We already know that $X \to \Spec(\mathbf{Z})$ is separated and locally of finite presentation (Lemma \ref{lemma-quot-s-lfp}). We also know that $X \to \Spec(\mathbf{Z})$ satisfies the existence part of the valuative criterion, see Lemma \ref{lemma-quot-existence-part}. By the valuative criterion for properness, it suffices to prove our Quot space is quasi-compact, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-characterize-proper}. Thus it suffices to find a quasi-compact scheme $T$ and a surjective morphism $T \to X$. Let $m$ be the integer found in Varieties, Lemma \ref{varieties-lemma-bound-quotients-free}. Let $$ N = r{m + n \choose n} - P(m) $$ We will write $\mathbf{P}^n$ for $\mathbf{P}^n_\mathbf{Z} = \text{Proj}(\mathbf{Z}[T_0, \ldots, T_n])$ and unadorned products will mean products over $\Spec(\mathbf{Z})$. The idea of the proof is to construct a ``universal'' map $$ \Psi : \mathcal{O}_{T \times \mathbf{P}^n}(-m)^{\oplus N} \longrightarrow \mathcal{O}_{T \times \mathbf{P}^n}^{\oplus r} $$ over an affine scheme $T$ and show that every point of $X$ corresponds to a cokernel of this in some point of $T$. \medskip\noindent Definition of $T$ and $\Psi$. We take $T = \Spec(A)$ where $$ A = \mathbf{Z}[a_{i, j, E}] $$ where $i \in \{1, \ldots, r\}$, $j \in \{1, \ldots, N\}$ and $E = (e_0, \ldots, e_n)$ runs through the multi-indices of total degree $|E| = \sum_{k = 0, \ldots n} e_k = m$. Then we define $\Psi$ to be the map whose $(i, j)$ matrix entry is the map $$ \sum\nolimits_{E = (e_0, \ldots, e_n)} a_{i, j, E} T_0^{e_0} \ldots T_n^{e_n} : \mathcal{O}_{T \times \mathbf{P}^n}(-m) \longrightarrow \mathcal{O}_{T \times \mathbf{P}^n} $$ where the sum is over $E$ as above (but $i$ and $j$ are fixed of course). \medskip\noindent Consider the quotient $\mathcal{Q} = \Coker(\Psi)$ on $T \times \mathbf{P}^n$. By More on Flatness, Lemma \ref{flat-lemma-generic-flatness-stratification} there exists a $t \geq 0$ and closed subschemes $$ T = T_0 \supset T_1 \supset \ldots \supset T_t = \emptyset $$ such that the pullback $\mathcal{Q}_p$ of $\mathcal{Q}$ to $(T_p \setminus T_{p + 1}) \times \mathbf{P}^n$ is flat over $T_p \setminus T_{p + 1}$. Observe that we have an exact sequence $$ \mathcal{O}_{(T_p \setminus T_{p + 1}) \times \mathbf{P}^n}(-m)^{\oplus N} \to \mathcal{O}_{(T_p \setminus T_{p + 1}) \times \mathbf{P}^n}^{\oplus r} \to \mathcal{Q}_p \to 0 $$ by pulling back the exact sequence defining $\mathcal{Q} = \Coker(\Psi)$. Therefore we obtain a morphism $$ \coprod (T_p \setminus T_{p + 1}) \longrightarrow \Quotfunctor_{\mathcal{O}^{\oplus r}/\mathbf{P}/\mathbf{Z}} \supset \Quotfunctor^P_{\mathcal{O}^{\oplus r}/\mathbf{P}/\mathbf{Z}} = X $$ Since the left hand side is a Noetherian scheme and the inclusion on the right hand side is open, it suffices to show that any point of $X$ is in the image of this morphism. \medskip\noindent Let $k$ be a field and let $x \in X(k)$. Then $x$ corresponds to a surjection $\mathcal{O}_{\mathbf{P}^n_k}^{\oplus r} \to \mathcal{F}$ of coherent $\mathcal{O}_{\mathbf{P}^n_k}$-modules such that the Hilbert polynomial of $\mathcal{F}$ is $P$. Consider the short exact sequence $$ 0 \to \mathcal{K} \to \mathcal{O}_{\mathbf{P}^n_k}^{\oplus r} \to \mathcal{F} \to 0 $$ By Varieties, Lemma \ref{varieties-lemma-bound-quotients-free} and our choice of $m$ we see that $\mathcal{K}$ is $m$-regular. By Varieties, Lemma \ref{varieties-lemma-m-regular-globally-generated} we see that $\mathcal{K}(m)$ is globally generated. By Varieties, Lemma \ref{varieties-lemma-m-regular-up} and the definition of $m$-regularity we see that $H^i(\mathbf{P}^n_k, \mathcal{K}(m)) = 0$ for $i > 0$. Hence we see that $$ \dim_k H^0(\mathbf{P}^n_k, \mathcal{K}(m)) = \chi(\mathcal{K}(m)) = \chi(\mathcal{O}_{\mathbf{P}^n_k}(m)^{\oplus r}) - \chi(\mathcal{F}(m)) = N $$ by our choice of $N$. This gives a surjection $$ \mathcal{O}_{\mathbf{P}^n_k}^{\oplus N} \longrightarrow \mathcal{K}(m) $$ Twisting back down and using the short exact sequence above we see that $\mathcal{F}$ is the cokernel of a map $$ \Psi_x : \mathcal{O}_{\mathbf{P}^n_k}(-m)^{\oplus N} \to \mathcal{O}_{\mathbf{P}^n_k}^{\oplus r} $$ There is a unique ring map $\tau : A \to k$ such that the base change of $\Psi$ by the corresponding morphism $t = \Spec(\tau) : \Spec(k) \to T$ is $\Psi_x$. This is true because the entries of the $N \times r$ matrix defining $\Psi_x$ are homogeneous polynomials $\sum \lambda_{i, j, E} T_0^{e_0} \ldots T_n^{e_n}$ of degree $m$ in $T_0, \ldots, T_n$ with coefficients $\lambda_{i, j, E} \in k$ and we can set $\tau(a_{i, j, E}) = \lambda_{i, j, E}$. Then $t \in T_p \setminus T_{p + 1}$ for some $p$ and the image of $t$ under the morphism above is $x$ as desired.