e^{{\mathcal{W}}_P^\varepsilon(u)}f_\eta(\|Pu\|_{L^2}^2) - |u^{\otimes k}\rangle \langle u^{\otimes k} | e^{{\mathcal{W}}^\varepsilon(u)}f_\eta(\|u\|_{L^2}^2) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \\ &\quad \leqslant 2{\mathcal{K}}^{2k}e^{\varepsilon^{-2}\|w\|_{L^\infty}^2{\mathcal{K}}^6}\|f_\eta\|_{L^\infty}. \end{align}\] Therefore, by the dominated convergence theorem, the second term on the right-hand side of 138 converges to zero as \(\Lambda_e \to \infty\). The first term can be treated in exactly the same way and also converges to zero. Hence, for every given \(\varepsilon\in (0,1)\), \[\begin{align} \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u) \xrightarrow{\Lambda_e \to \infty} \int_{{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u) ,\qquad \mathrm{in} \quad {\mathfrak{S}}^1({\mathfrak{H}}^{(k)}). \end{align}\] Together with 137 , this proves 135 by letting \(\overline{\Lambda}_e \to \infty\). Finally, combining 123 , 133 , and 135 , and letting \(\tau \to \infty\) and \(\Lambda_e \to \infty\), we conclude the proof of ?? . ◻

Proof of Theorem 5. We first observe that Proposition 8 applies directly in the present setting. Indeed, in Theorem 5, both \(\varepsilon>0\) and the cutoff \(g\in C_c^\infty([0,\infty);\mathbb{R}_+)\) are fixed, so that the only remaining limit is \(\tau \to \infty\). Therefore, the constraints in Proposition 8 are automatically satisfied, and we obtain \[\begin{align} \left| \frac{\mathcal{Z}^{g}_{\tau}}{\mathcal{Z}^{g}_{\tau,0}} - \frac{\int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon} g( \|u\|^2_{L^2}){\mathord{{\rm d}}}\mu_{0}(u) }{\int_{{\mathfrak{H}}} g( \|u\|^2_{L^2}) {\mathord{{\rm d}}}\mu_{0}(u) } \right| \to 0. \end{align}\] Together with ?? , this implies \[\begin{align} \frac{\mathcal{Z}^{g}_{\tau}}{\mathcal{Z}_{\tau,0}} = \frac{\mathcal{Z}^{g}_{\tau}}{\mathcal{Z}^{g}_{\tau,0}} \cdot \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \Gamma_{\tau,0} g\left( {\mathcal{N}}/\tau \right) \right) \xrightarrow{\tau \to \infty} \int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon} {\mathord{{\rm d}}}\mu^g_{0}(u) \cdot \int_{{\mathfrak{H}}} g( \|u\|^2_{L^2}) {\mathord{{\rm d}}}\mu_{0}(u) = \int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon} g( \|u\|^2_{L^2}){\mathord{{\rm d}}}\mu_{0}(u) . \end{align}\] The convergence of the reduced density matrices is obtained exactly as in the proof of ?? . ◻

Proof. The proof is divided into three steps, following the picture introduced in Section 3. Throughout the proof, we fix \({\mathcal{K}}_s \in ({\mathcal{K}},{\mathcal{K}}_c)\) and assume that \(\tau\) and \(\Lambda_e\) are sufficiently large. We also let \(\eta\in \big(0,\frac{1}{2}{\mathcal{K}}^2\big)\) and \(\varepsilon\in(0,1)\) such that \[\begin{align} \label{refined:relation} \eta^{-1} \leqslant\Lambda_e^{\frac{1}{8}}, \qquad \varepsilon^{-3}\leqslant\Lambda_e^{\frac{1}{8}}, \qquad \Lambda_e\leqslant\tau^{\frac{1}{4}}. \end{align}\tag{139}\] We first establish the lower bound on the free energy.

Step 1: Free energy lower bound in the mass-subcritical regime. The key point in the subcritical regime is that the auxiliary mass parameter \({\mathcal{K}}_s\) appearing in the tail estimate for the interacting lower symbol may be chosen strictly below the critical mass \({\mathcal{K}}_c\). The large-mass contribution is still absorbed by Lemma 5, while the contribution from the complementary region can be controlled by the uniform Oh-Sosoe-Tolomeo bound in [1], rather than by the rough pointwise exponential estimate used in the critical case. Since the estimate 103 from Subsection 3.3 is independent of pointwise exponential bounds, it gives, uniformly for all parameters satisfying 139 , \[\begin{align} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} &\geqslant{\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu_{P,0}^{\tau^{-1}})- \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u)-C\Lambda_e^{-\frac{1}{8}} \nonumber \\ & \geqslant-\log \left( \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s ^2 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u) \right)- \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P \mathbb{1}_{\{\|u\|_{L^2}^2 > {\mathcal{K}}_s^2 \}} {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u) -C\Lambda_e^{-\frac{1}{8}} . \end{align}\] For the mass-tail contribution, we apply Lemma 5 to the Gibbs state \(\Gamma_\tau^{f_\eta}\) with \(R={\mathcal{K}}_s^2\). This yields, again uniformly for all parameters satisfying 139 , \[\begin{align} \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P \mathbb{1}_{\{\|u\|_{L^2}^2 > {\mathcal{K}}_s^2\}} {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u) & \leqslant\varepsilon^{-2} \|w\|^2_{L^\infty} \int_{\{\|Pu\|_{L^2}^2>{\mathcal K}_s^2\}}\|Pu\|_{L^2}^{6}{\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u) \\&\leqslant C({\mathcal{K}},{\mathcal{K}}_s,w)\varepsilon^{-2} e^{-c({\mathcal{K}},{\mathcal{K}}_s)\tau} \leqslant C\Lambda_e^{-\frac{1}{8}}. \end{align}\] Combining the preceding two estimates, we obtain \[\begin{align} \label{variation1} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} \geqslant -\log \left( \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u) \right)- C\Lambda_e^{-\frac{1}{8}}. \end{align}\tag{140}\] It remains to compare the integral with respect to the free lower-symbol \(\mu_{P,0}^{\tau^{-1}}\) with the corresponding integral under the projected truncated Gaussian measure \(\mu^{f_\eta}_{0,P}\). We decompose \[\begin{align} \label{lower:tail} \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u) & \leqslant\int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) + \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } | {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}- {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}| . \end{align}\tag{141}\] The following comparison is the only point where the subcritical argument differs from the lower-bound proof in Subsection 3.3. Since \({\mathcal{K}}_s<{\mathcal K}_c\), the uniform projected Oh-Sosoe-Tolomeo bound gives a sufficiently small constant \(\varsigma>0\) such that \[\begin{align} \label{mian:appendix32A} \sup_{\Lambda_e>0} \int_{{\mathfrak{H}}} e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s \}} {\mathord{{\rm d}}}\mu_{0}(u) <\infty. \end{align}\tag{142}\] For completeness, this estimate is proved in Appendix 8. The \(L^1\)-comparison estimate 105 is also independent of the pointwise exponential bound, and hence remains valid here: \(\|\mu_{P,0}^{\tau^{-1}}- \mu^{f_\eta}_{0,P} \|_{L^1(P{\mathfrak{H}})}\leqslant C\Lambda_e^{-\frac{1}{4}}\) whenever \(\eta^{-1}\leqslant\Lambda_e^{\frac{1}{8}}\leqslant\tau^{\frac{1}{32}}\). Hölder’s inequality then gives \[\begin{align} \label{Holder:variation} & \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } | {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}- {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}| \nonumber \\ &\leqslant\left( \int_{P{\mathfrak{H}}} e^{(1+\varsigma) {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } |{\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}- {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}|\right)^{\frac{1}{1+\varsigma}} \|\mu_{P,0}^{\tau^{-1}}- \mu^{f_\eta}_{0,P} \|_{L^1(P{\mathfrak{H}})}^{\frac{\varsigma}{1+\varsigma}} \nonumber \\ &\leqslant\left( \int_{P{\mathfrak{H}}}\big( \mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} e^{(1+\varsigma) {\mathcal{W}}_P^\varepsilon} + \mathbb{1}_{\{\|u\|_{L^2}^2 > {\mathcal{K}}_s^2 \}} \big)({\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}+ {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}) \right)^{\frac{1}{1+\varsigma}} \Lambda_e^{-\frac{\varsigma}{4(1+\varsigma)}} \nonumber \\ &\leqslant\left( \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} e^{(1+\varsigma) {\mathcal{W}}_P^\varepsilon} ({\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}+ {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}) +2 \right)^{\frac{1}{1+\varsigma}} \Lambda_e^{-\frac{\varsigma}{4(1+\varsigma)}}. \end{align}\tag{143}\] It remains to prove that the first factor on the right-hand side of 143 is uniformly bounded. This is precisely where the mass-subcritical condition enters. By [2] and \(\|w^\varepsilon\|_{L^1}=1\), for every \(\varepsilon>0\) we have \[\begin{align} \label{lemma2463:RS25} {\mathcal{W}}^\varepsilon_P\leqslant\frac{1}{6} \|w^\varepsilon\|_{L^1}^2 \|Pu\|_{L^6}^6=\frac{1}{6} \|Pu\|_{L^6}^6. \end{align}\tag{144}\] Together with 142 , this implies \[\begin{align} \label{uniform:subcritical} \sup_{\varepsilon>0,\, \Lambda_e>0} \int_{\{\|Pu\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \} } e^{(1+\varsigma) {\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{0,P} \leqslant\sup_{\Lambda_e>0} \int_{ \{\|Pu\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} e^{\frac{1+\varsigma}{6} \|Pu\|_{L^6}^6 }{\mathord{{\rm d}}}\mu_{0,P} <\infty . \end{align}\tag{145}\] We next establish the corresponding exponential moment bound for the lower symbol \(\mu_{P,0}^{\tau^{-1}}\). Combining 106 –108 , we obtain \[\begin{align} {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u) &= \left( \frac{\tau}{\pi}\right)^J \frac{1}{\mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \big(\Gamma_{\tau,0}f_\eta\big(\frac{{\mathcal{N}}}{\tau}\big)\big)} \left\langle \xi(\sqrt{\tau}u),\mathrm {Tr}_{{\mathfrak{F}}(Q{\mathfrak{H}})}\big( {\mathcal{U}}\Gamma_{\tau,0}f_\eta\big({\mathcal{N}}/ \tau\big) {\mathcal{U}}^*\big) \xi(\sqrt{\tau}u) \right\rangle {\mathord{{\rm d}}}u \\ &\leqslant C \left( \frac{\tau}{\pi}\right)^J \left\langle \xi(\sqrt{\tau}u), \big(\Gamma_{\tau,0,P}f_\eta\big({\mathcal{N}}_P/ \tau\big)+ \eta^{-1}\Lambda_e^{-\frac{1}{2}} \Gamma_{\tau,0,P} \big) \xi(\sqrt{\tau}u) \right\rangle {\mathord{{\rm d}}}u \\ &\leqslant C \left( \frac{\tau}{\pi}\right)^J \left\langle \xi(\sqrt{\tau}u), \Gamma_{\tau,0,P} \xi(\sqrt{\tau}u) \right\rangle {\mathord{{\rm d}}}u \leqslant C e^{\frac{1}{\tau}\Lambda_e^2\|Pu\|_{L^2}^2} {\mathord{{\rm d}}}\mu_{0,P}(u). \end{align}\] Hence, using 145 together with the constraint \(\Lambda_e \leqslant\tau^{\frac{1}{4}}\), we obtain \[\begin{align} \label{uniform:subcritical2} \sup_{\substack{ \varepsilon>0, \\ 0< \Lambda_e \leqslant\tau^{1/4} }} & \int_{\{\|Pu\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \} } e^{(1+\varsigma) {\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}} \leqslant C \sup_{0< \Lambda_e \leqslant\tau^{1/4} } \int_{ \{\|Pu\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} e^{\frac{1+\varsigma}{6} \|Pu\|_{L^6}^6 } e^{\frac{1}{\tau}\Lambda_e^2\|Pu\|_{L^2}^2} {\mathord{{\rm d}}}\mu_{0,P} \nonumber \\ & \leqslant C e^{{\mathcal{K}}_s^2} \sup_{ \Lambda_e >0} \int_{ \{\|Pu\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} e^{\frac{1+\varsigma}{6} \|Pu\|_{L^6}^6 } {\mathord{{\rm d}}}\mu_{0,P} <\infty . \end{align}\tag{146}\] Combining 143 , 145 , and 146 , we obtain \[\begin{align} \label{observable:against:difference} \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } | {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}- {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}| \leqslant C \Lambda_e^{-\frac{\varsigma}{4(1+\varsigma)}}. \end{align}\tag{147}\] Inserting 147 into 140 and 141 , we conclude that, for all sufficiently large \(\tau\) and \(\Lambda_e\), and all \(\eta\in \big(0,\frac{1}{2}{\mathcal{K}}^2\big)\) and \(\varepsilon\in(0,1)\) satisfying 139 , \[\begin{align} \label{lower:bound:sub} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} \geqslant -\log \left( \int_{P{\mathfrak{H}}} e^{ {\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) \right)- C\Lambda_e^{-\frac{\varsigma}{4(1+\varsigma)}}. \end{align}\tag{148}\] This proves the desired projected lower bound in the mass-subcritical regime.

Step 2: Free energy upper bound in the mass-subcritical regime. We now prove the matching upper bound. Compared with Section 3, two modifications are needed in the mass-subcritical regime. The first concerns the localization error in the relative entropy estimate of Lemma 4. The second occurs at the final stage of the semiclassical analysis, namely in 94 and 95 , where one has to control the error produced by the cutoff. In both places, the pointwise exponential bounds from the critical argument are replaced by the uniform moment bounds available in the subcritical regime. We also note that Lemma 3 remains valid here, since its proof does not use pointwise exponential bounds. We begin with the corresponding refinement of Lemma 4.

We now return to the projected semiclassical estimate. Since the estimates 92 and 93 do not rely on the pointwise exponential bound, they remain valid here. Therefore, for all parameters satisfying 139 , we have \[\begin{align} \label{uniform:frequencey:cutoff} \frac{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}\left(e^{-\mathbb{H}_{\tau,P}}f_\eta(\mathcal{N}_P/\tau) \right) }{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}(e^{-\mathbb{H}_{\tau,0,P}})} & \geqslant\left( 1-\Lambda_e^{-1} \right) \int_{P{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{0,P}(u) \nonumber \\ &\qquad - C_{{\mathcal{K}}_c,w} \frac{ \Lambda_e}{\sqrt{\tau}} \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2}\leqslant{\mathcal{K}}\} } e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{0,P}(u) . \end{align}\tag{154}\] Using the uniform estimate 145 once more, we obtain \[\begin{align} \sup_{\varepsilon, \, \Lambda_e>0} \int_{P{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{0,P}(u) \leqslant\sup_{\Lambda_e>0} \int_{P{\mathfrak{H}}} \mathbb{1}_{\{\|Pu\|_{L^2} \leqslant{\mathcal{K}}\}} e^{\frac{1}{6} \|Pu\|_{L^6}^6 } {\mathord{{\rm d}}}\mu_{0,P}(u) <\infty. \end{align}\] Substituting this bound into 154 yields \[\begin{align} \label{semi:subcritical:lower} \frac{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}\left(e^{-\mathbb{H}_{\tau,P}}f_\eta(\mathcal{N}_P/\tau) \right) }{\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})}(e^{-\mathbb{H}_{\tau,0,P}})} \geqslant\int_{P{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) e^{{\mathcal{W}}_P^\varepsilon} {\mathord{{\rm d}}}\mu_{0,P}(u) - C\Lambda_e^{-1}. \end{align}\tag{155}\] We then combine 155 with Lemmas 3 and 7, and repeat the argument leading to 96 –99 . The pointwise exponential bounds are not used in these estimates. Hence, for all sufficiently large \(\tau\) and \(\Lambda_e\), and all \(\eta\in \big(0,\frac{1}{2}{\mathcal{K}}^2\big)\), \(\varepsilon\in(0,1)\) satisfying 139 , we have \[\begin{align} \label{upper:bound:sub} -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{{\mathcal{Z}}^{f_\eta}_{\tau,0}} \leqslant- \log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) \right) + O\left( \Lambda_e^{-\frac{1}{8}}\right). \end{align}\tag{156}\] This proves the desired projected upper bound in the mass-subcritical regime.

Step 3: Conclusion of Proposition 15. The preceding two steps identify the projected free energy. It remains to remove the projection. This is done by the Cauchy argument from Proposition 14: the projected Hartree partition functions form a Cauchy family as \(\Lambda_e\to\infty\), and their limit is the corresponding full partition function. This argument is independent of the pointwise exponential bounds. Hence, combining 148 and 156 , we obtain that, for all sufficiently large \(\Lambda_e>0\) and all \(\eta\in \big(0,\frac{1}{2}{\mathcal{K}}^2\big)\), \(\varepsilon\in(0,1)\) satisfying \(\eta^{-1}\leqslant\Lambda_e^{\frac{1}{8}}\) and \(\varepsilon^{-3}\leqslant\Lambda_e^{\frac{1}{8}}\), \[\begin{align} \left| \log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u)\right) -\log \left( \int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0}(u)\right)\right| \leqslant O\left( \Lambda_e^{-\sigma} \right), \end{align}\] for some sufficiently small \(\sigma>0\), depending only on the exponent \(\varsigma\) in the projected Oh-Sosoe-Tolomeo estimate 142 . Under the assumptions \(\eta\geqslant\tau^{-\frac{1}{64}}\) and \(\varepsilon\geqslant\tau^{-\frac{1}{96}}\), we may choose \(\Lambda_e\) so that 139 holds and \(\Lambda_e\to\infty\). Therefore, \[\begin{align} & \left| -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} + \log \left( \int_{ {\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_0 (u)\right) \right| \leqslant\left| -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} + \log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (u)\right) \right| \\ &\qquad \qquad \qquad \qquad + \left| \log \left( \int_{P {\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (u)\right) -\log \left( \int_{ {\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_0 (u)\right) \right| \leqslant O\left( \Lambda_e^{-\sigma} \right). \end{align}\] Using 144 and the elementary inequality \(|e^x-e^y| \leqslant e^y(e^{|x-y|}-1)\), \(x,y\in\mathbb{R}\), we then obtain \[\begin{align} \left|\frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} -\int_{ {\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_0 (u) \right| & \leqslant\int_{ {\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_0 (u) \big(e^{C\Lambda_e^{-\sigma}}-1\big) \\ &\leqslant C\Lambda_e^{-\sigma} \int_{ \{\|u\|_{L^2} \leqslant{\mathcal{K}}\}} e^{\frac{1}{6} \|u\|_{L^6}^6} {\mathord{{\rm d}}}\mu_0 (u) \leqslant C \Lambda_e^{-\sigma}, \end{align}\] where the last step follows from the subcritical integrability result of [1]. Letting \(\tau\to\infty\) and \(\Lambda_e\to\infty\) proves ?? .

We next prove the convergence of the reduced density matrices. The argument follows the proof of ?? ; we only record the changes needed in the subcritical regime. First, we claim convergence to the projected Hartree density matrices. Namely, there exists \(\beta>0\), independent of \(\varepsilon,\eta,\tau\) and \(\Lambda_e\), such that \[\begin{align} \label{sub:dm:projected-convergence} \left\|\frac{k!}{\tau^k}((\Gamma^{f_\eta}_\tau)_P)^{(k)} -\int_{P{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|{\mathord{{\rm d}}}\mu_P^{\varepsilon,f_\eta}(u)\right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant C_{k,{\mathcal{K}}}\Lambda_e^{-\beta}, \end{align}\tag{157}\] where the projected Hartree measure is defined by \({\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u) := \frac{e^{{\mathcal{W}}_P^\varepsilon(u)}{\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u)}{\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(v)}{\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(v)}\) and \(((\Gamma^{f_\eta}_\tau)_P)^{(k)}=P^{\otimes k}(\Gamma^{f_\eta}_\tau)^{(k)}P^{\otimes k}\). Since the estimates 124 –128 do not use pointwise exponential bounds, they remain valid under the present constraint 139 . Thus, \[\begin{align} & \left\|\frac{k!}{\tau^k}((\Gamma^{f_\eta}_\tau)_P)^{(k)} -\int_{P{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|{\mathord{{\rm d}}}\mu_P^{\varepsilon,f_\eta}(u)\right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \\ &\quad \leqslant\left\|\frac{k!}{\tau^k}((\Gamma^{f_\eta}_\tau)_P)^{(k)}- \int_{P{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|{\mathord{{\rm d}}}\mu^{\tau^{-1}}_{P,\tau}(u)\right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \\ &\quad + \left\| \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\tau^{-1}}_{P,\tau}(u)- \int_{P{\mathfrak{H}}} |u^{\otimes k}\rangle \langle u^{\otimes k} | {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}_P(u) \right\| _{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant C_{k,{\mathcal{K}}}\big( \Lambda_e^{-1}+ \|\mu^{\tau^{-1}}_{P,\tau}-\mu^{\varepsilon,f_\eta}_P\|_{L^1}^{1/2}\big). \end{align}\] It remains to control \(\|\mu^{\tau^{-1}}_{P,\tau}-\mu^{\varepsilon,f_\eta}_P\|_{L^1}\) without invoking the pointwise exponential bounds. Combining the lower bound argument leading to 140 and 148 with the upper bound 156 , we obtain \[\begin{align} \label{density32matrix32convergence42} - \log & \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) } {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u) \right) + C \Lambda_e^{-\frac{1}{8}} \geqslant -\log \frac{{\mathcal{Z}}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} \nonumber \\ & \geqslant{\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu_{P,0}^{\tau^{-1}})- \int_{P{\mathfrak{H}}} {\mathcal{W}}^\varepsilon_P \mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} {\mathord{{\rm d}}}\mu_{P,\tau}^{\tau^{-1}}(u)-C\Lambda_e^{-\frac{1}{8}} \nonumber \\ & = {\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu') -\log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u) \mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u)\right)-C\Lambda_e^{-\frac{1}{8}} \nonumber \\ & \geqslant{\mathcal{H}}_{\mathrm{cl}} (\mu_{P,\tau}^{\tau^{-1}}, \mu')-\log \left( \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(u)} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P}(u)\right)- C\Lambda_e^{-\frac{\varsigma}{4(1+\varsigma)}}, \end{align}\tag{158}\] where \({\mathord{{\rm d}}}\mu'(u):=\frac{ \mathrm{\exp}\big({\mathcal{W}}_P^\varepsilon(u)\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} \big) {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(u)}{\int_{P{\mathfrak{H}}} \mathrm{\exp}\big({\mathcal{W}}_P^\varepsilon(v)\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} \big) {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v)}\). Pinsker’s inequality gives \[\begin{align} \|\mu^{\tau^{-1}}_{P,\tau}-\mu'\|_{L^1} \leqslant\sqrt{2}{\mathcal{H}}_{\mathrm{cl}}\big(\mu^{\tau^{-1}}_{P,\tau},\mu'\big)^{\frac{1}{2}} \leqslant C\Lambda_e^{-\frac{\varsigma}{8(1+\varsigma)}}. \end{align}\] We now compare \(\mu'\) with \(\mu^{\varepsilon,f_\eta}_P\). By the same calculation as in 132 , \[\begin{align} \| \mu' - \mu^{\varepsilon,f_\eta}_P & \|_{L^1(P{\mathfrak{H}})} \leqslant\frac{1}{\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v)} \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } \big| {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}- {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} \big| \nonumber \\ & +\frac{1}{\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v)} \left| \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v) - \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon(v)} {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (v) \right| \\ & \leqslant C \int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|u\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } \big| {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}- {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} \big| + C \int_{P{\mathfrak{H}}} \big| e^{{\mathcal{W}}_P^\varepsilon(v)\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } - e^{{\mathcal{W}}_P^\varepsilon(v) } \big| {\mathord{{\rm d}}}\mu^{f_\eta}_{0,P} (v) , \end{align}\] where we used \(\int_{P{\mathfrak{H}}} e^{{\mathcal{W}}_P^\varepsilon\mathbb{1}_{\{\|v\|_{L^2}^2 \leqslant{\mathcal{K}}_s^2 \}} } {\mathord{{\rm d}}}\mu_{P,0}^{\tau^{-1}}(v)\geqslant 1\) in the last line. The second term on the right-hand side vanishes because \(\mu^{f_\eta}_{0,P}\) is supported on \(\{\|v\|_{L^2}^2\leqslant{\mathcal{K}}^2\}\) and \({\mathcal{K}}_s^2>{\mathcal K}^2\). The first term is exactly 147 , and is bounded by \(C\Lambda_e^{-\frac{\varsigma}{4(1+\varsigma)}}\). Consequently, \(\|\mu^{\tau^{-1}}_{P,\tau}-\mu^{\varepsilon,f_\eta}_P\|_{L^1}^{1/2}\leqslant C\Lambda_e^{-\beta}\) for some sufficiently small \(\beta>0\). This proves the localized convergence 157 .

The removal of the projection on the quantum side is unchanged, since it relies only on Lemma 3; the assumptions \(\varepsilon^{-3}\leqslant\Lambda_e^{1/8}\) and \(\Lambda_e\leqslant\tau^{1/4}\) are precisely those needed for that lemma. Therefore, \[\begin{align} \label{sub:dm:quantum-projection-removal} \left\|\frac{k!}{\tau^k}(\Gamma^{f_\eta}_\tau)^{(k)}-\frac{k!}{\tau^k}((\Gamma^{f_\eta}_\tau)_P)^{(k)}\right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant C_{k,{\mathcal{K}}}\Lambda_e^{-\frac{1}{8}}. \end{align}\tag{159}\] The removal of the projection on the classical side uses only 157 , 159 , and the Cauchy argument of Proposition 14. These ingredients are independent of the pointwise exponential bounds. Thus, under the constraint 139 , \[\begin{align} \label{sub:dm:classical-projection-removal} \left\|\int_{P{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|{\mathord{{\rm d}}}\mu_P^{\varepsilon,f_\eta}(u) -\int_{{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|{\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u)\right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant C_{k,{\mathcal{K}}}\Lambda_e^{-\beta}. \end{align}\tag{160}\] Combining 157 , 159 , and 160 , we obtain \[\begin{align} \left\|\frac{k!}{\tau^k}(\Gamma^{f_\eta}_\tau)^{(k)}- \int_{{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|{\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u)\right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \leqslant C_{k,{\mathcal{K}}}\Lambda_e^{-\beta}. \end{align}\] Finally, under the hypotheses \(\eta\geqslant\tau^{-1/64}\) and \(\varepsilon\geqslant\tau^{-1/96}\), we may choose \(\Lambda_e\) so that 139 holds and \(\Lambda_e\to\infty\). This proves ?? and completes the proof of Proposition 15. ◻

Proof. First, for any \(\varepsilon,\eta>0\), we have \[\label{hartree:to:phi61:1} \begin{align} & \left| \int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u)} f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0 (u)- \int_{{\mathfrak{H}}} e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}}{\mathord{{\rm d}}}\mu_{0}(u) \right| \\ &\leqslant\int_{{\mathfrak{H}}} \left|e^{{\mathcal{W}}^\varepsilon(u)} -e^{{\mathcal{W}}(u)} \right| f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0 (u)+\int_{{\mathfrak{H}}} e^{{\mathcal{W}}(u)} \left| \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}} - f_\eta(\|u\|_{L^2}^2) \right| {\mathord{{\rm d}}}\mu_{0}(u) \\ &\leqslant\int_{{\mathfrak{H}}} \left|e^{{\mathcal{W}}^\varepsilon(u)} -e^{{\mathcal{W}}(u)} \right| \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}_c \}} {\mathord{{\rm d}}}\mu_0 (u)+\int_{{\mathfrak{H}}} e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ {\mathcal{K}}^2 -\eta\leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_{0}(u) . \end{align}\tag{161}\] We aim to show that both terms on the right-hand side of 161 vanish as \(\varepsilon, \eta \to 0\). Concerning the first term, we claim that \[\begin{align} \label{hartree:to:phi61:1:a46s46cv} \lim_{\varepsilon\to 0} e^{{\mathcal{W}}^\varepsilon(u)} = e^{{\mathcal{W}}(u)} , \qquad \mu_0\mathrm{-almost \,\, surely}. \end{align}\tag{162}\] By a fundamental factorization and Hölder’s inequality, we deduce that for any \(\varepsilon>0\) \[\begin{align} \label{dominated32control321} \left| {\mathcal{W}}^\varepsilon(u)-{\mathcal{W}}(u) \right| &=\frac{1}{6} \left|\int_{\mathbb{T}} \left(w^\varepsilon*|u|^2 -|u|^2 \right) \left(w^\varepsilon*|u|^2 +|u|^2 \right)|u|^2 {\mathord{{\rm d}}}x \right| \nonumber \\&\leqslant\left\| w^\varepsilon*|u|^2-|u|^2 \right\|_{L^3} \left(\left\| w^\varepsilon*|u|^2 \right\|_{L^3} + \|u\|_{L^6}^2\right) \|u\|_{L^6}^2 \nonumber \\ &\leqslant\left\| w^\varepsilon*|u|^2-|u|^2 \right\|_{L^3} \left(\left\| w^\varepsilon\right\|_{L^1} \|u\|_{L^6}^2+ \|u\|_{L^6}^2\right) \|u\|_{L^6}^2 \nonumber \\ &=2 \left\| w^\varepsilon*|u|^2-|u|^2 \right\|_{L^3} \|u\|_{L^6}^4, \end{align}\tag{163}\] where we used Young’s inequality in the third line and the fact \(\| w^\varepsilon\|_{L^1} =1\) in the last line. By the Sobolev embedding \(H^{\frac{1}{3}}\subset L^6\), we have that \(|u|^2 \in L^3\), \(\mu_0\)-almost surely. Then, it follows that \[\begin{align} \label{dominated32control322} \| w^\varepsilon*|u|^2-|u|^2 \|_{L^3} \to0 , \qquad \mu_0-\mathrm{almost \, \, surely}. \end{align}\tag{164}\] By combining 163 , 164 together with the fact that \(\|u \|_{L^6}< \infty\) \(\mu_0\)-almost surely, we obtain \[\begin{align} \label{a46s46cv} \lim_{\varepsilon\to 0} {\mathcal{W}}^\varepsilon(u) = {\mathcal{W}}(u) , \qquad \mu_0\mathrm{-almost \,\, surely}. \end{align}\tag{165}\] Then, the claim 162 follows from the continuity of the exponential function. Combining 144 with [1] yields that for any \(\varepsilon>0\) \[\begin{align} \label{uniform32integrability1} \left| e^{{\mathcal{W}}^\varepsilon(u)} - e^{{\mathcal{W}}(u)} \right| \mathbb{1}_{\{\|u\|_{L^2}\leqslant{\mathcal{K}}_c \}} \leqslant 2 e^{\frac{1}{6}\|u\|_{L^6}^6} \mathbb{1}_{\{\|u\|_{L^2}\leqslant{\mathcal{K}}_c \}} \in L^1({\mathord{{\rm d}}}\mu_0). \end{align}\tag{166}\] In view of 162 , 166 , and the dominated convergence theorem, it follows that \[\begin{align} \label{hartree:to:phi61:1:cv} \lim_{\varepsilon\to 0} \int_{{\mathfrak{H}}} \left|e^{{\mathcal{W}}^\varepsilon(u)} -e^{{\mathcal{W}}(u)} \right| \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}_c \}} {\mathord{{\rm d}}}\mu_0 (u)=0. \end{align}\tag{167}\]

Next, we turn to the second term on the right-hand side of 161 and show that \[\begin{align} \label{f:to:indicator} \lim_{\eta \to 0}\int_{{\mathfrak{H}}} e^{\frac{1}{6} \|u\|_{L^6}^6} \mathbb{1}_{\{ {\mathcal{K}}^2-\eta \leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2\}} {\mathord{{\rm d}}}\mu_{0} (u) =0, \end{align}\tag{168}\] for any \({\mathcal{K}}\in(0,{\mathcal{K}}_c]\). Consider the measure \({\mathord{{\rm d}}}\mu_{\Phi}(u):= \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}} {\mathord{{\rm d}}}\mu_{0} (u)\). It follows from [1] that \[\begin{align} e^{\frac{1}{6} \|u\|_{L^6}^6} \in L^1({\mathord{{\rm d}}}\mu_{\Phi}). \end{align}\] To derive 168 , it suffices to show that \[\begin{align} \lim_{\eta \to 0} \mu_{\Phi} \left( \left\{ {\mathcal{K}}^2-\eta \leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2\right\} \right)=0. \end{align}\] Since \(\mu_{\Phi} ( \{ {\mathcal{K}}^2-\eta \leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2 \} ) = \mu_0 ( \{ {\mathcal{K}}^2-\eta \leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2 \} \cap \{ \|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2 \} ) = \mu_0 ( \{ {\mathcal{K}}^2-\eta \leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2 \} )\), this reduces to proving that \[\begin{align} \lim_{\eta \to 0} \mu_0 \left( \left\{ {\mathcal{K}}^2-\eta \leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2\right\} \right)=0. \end{align}\] Under the Gaussian free field measure \(\mu_0\), we can write \(\|u\|_{L^2}^2=\sum_{j=1}^\infty |\alpha_j|^2\), where \(\{\alpha_j\}_{j\geqslant 1}\) denotes a sequence of independent complex-valued Gaussian random variables. Specifically, \(\mathrm{Re} \, \alpha_j\) and \(\mathrm{Im} \,\alpha_j\) are independent real-valued mean-zero Gaussian random variables with variance \((2\lambda_j)^{-1}\). Consequently, \(\{|\alpha_j|^2\}_{j\geqslant 1}\) are independent exponential random variables with density \(f_j(x)= \lambda_j e^{-\lambda_j x}\mathbb{1}_{\{x\geqslant 0\}}\). Define the random variable \(S_N=\sum_{j=1}^N |\alpha_j|^2\), it follows that \(S_N\xrightarrow{N \to \infty} S_\infty:=\|u\|_{L^2}^2\) \(\mu_0\)-almost surely. Our objective is to establish that \(S_\infty\) admits a continuous density with respect to the Lebesgue measure. To this end, we first observe that the characteristic function of the random variable \(|\alpha_j|^2\) is given by \[\begin{align} \phi_j(t)= \int e^{it |\alpha_j|^2} {\mathord{{\rm d}}}\mu_0 =\frac{\lambda_j}{\lambda_j-it}, \qquad t \in \mathbb{R}. \end{align}\] Since \(S_N\) converges to \(S_\infty\) \(\mu_0\)-almost surely, the dominated convergence theorem and the independence of \(|\alpha_j|^2\) imply that the characteristic function of \(S_\infty\) is given by \[\begin{align} \phi_{S_\infty}(t)=\lim_{N\to \infty} \phi_N(t)=\lim_{N\to \infty} \prod_{j=1}^N \int e^{it |\alpha_j|^2} {\mathord{{\rm d}}}\mu_0= \prod_{j=1}^\infty \frac{\lambda_j}{\lambda_j-it}, \qquad t \in \mathbb{R}. \end{align}\] We then verify \[\begin{align} \int_{\mathbb{R}} | \phi_{S_\infty}(t) |{\mathord{{\rm d}}}t &\leqslant\int_{\mathbb{R}} \prod_{j=1}^\infty \left| \frac{\lambda_j}{\lambda_j-it}\right| {\mathord{{\rm d}}}t \leqslant\int_{\mathbb{R}} \prod_{j=1}^\infty \frac{\lambda_j}{\sqrt{\lambda_j^2+t^2} }{\mathord{{\rm d}}}t \\ &\leqslant\int_{|t|\leqslant 1}1 {\mathord{{\rm d}}}t+ \int_{|t|> 1} \frac{\lambda_1\lambda_2}{\sqrt{(\lambda_1^2+t^2)(\lambda_2^2+t^2)}} {\mathord{{\rm d}}}t <\infty. \end{align}\] By the inversion formula of the characteristic function (see, e.g., [70]), we obtain that \(S_\infty\) has bounded continuous density \(f_{S_\infty}\). Then, we obtain \[\begin{align} \lim_{\eta \to 0}\mu_0 \left( \left\{ {\mathcal{K}}^2-\eta \leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2\right\} \right) =\lim_{\eta \to 0} \int_{{\mathcal{K}}^2 -\eta}^{{\mathcal{K}}^2} f_{S_\infty}(x) {\mathord{{\rm d}}}x=0. \end{align}\] Combining 161 , 167 and 168 , we complete the proof of Proposition 16. ◻

Proof of 30 and 32 . Let \({\mathcal{K}}\in (0,{\mathcal{K}}_c]\) be arbitrary. Throughout this proof, and in the subsequent arguments, let the cutoff \(f_\eta\) satisfy Assumption 2 with cutoff parameter \({\mathcal{K}}\). Combining Proposition 8 in the critical case, Proposition 15 in the subcritical case, and Proposition 16, and then letting \(\tau \to \infty\) and \(\varepsilon,\eta \to 0\), we obtain \[\begin{align} \label{proof32of32main1} \left| \frac{\mathcal{Z}^{f_\eta}_{\tau}}{\mathcal{Z}^{f_\eta}_{\tau,0}} - \frac{\int_{{\mathfrak{H}}} e^{\frac{1}{6} \|u\|_{L^6}^6} \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}} {\mathord{{\rm d}}}\mu_{0}(u) }{\int_{{\mathfrak{H}}} \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}} {\mathord{{\rm d}}}\mu_{0}(u) }\right| \to 0. \end{align}\tag{170}\] The only point requiring comment is the admissible choice of the interaction scale. If \({\mathcal{K}}<{\mathcal K}_c\), then we may take \(\varepsilon\geqslant\tau^{-\frac{1}{96}} \to 0\) whereas at the threshold \({\mathcal{K}}={\mathcal{K}}_c\), the argument requires \(\varepsilon\gtrsim ( \log \tau)^{-\frac{1}{2}} \to 0\). It remains to identify the denominator with the corresponding quantum free partition function. Combining 169 with ?? , we have, as \(\tau \to \infty\) and \(\eta\to 0\), \[\begin{align} &\left| \frac{\mathcal{Z}^{f_\eta}_{\tau,0}}{{\mathcal{Z}}_{\tau,0}} - \int_{\mathfrak{H}} \mathbb{1}_{\{\|u\|_{L^2}\leqslant{\mathcal{K}}\}} \, {\mathord{{\rm d}}}\mu_0 (u) \right| \\ &\leqslant\left| \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \Gamma_{\tau,0} f_\eta\left( {\mathcal{N}}/\tau\right) \right)-\int_{{\mathfrak{H}}} f_\eta(\|u\|^2_{L^2}) {\mathord{{\rm d}}}\mu_0(u) \right| +\left| \int_{{\mathfrak{H}}} \mathbb{1}_{\{ \|u\|_{L^2} \leqslant{\mathcal{K}}\}} -f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_{0} (u) \right| \to 0. \end{align}\] Together with 170 , this proves 30 and 32 . ◻

Proof of 31 and 33 . Let \({\mathcal{K}}\in(0,{\mathcal{K}}_c]\) be arbitrary. By the triangle inequality, \[\begin{align} \label{triangle32decomposition32final32trace32norm} & \left\| \frac{k!}{\tau^k}(\Gamma^{f_\eta}_{\tau})^{(k)} -\int_{{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|\,{\mathord{{\rm d}}}\mu^{{\mathcal{K}}}(u) \right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \nonumber \\ &\qquad \leqslant\left\| \frac{k!}{\tau^k}(\Gamma^{f_\eta}_{\tau})^{(k)} - \int_{{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|\,{\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u) \right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \nonumber \\ &\qquad \qquad +\left\| \int_{{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|\,{\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u) - \int_{{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|\,{\mathord{{\rm d}}}\mu^{{\mathcal{K}}}(u) \right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} . \end{align}\tag{171}\] The first term on the right-hand side of 171 tends to \(0\) by ?? in the critical regime and by ?? in the subcritical regime. We now estimate the second term. By the same calculations as in 138 , we deduce for all \(k\geqslant 1\), \[\begin{align} \label{density32matrix32convergence4:new} & \left\| \int_{{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|\,{\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u) - \int_{{\mathfrak{H}}}|u^{\otimes k}\rangle\langle u^{\otimes k}|\,{\mathord{{\rm d}}}\mu^{{\mathcal{K}}}(u) \right\|_{{\mathfrak{S}}^1({\mathfrak{H}}^{(k)})} \\&\quad \leqslant\frac{1}{ \int_{{\mathfrak{H}}} e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}} {\mathord{{\rm d}}}\mu_{0}(u)} \int_{{\mathfrak{H}}} \|u\|_{L^2}^{2k} \left| e^{{\mathcal{W}}^\varepsilon(u)} f_\eta(\|u\|_{L^2}^2) - e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}} \right| {\mathord{{\rm d}}}\mu_{0}(u)\nonumber \\ &\quad + \frac{\int_{{\mathfrak{H}}} \|u\|_{L^2}^{2k} {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u)}{ \int_{{\mathfrak{H}}} e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}} {\mathord{{\rm d}}}\mu_{0}(u)} \int_{{\mathfrak{H}}} \left| e^{{\mathcal{W}}^\varepsilon(u)} f_\eta(\|u\|_{L^2}^2) - e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}} \right| {\mathord{{\rm d}}}\mu_{0}(u). \nonumber \end{align}\tag{172}\] By \(\int_{{\mathfrak{H}}} \|u\|_{L^2}^{2k} {\mathord{{\rm d}}}\mu^{\varepsilon,f_\eta}(u)\leqslant{\mathcal{K}}^{2k}\) and Proposition 16, it only remains to show that as \(\varepsilon,\eta \to 0\), \[\label{density32matrix32convergence41} \begin{align} & \int_{{\mathfrak{H}}} \|u\|_{L^2}^{2k} \left| e^{{\mathcal{W}}^\varepsilon(u)} f_\eta(\|u\|_{L^2}^2) - e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}\}} \right| {\mathord{{\rm d}}}\mu_{0}(u) \\ & \leqslant\int_{{\mathfrak{H}}} \|u\|_{L^2}^{2k} \left|e^{{\mathcal{W}}^\varepsilon(u)} -e^{{\mathcal{W}}(u)} \right| \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}_c \}} {\mathord{{\rm d}}}\mu_0 (u) +\int_{{\mathfrak{H}}} \|u\|_{L^2}^{2k} e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ {\mathcal{K}}^2 -\eta\leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_{0}(u) \to0. \end{align}\tag{173}\] Repeating the arguments as in the proof of Proposition 16, one can check that it holds \(\mu_0\)-almost surely, \[\begin{align} \lim_{ \varepsilon\to 0} e^{{\mathcal{W}}^\varepsilon(u)} \|u\|_{L^2}^{2k} = e^{{\mathcal{W}}(u)} \|u\|_{L^2}^{2k} , \end{align}\] for all \(k\geqslant 1\). Also, it follows from 166 that \[\begin{align} \|u\|_{L^2}^{2k} \left| e^{{\mathcal{W}}^\varepsilon(u)} - e^{{\mathcal{W}}(u)} \right| \mathbb{1}_{\{\|u\|_{L^2}\leqslant{\mathcal{K}}_c \}} \leqslant C_{k,{\mathcal{K}}_c} e^{\frac{1}{6}\|u\|_{L^6}^6} \mathbb{1}_{\{\|u\|_{L^2}\leqslant{\mathcal{K}}_c \}} \in L^1({\mathord{{\rm d}}}\mu_0). \end{align}\] Hence, by using the dominated convergence theorem, we obtain that as \(\varepsilon\to 0\) \[\begin{align} \label{hartree:to:phi61:1:cv:new} \int_{{\mathfrak{H}}} \|u\|_{L^2}^{2k} \left|e^{{\mathcal{W}}^\varepsilon(u)} -e^{{\mathcal{W}}(u)} \right| \mathbb{1}_{\{ \|u\|_{L^2}\leqslant{\mathcal{K}}_c \}} {\mathord{{\rm d}}}\mu_0 (u)\to 0. \end{align}\tag{174}\] Invoking again 168 implies that as \(\eta\to 0\) \[\begin{align} \int_{{\mathfrak{H}}} \|u\|_{L^2}^{2k} e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ {\mathcal{K}}^2 -\eta\leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_{0}(u)\lesssim_{{\mathcal{K}},k} \int_{{\mathfrak{H}}} e^{{\mathcal{W}}(u)} \mathbb{1}_{\{ {\mathcal{K}}^2 -\eta\leqslant\|u\|_{L^2}^2 \leqslant{\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_{0}(u) \to0. \end{align}\] This, together with 174 , implies 173 . Hence, we complete the proof of 31 and 33 . ◻

Proof of Theorem 6. To apply Proposition 8, we first choose the cutoff \(f_\eta\) as in Assumption 2, with cutoff parameter \({\mathcal{K}}> {\mathcal{K}}_c\). For \(\eta>0\) sufficiently small, we have \({\mathcal{K}}^2- \eta > \frac{1}{2}({\mathcal{K}}^2_c+{\mathcal{K}}^2)\), and hence \(\mathbb{1}_{[0,\frac{1}{2}({\mathcal{K}}^2_c+{\mathcal{K}}^2) ] } (x)\leqslant f_\eta (x)\leqslant g(x)\). It follows from Proposition 8 and ?? that, for \(1>\varepsilon\geqslant M \left(\log \tau\right)^{-\frac{1}{2}}\) and \(\eta \in [\tau^{-\frac{1}{64}}, \frac{1}{2} {\mathcal{K}}^2)\), as \(\tau \to \infty\), \[\begin{align} \label{blow:up11} \frac{\mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})}\left(e^{-\mathbb{H}_\tau} g({\mathcal{N}}/\tau) \right)}{\mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left(e^{-\mathbb{H}_{\tau,0}} \right)}& \geqslant\frac{\mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})}\left(e^{-\mathbb{H}_\tau} f_\eta({\mathcal{N}}/\tau) \right)}{\mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left(e^{-\mathbb{H}_{\tau,0}} \right)} \to \int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u)} f_\eta(\|u\|_{L^2}^2) {\mathord{{\rm d}}}\mu_0(u) \\ & \qquad \qquad \geqslant\int_{{\mathfrak{H}}} e^{{\mathcal{W}}^\varepsilon(u)} \mathbb{1}_{\{\|u\|^2_{L^2}\leqslant\frac{1}{2}{\mathcal{K}}^2_c+\frac{1}{2} {\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_0(u) =: {\mathcal{Z}}^\varepsilon. \nonumber \end{align}\tag{175}\] To conclude the proof, it remains to verify that \(\lim_{\varepsilon\to 0} {\mathcal{Z}}^\varepsilon=\infty\). Given \(\varepsilon\geqslant 0\) and \(R>0\), we define an auxiliary function \[\begin{align} F_{\varepsilon,R}(u):= \begin{cases} e^{{\mathord{{\rm min}}}\left\{ {\mathcal{W}}^\varepsilon(u), R \right\}}, &\varepsilon>0,\\ e^{{\mathord{{\rm min}}}\left\{ {\mathcal{W}}(u), R \right\}}, & \varepsilon=0. \end{cases} \end{align}\] Then, for any \(\varepsilon>0\) and \(R>0\), it holds that \[\begin{align} \label{Zeps:below} {\mathcal{Z}}^\varepsilon\geqslant\int_{ {\mathfrak{H}}} F_{\varepsilon,R}(u) \mathbb{1}_{\{\|u\|^2_{L^2}\leqslant\frac{1}{2}{\mathcal{K}}^2_c+\frac{1}{2} {\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_0(u). \end{align}\tag{176}\] Observe that \(F_{0,R}\) increases to \(e^{{\mathcal{W}}(u)}\) almost surely. By the monotone convergence theorem, we obtain \[\begin{align} \int_{ {\mathfrak{H}}} F_{0,R}(u) \mathbb{1}_{\{\|u\|^2_{L^2}\leqslant\frac{1}{2}{\mathcal{K}}^2_c+\frac{1}{2} {\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_0(u) \xrightarrow{R\to \infty} \int_{ {\mathfrak{H}}} e^{{\mathcal{W}}(u)} \mathbb{1}_{\{\|u\|^2_{L^2}\leqslant\frac{1}{2}{\mathcal{K}}^2_c+\frac{1}{2} {\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_0(u) =\infty , \end{align}\] where the last equality follows from the non-normalizability of the focusing \(\Phi^6_1\) measure above the optimal threshold, see [37]. Hence, for any \(A>0\), there exists \(R(A)\) such that \[\begin{align} \label{blow:up2} \int_{ {\mathfrak{H}}} F_{0,R(A)}(u) \mathbb{1}_{\{\|u\|^2_{L^2}\leqslant\frac{1}{2}{\mathcal{K}}^2_c+\frac{1}{2} {\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_0(u) \geqslant 2A. \end{align}\tag{177}\] From 165 , we deduce that \(F_{\varepsilon,R(A)}(u)\xrightarrow{\varepsilon\to 0} e^{{\mathord{{\rm min}}}\left\{ {\mathcal{W}}(u), R(A) \right\}}\), \(\mu_0\)-almost surely. Since \(F_{\varepsilon,R(A)}\leqslant e^{R(A)}\), the dominated convergence theorem yields \[\begin{align} \int_{ {\mathfrak{H}}} F_{\varepsilon,R(A)}(u) \mathbb{1}_{\{\|u\|^2_{L^2}\leqslant\frac{1}{2}{\mathcal{K}}^2_c+\frac{1}{2} {\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_0(u) \xrightarrow{\varepsilon\to 0} \int_{ {\mathfrak{H}}} F_{0,R(A)}(u) \mathbb{1}_{\{\|u\|^2_{L^2}\leqslant\frac{1}{2}{\mathcal{K}}^2_c+\frac{1}{2} {\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_0(u). \end{align}\] Together with 176 and 177 , this implies that for any \(A>0\), there exists \(\varepsilon(A)>0\) such that for any \(\varepsilon\in(0,\varepsilon(A))\) \[\begin{align} {\mathcal{Z}}^\varepsilon\geqslant\int_{ {\mathfrak{H}}} F_{\varepsilon,R(A)}(u) \mathbb{1}_{\{\|u\|^2_{L^2}\leqslant\frac{1}{2}{\mathcal{K}}^2_c+\frac{1}{2} {\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_0(u) \geqslant\frac{1}{2} \int_{ {\mathfrak{H}}} F_{0,R(A)}(u) \mathbb{1}_{\{\|u\|^2_{L^2}\leqslant\frac{1}{2}{\mathcal{K}}^2_c+\frac{1}{2} {\mathcal{K}}^2 \}} {\mathord{{\rm d}}}\mu_0(u) \geqslant A. \end{align}\] Hence, we obtain \(\lim_{\varepsilon\to 0} {\mathcal{Z}}^\varepsilon=\infty\). This completes the proof of Theorem 6. ◻

Proof. Since \(P{\mathfrak{H}}=\mathrm{span}\{u_1,\ldots,u_J\}\), we identify \({\mathfrak{F}}(P{\mathfrak{H}})\simeq\bigotimes_{j=1}^J {\mathfrak{F}}(\mathbb{C} u_j)\). For the corresponding occupation-number basis, we write \[|n_1,\ldots,n_J \rangle =|n_1\rangle_1 \otimes \cdots \otimes |n_J\rangle_J ,\qquad |n_j\rangle_j= \frac{1}{\sqrt{n_j!}}(a^\dagger_j)^{n_j} |0\rangle .\] Then, \(a_j^\dagger a_j |n_j\rangle_j =n_j |n_j\rangle_j\), and it follows that \[\label{number:hit:on:basis} \begin{align} f_\eta\left(\frac{{\mathcal{N}}_P}{\tau}\right) \big|n_1,\ldots,n_J \big\rangle &= f_\eta \left( \frac{\sum_{j=1}^J n_j}{\tau} \right) \big|n_1,\ldots,n_J \big\rangle , \\ e^{-\sum_{j=1}^J\lambda_ja^\dagger_ja_j/\tau}\big|n_1,\ldots,n_J \big\rangle &= e^{-\sum_{j=1}^J \lambda_jn_j/\tau} \big|n_1,\ldots,n_J \big\rangle. \end{align}\tag{179}\] For \(u=\sum_{j=1}^J \alpha_j u_j\in P{\mathfrak{H}}\), the coherent state factorizes into one-mode states: \[\begin{align} \label{eq:extension} \xi(u)=\bigotimes_{j=1}^J \xi(\alpha_j u_j)=e^{-\sum_{j=1}^J |\alpha_j|^2/2} \sum_{n_1,\ldots,n_J\geqslant 0} \prod_{j=1}^J \frac{\alpha_j^{n_j}}{\sqrt{n_j!}} |n_1,\ldots,n_J \rangle . \end{align}\tag{180}\] Using the resolution of the identity in coherent states 43 , together with 179 and 180 , we obtain \[\begin{align} &\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( \Gamma_{\tau,0,P} f_\eta ({\mathcal{N}}_P/\tau)\right) =\prod_{j=1}^J \frac{\tau}{\pi} (1-e^{-\lambda_j/\tau}) \int_{P{\mathfrak{H}}} \left\langle \xi \big( \sqrt{\tau}u \big), e^{-\mathbb{H}_{\tau,0,P}}f_\eta ({\mathcal{N}}_P/\tau) \xi \big( \sqrt{\tau}u \big) \right\rangle \,{\mathord{{\rm d}}}u \\ &= \int_{P{\mathfrak{H}}} \prod_{j=1}^J \frac{\tau}{\pi} (1-e^{-\lambda_j/\tau}) e^{-\tau |\alpha_j|^2} \sum_{n_1,\ldots,n_J\geqslant 0} f_\eta \left( \frac{\sum_{j=1}^J n_j}{\tau} \right) \prod_{j=1}^J \frac{(\tau|\alpha_j|^2e^{-\lambda_j/\tau})^{n_j}}{n_j!} \,{\mathord{{\rm d}}}u \\ &= \int_{P{\mathfrak{H}}} \prod_{j=1}^J \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} \sum_{n_1,\ldots,n_J\geqslant 0} f_\eta \left( \frac{\sum_{j=1}^J n_j}{\tau} \right) \prod_{j=1}^J e^{-\tau|\alpha_j|^2e^{-\lambda_j/\tau}} \frac{(\tau|\alpha_j|^2e^{-\lambda_j/\tau})^{n_j}}{n_j!} \,{\mathord{{\rm d}}}u, \end{align}\] where we defined \(q_j:= \tau(1-e^{-\lambda_j/\tau})\). Let \(\{\mathbf{X}_j\}_{j=1}^J\) be independent Poisson random variables with means \(\tau |\alpha_j|^2e^{-\lambda_j/\tau}\), respectively. Then the preceding identity becomes \[\begin{align} \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( \Gamma_{\tau,0,P} f_\eta ({\mathcal{N}}_P/\tau)\right) = \int_{P{\mathfrak{H}}} \mathbf{E} \left[ f_\eta \left(\frac{\sum_{j=1}^J \mathbf{X}_j}{\tau} \right)\right] \prod_{j=1}^J \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} {\mathord{{\rm d}}}u. \end{align}\] We compare this expression with the classical Gaussian integral in three elementary steps: the Poisson variables are first replaced by their means; the means are then replaced by \(\sum_j |\alpha_j|^2\); finally, the Gaussian density with variances \(q_j^{-1}\) is compared with the density with variances \(\lambda_j^{-1}\). This gives \[\begin{align} \Big| &\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( \Gamma_{\tau,0,P} f_\eta ({\mathcal{N}}_P/\tau)\right) - \int_{{\mathfrak{H}}} f_{\eta} (\|Pu\|_{L^2}^2){\mathord{{\rm d}}}\mu_{0}(u) \Big| \\ &\leqslant\int_{P{\mathfrak{H}}} \mathbf{E} \Big| f_\eta \Big( \frac{1}{\tau} \sum_{j=1}^J \mathbf{X}_j \Big) - f_\eta \Big(\sum_{j=1}^J |\alpha_j|^2 e^{-\lambda_j/\tau} \Big) \Big| \prod_{j=1}^J \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} {\mathord{{\rm d}}}u \\ &\quad + \int_{P{\mathfrak{H}}} \Big| f_\eta \Big(\sum_{j=1}^J |\alpha_j|^2 e^{-\lambda_j/\tau} \Big) - f_\eta \Big(\sum_{j=1}^J |\alpha_j|^2 \Big) \Big| \prod_{j=1}^J \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} {\mathord{{\rm d}}}u \\ &\quad + \int_{P{\mathfrak{H}}} f_\eta \Big(\sum_{j=1}^J |\alpha_j|^2 \Big) \Big| \prod_{j=1}^J \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} - \prod_{j=1}^J \frac{\lambda_j}{\pi} e^{-\lambda_j |\alpha_j|^2} \Big| {\mathord{{\rm d}}}u =: {\mathcal{E}}_{\mathrm{Pois}}+{\mathcal{E}}_{\mathrm{cut}}+ {\mathcal{E}}_{\mathrm{dens}}. \end{align}\] We now estimate these three errors separately. Since \(\sum_{j=1}^J \mathbf{X}_j\) is Poisson with mean \(\tau \sum_{j=1}^J |\alpha_j|^2e^{-\lambda_j/\tau}\), Assumption 2 and Hölder’s inequality imply \[\begin{align} {\mathcal{E}}_{\mathrm{Pois}} & \leqslant\frac{\|f_\eta'\|_{L^\infty} }{\tau} \int_{P{\mathfrak{H}}} \mathbf{E} \Big| \sum_{j=1}^J \mathbf{X}_j - \tau \sum_{j=1}^J |\alpha_j|^2 e^{-\lambda_j/\tau} \Big| \prod_{j=1}^J \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} {\mathord{{\rm d}}}u \\ &\leqslant\frac{\eta^{-1} }{\tau} \int_{P{\mathfrak{H}}} \left( \mathbf{E} \Big| \sum_{j=1}^J \mathbf{X}_j - \tau \sum_{j=1}^J |\alpha_j|^2 e^{-\lambda_j/\tau} \Big|^2 \right)^{1/2} \prod_{j=1}^J \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} {\mathord{{\rm d}}}u \\ &\leqslant\frac{\eta^{-1} }{\sqrt{\tau}} \int_{P{\mathfrak{H}}} \Big( \sum_{j=1}^J |\alpha_j|^2 e^{-\lambda_j/\tau} \Big)^{1/2} \prod_{j=1}^J \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} {\mathord{{\rm d}}}u \leqslant\frac{\eta^{-1} }{\sqrt{\tau}} \Big( \sum_{j=1}^J e^{-\lambda_j/\tau} \frac{q_j}{\pi} \int_{\mathbb{C}} |\alpha_j|^2 e^{-q_j |\alpha_j|^2} {\mathord{{\rm d}}}\alpha_j \Big)^{1/2} \\ & = \frac{\eta^{-1} }{\sqrt{\tau}} \Big( \sum_{j=1}^J q_j^{-1}e^{-\lambda_j/\tau} \Big)^{1/2} = \frac{\eta^{-1} }{\sqrt{\tau}} \Big(\sum_{j=1}^J \frac{1}{\tau(e^{\lambda_j/\tau}-1)} \Big)^{1/2} \leqslant\frac{\eta^{-1} }{\sqrt{\tau}} \mathrm {Tr}(Ph^{-1}) ^{1/2} \leqslant C \tau^{-\frac{1}{4}}. \end{align}\] Here the last inequality follows from \(\eta^{-1}\leqslant\Lambda_e^{1/8} \leqslant\tau^{1/32}\) and \(\mathrm {Tr}(Ph^{-1}) \leqslant\mathrm {Tr}(h^{-1})\lesssim 1\). The second error \({\mathcal{E}}_{\mathrm{cut}}\) measures the difference between \(e^{-\lambda_j/\tau}\) and \(1\) inside the cutoff. Using the Lipschitz bound on \(f_\eta\), we obtain \[\begin{align} {\mathcal{E}}_{\mathrm{cut}} & \leqslant\|f_\eta' \|_{L^\infty} \sum_{j=1}^J (1-e^{-\lambda_j/\tau} ) \frac{q_j}{\pi} \int_{\mathbb{C}} |\alpha_j|^2 e^{-q_j |\alpha_j|^2} {\mathord{{\rm d}}}\alpha_j \\ & \leqslant\eta^{-1} \sum_{j=1}^J (1-e^{-\lambda_j/\tau} ) q_j^{-1} \leqslant\eta^{-1} \sqrt{\Lambda_e} \tau^{-1} \leqslant C \tau^{-\frac{1}{4}}. \end{align}\] For the last error \({\mathcal{E}}_{\mathrm{dens}}\), we first reduce the \(J\)-dimensional total variation to one-dimensional comparisons by telescoping the product densities: \[\begin{align} \label{J-dimensional:variation} \int_{P{\mathfrak{H}}}& \Big| \prod_{j=1}^J \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} - \prod_{j=1}^J \frac{\lambda_j}{\pi} e^{-\lambda_j |\alpha_j|^2} \Big| {\mathord{{\rm d}}}u \nonumber \\ &\leqslant\sum_{k=1}^J \int_{\mathbb{C}^J} \Big( \prod_{j<k} \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} \Big) \Big| \frac{q_k}{\pi} e^{-q_k |\alpha_k|^2} - \frac{\lambda_k}{\pi} e^{-\lambda_k |\alpha_k|^2} \Big| \Big( \prod_{j>k} \frac{\lambda_j}{\pi} e^{-\lambda_j |\alpha_j|^2} \Big) {\mathord{{\rm d}}}\alpha_1 \cdots {\mathord{{\rm d}}}\alpha_J \\ &= \sum_{j=1}^J \int_{\mathbb{C}} \Big| \frac{q_j}{\pi} e^{-q_j |\alpha_j|^2} - \frac{\lambda_j}{\pi} e^{-\lambda_j |\alpha_j|^2} \Big| {\mathord{{\rm d}}}\alpha_j = \sum_{j=1}^J \int_{0}^\infty \Big| \frac{q_j}{\lambda_j} e^{-q_j r /\lambda_j} -e^{-r} \Big| {\mathord{{\rm d}}}r. \nonumber \end{align}\tag{181}\] In the last step we used polar coordinates and the change of variables \(r=\lambda_j |\alpha_j|^2\). By the elementary inequalities \(\frac{x}{1+x}\leqslant 1- e^{-x}\leqslant x\), \(x \geqslant 0\), and by \(\lambda_j \leqslant\Lambda_e \leqslant\tau\), we have \[\begin{align} \frac{1}{2} \leqslant\frac{1}{1+\lambda_j/\tau} \leqslant\frac{q_j}{\lambda_j} =\frac{1- e^{-\lambda_j/\tau}}{\lambda_j/\tau} \leqslant 1. \end{align}\] Combining this bound with the mean value theorem gives \[\begin{align} & \int_{0}^\infty \Big| \frac{q_j}{\lambda_j} e^{-q_j r /\lambda_j} -e^{-r} \Big| {\mathord{{\rm d}}}r \leqslant\Big| 1-\frac{q_j}{\lambda_j} \Big| \int_{0}^\infty \sup_{s\in[\frac{1}{2},1]} |1-sr| e^{-sr } {\mathord{{\rm d}}}r \\ &\quad \leqslant\Big| 1-\frac{q_j}{\lambda_j} \Big| \int_{0}^\infty (1+r) e^{-r/2 } {\mathord{{\rm d}}}r \leqslant C \Big| 1-\frac{q_j}{\lambda_j} \Big| =C\Big| \frac{\lambda_j/\tau-1+e^{-\lambda_j/\tau}}{\lambda_j/\tau}\Big| \leqslant\frac{C\lambda_j}{2\tau}, \end{align}\] where we used \(0\leqslant x-1+e^{-x}\leqslant x^2/2\), for \(x\geqslant 0\) in the last inequality. Inserting this estimate into 181 , we obtain \[\begin{align} {\mathcal{E}}_{\mathrm{dens}} \leqslant C\tau^{-1} \sum_{j=1}^J \lambda_j \leqslant C\Lambda_e^{\frac{3}{2}} \tau^{-1} \leqslant C\tau^{-\frac{1}{4}}. \end{align}\] Combining the estimates on \({\mathcal{E}}_{\mathrm{Pois}}\), \({\mathcal{E}}_{\mathrm{cut}}\), \({\mathcal{E}}_{\mathrm{dens}}\) proves 178 . ◻

Proof of Proposition 17. For \(\tau\) sufficiently large, since \(\eta \geqslant\tau^{-\frac{1}{64}}\), we may choose \(\Lambda_e \to \infty\) such that \(\eta^{-1} \leqslant\Lambda_e^{\frac{1}{8}} \leqslant\tau^{\frac{1}{32}}\). Combining 178 , 182 and 183 , we obtain \[\begin{align} &\Big|\mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( \Gamma_{\tau,0} f_\eta ({\mathcal{N}}/\tau)\right) - \int_{\mathfrak{H}} f_\eta(\|u\|_{L^2}^2) \,{\mathord{{\rm d}}}\mu_0(u) \Big| \\ &\leqslant \Big| \mathrm {Tr}_{{\mathfrak{F}}({\mathfrak{H}})} \left( f_\eta \left({\mathcal{N}}/\tau\right) \Gamma_{\tau,0} \right) -\mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \Big( f_\eta({\mathcal{N}}_P/\tau) \Gamma_{\tau,0,P} \Big) \Big| \\ &\quad + \Big| \mathrm {Tr}_{{\mathfrak{F}}(P{\mathfrak{H}})} \left( \Gamma_{\tau,0,P} f_\eta ({\mathcal{N}}_P/\tau)\right) - \int_{\mathfrak{H}} f_\eta (\|Pu\|_{L^2}^2) \, {\mathord{{\rm d}}}\mu_0 (u) \Big| \\ &\quad +\Big| \int_{{\mathfrak{H}}} f_\eta\left( \|Pu\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0}(u)- \int_{{\mathfrak{H}}} f_\eta\left( \|u\|_{L^2}^2 \right) {\mathord{{\rm d}}}\mu_{0}(u) \Big| \leqslant C\Lambda_e^{-\frac{1}{8}} + C\tau^{-\frac{1}{4}}. \end{align}\] Letting \(\tau \to \infty\) and \(\Lambda_e \to \infty\) proves ?? . The convergence in ?? follows by the same argument with the fixed test function \(g\in C^\infty_c([0,\infty);\mathbb{R}_+)\). Since \(g\) does not depend on \(\eta\), the preceding estimates are simpler and no \(\eta\)-dependent loss has to be tracked. This completes the proof of the proposition. ◻

Proof of 142 . We first split the integral into the regions \(\{\|Pu\|_{L^6}\leqslant 1\}\) and \(\{\|Pu\|_{L^6}> 1\}\): \[\begin{align} \int_{{\mathfrak{H}}} e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s \}} {\mathord{{\rm d}}}\mu_{0}(u) &= \int_{{\mathfrak{H}}} e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s , \, \|Pu\|_{L^6}\leqslant 1 \}} {\mathord{{\rm d}}}\mu_{0}(u) \\ &\quad +\int_{{\mathfrak{H}}} e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s , \, \|Pu\|_{L^6}> 1 \}} {\mathord{{\rm d}}}\mu_{0}(u) . \end{align}\] The first contribution is trivially bounded by \(e^{\frac{1+\varsigma}{6}}\), so it remains to control the latter region. For \(k\geqslant 1\), define \(E_k\) by \[\label{def:Ek} \begin{align} E_k&:=\{\|(Pu)_{\geqslant 0}\|_{L^6 } >1 , \ldots,\|(Pu)_{\geqslant k-1}\|_{L^6 } >1 , \|(Pu)_{\geqslant k}\|_{L^6 } \leqslant 1 \} \\ &\quad \subset \{ \|(Pu)_{\geqslant k-1}\|_{L^6 } >1 , \|(Pu)_{\geqslant k}\|_{L^6 } \leqslant 1 \}. \end{align}\tag{185}\] The sets \(E_k\) are disjoint. Since \((Pu)_{\geqslant k}=0\) for \(k> J:=\mathrm {Tr}[P]\), then \(E_k=\emptyset\) for any \(k> J\). Thus, the \(E_k\) form a disjoint decomposition of \(\{ \|(Pu)\|_{L^6 } >1 \}\), and it is harmless to sum over all \(k\geqslant 1\), since only finitely many terms are non-zero: \[\label{series32uniform32bound} \int_{{\mathfrak{H}}} e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s , \, \|Pu\|_{L^6}> 1 \}} {\mathord{{\rm d}}}\mu_{0}(u)= \sum_{k\geqslant 1} \int_{{\mathfrak{H}}} e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s , \, E_k \}} {\mathord{{\rm d}}}\mu_{0}(u) .\tag{186}\] As in [1], Hölder’s inequality and Young’s inequality imply that, for \(u\) satisfying \(\|(Pu)_{\geqslant k}\|_{L^6}\leqslant 1\) and some small \(\gamma>0\) (to be chosen later): \[\begin{align} \int_{\mathbb{T}} |Pu|^6 {\mathord{{\rm d}}}x&=\int_{\mathbb{T}} |(Pu)_{\leqslant k-1}+(Pu)_{\geqslant k}|^6 {\mathord{{\rm d}}}x \leqslant\sum_{\ell=0}^6 {6 \choose \ell} \int_{\mathbb{T}}|(Pu)_{\leqslant k-1}|^{6-\ell}|(Pu)_{\geqslant k}|^{\ell} {\mathord{{\rm d}}}x \nonumber \\ & \leqslant\int_{\mathbb{T}} |(Pu)_{\leqslant k-1}|^6 {\mathord{{\rm d}}}x+ \sum_{\ell=1}^6 {6 \choose \ell} \left( \frac{6-\ell}{6} \gamma \|(Pu)_{\leqslant k-1}\|_{L^6}^{6} + \frac{\ell}{6}\gamma^{-\frac{6-\ell}{\ell}} \right) \\ &\leqslant\left( 1 +(2^6-1)\gamma \right) \int_{\mathbb{T}} |(Pu)_{\leqslant k-1}|^6 {\mathord{{\rm d}}}x +C_6(\gamma). \nonumber \end{align}\] Combining this bound with 185 yields \[\begin{align} \label{bound:Pu:L6} \int_{{\mathfrak{H}}} & e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s , \, E_k \}} {\mathord{{\rm d}}}\mu_{0}(u) \\ & \leqslant\int_{{\mathfrak{H}}} e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s , \, \|(Pu)_{\geqslant k-1}\|_{L^6 } >1 , \|(Pu)_{\geqslant k}\|_{L^6 } \leqslant 1 \}} {\mathord{{\rm d}}}\mu_{0}(u) \nonumber \\ & \leqslant e^{C_6(\gamma)} \int_{{\mathfrak{H}}} \mathrm{exp}\Big(\frac{(1+\varsigma)(1+63\gamma)}{6}\int_{\mathbb{T}} |(Pu)_{\leqslant k-1}|^6 {\mathord{{\rm d}}}x\Big) \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s , \, \|(Pu)_{\geqslant k-1}\|_{L^6 } >1 \}} {\mathord{{\rm d}}}\mu_{0}(u) .\nonumber \end{align}\tag{187}\] Applying 184 to \((Pu)_{\leqslant k-1}\) with some small \(m>0\) (to be chosen later), and using Hölder’s inequality, we deduce from 187 that \[\begin{align} \label{bound:Pu:L6a} & \int_{{\mathfrak{H}}} e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s , \, E_k \}} {\mathord{{\rm d}}}\mu_{0}(u) \\ & \leqslant e^{C_6(\gamma)+C(m){\mathcal{K}}_s^6} \int_{\{ \|(Pu)_{\geqslant k-1}\|_{L^6 } >1 \}} \mathrm{exp} \Big(\frac{(C_{\mathrm{GNS}}(1,6)+m){\mathcal{K}}_s^4(1+\varsigma)(1+63\gamma)}{6}\int_{\mathbb{T}} |\partial_x(Pu)_{\leqslant k-1}|^2 {\mathord{{\rm d}}}x\Big) {\mathord{{\rm d}}}\mu_{0}(u) \nonumber \\ &\leqslant e^{C_6(\gamma)+C(m){\mathcal{K}}_s^6} \left( \int_{{\mathfrak{H}}} \mathrm{exp} \Big( \big(3\|Q\|^{-4}_{L^2(\mathbb{R})}+m\big)\frac{{\mathcal{K}}_s^4(1+\varsigma)(1+63\gamma)(1+\iota)}{6}\int_{\mathbb{T}} |\partial_x(Pu)_{\leqslant k-1}|^2 {\mathord{{\rm d}}}x\Big) {\mathord{{\rm d}}}\mu_{0}(u) \right)^{\frac{1}{1+\iota}} \nonumber \\ &\qquad \times \left[ \mu_0\left( \|(Pu)_{\geqslant k-1}\|_{L^6 } >1 \right) \right]^{\frac{\iota}{1+\iota}}. \nonumber \end{align}\tag{188}\] This is the only point where the extra factor \(1+\varsigma\) enters. Since \({\mathcal{K}}_s <\|Q\|_{L^2(\mathbb{R})}\), we can choose \(m,\varsigma,\gamma,\iota>0\) sufficiently small such that \[\begin{align} \frac{{\mathcal{K}}_s^4}{\|Q\|^4_{L^2(\mathbb{R})}}\left( 1+\frac{m}{3}\|Q\|^4_{L^2(\mathbb{R})} \right)(1+\varsigma)(1+63\gamma)(1+\iota) =:2c<1. \end{align}\] Writing \(u=\sum_{j\in\mathbb{Z}}\alpha_j e^{2\pi i j x}\) under \(\mu_0\), the first integral on the right-hand side of 188 is bounded by \[\begin{align} \label{uniform:projection1} &\int_{{\mathfrak{H}}}\exp\left(c \int_{\mathbb{T}}|\partial_x(Pu)_{\leqslant k-1}|^2\,{\mathord{{\rm d}}}x\right)\,{\mathord{{\rm d}}}\mu_0(u) \leqslant\int_{{\mathfrak{H}}}\exp\left(c \sum_{1\leqslant|j| \leqslant 2^{k-1}} (2\pi j)^2 |\alpha_j|^2 \right)\,{\mathord{{\rm d}}}\mu_0(u) \nonumber \\&\qquad=\prod_{ 1\leqslant|j|\leqslant 2^{k-1}}\frac{\lambda_j}{\lambda_j-c(2\pi j)^2} \leqslant\prod_{1\leqslant|j|\leqslant 2^{k-1}}(1-2c)^{-1} \leqslant(1-2c)^{-2^k}, \end{align}\tag{189}\] where in the second inequality we used \(c(2\pi j)^2 /\lambda_j \leqslant 2c\). For the second probability on the right-hand side of 188 , the Gaussian high-frequency \(L^6\) tail estimate from [1] gives \[\begin{align} \mu_0\left(\|u_{\geqslant k-1}\|_{L^6}>1\right) \leqslant C_0 \mathrm{exp} \Big(-C 2^{\frac{4}{3} k} \Big), \end{align}\] for some constants \(C_0,C>0\) independent of \(k\). The same bound remains valid uniformly after inserting the projection \(P\). Indeed, the proof of the Gaussian high-frequency tail estimate in [1] is uniform with respect to an additional upper frequency cutoff. Applying that argument to the truncated field \((Pu)_{\geqslant k-1}\) gives \[\begin{align} \label{uniform:projection} \mu_0\left(\|(Pu)_{\geqslant k-1}\|_{L^6}>1\right) \leqslant C_0 \mathrm{exp} \Big(-C 2^{\frac{4}{3} k} \Big). \end{align}\tag{190}\] Substituting 189 and 190 into 188 , we obtain \[\begin{align} \eqref{bound:Pu:L6a} \leqslant C_{\iota,\gamma,m,{\mathcal{K}}_s} \left( 1- 2c \right)^{-\frac{2^k}{1+\iota}} \mathrm{exp} \Big(-C \frac{\iota}{1+\iota}2^{\frac{4}{3} k} \Big) = C_{\iota,\gamma,m,{\mathcal{K}}_s} \mathrm{exp}\left( -C\frac{\iota}{1+\iota}2^{\frac{4}{3} k}+\frac{2^k}{1+\iota} \log \frac{1}{1-2c} \right). \end{align}\] The right-hand side is summable in \(k\), uniformly in \(P\), because the negative term has order \(2^{4k/3}\) whereas the positive term has order \(2^k\). Therefore \[\begin{align} &\sum_{k\geqslant 1} \int_{{\mathfrak{H}}} e^{\frac{1+\varsigma}{6}\|Pu\|_{L^6}^6} \mathbb{1}_{\{ \|Pu\|_{L^2} \leqslant{\mathcal{K}}_s , \, E_k \}} {\mathord{{\rm d}}}\mu_{0}(u) \leqslant C_{\iota,\gamma,m,{\mathcal{K}}_s} \sum_{k\geqslant 1} \mathrm{exp}\left( -C\frac{\iota}{1+\iota}2^{\frac{4}{3} k}+\frac{2^k}{1+\iota} \log \frac{1}{1-2c} \right)<\infty . \end{align}\] This proves 142 . ◻