Assume (1). Since $\Hom_{D(\mathcal{O}_X)}(G[-i], E) = \Hom_{D(\mathcal{O}_X)}(G, E[i])$ we see that this is zero for $i \ll 0$ by Lemma \ref{lemma-ext-from-perfect-into-bounded-QCoh}. This proves that (1) implies (2). \medskip\noindent Parts (2), (3), (4) are equivalent by the discussion in Cohomology, Section \ref{cohomology-section-global-RHom}. Part (5) and (6) are equivalent as $H^i(X, -) = H^i(R\Gamma(X, -))$ by definition. The equivalent conditions (2), (3), (4) are equivalent to the equivalent conditions (5), (6) by Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex} and the fact that $(G[-i])^\vee = G^\vee[i]$. \medskip\noindent It is clear that (7) implies (2). Conversely,  let us prove that the equivalent conditions (2) -- (6) imply (7). Recall that $G$ is a  classical generator for $D_{perf}(\mathcal{O}_X)$ by Remark \ref{remark-classical-generator}. For $P \in D_{perf}(\mathcal{O}_X)$ let $T(P)$ be the assertion that $R\Hom_X(P, E)$ is in $D^+(\mathbf{Z})$. Clearly, $T$ is inherited by direct sums, satisfies the 2-out-of-three property for distinguished triangles, is inherited by direct summands, and is perserved by shifts. Hence by Derived Categories, Remark \ref{derived-remark-check-on-generator} we see that (4) implies $T$ holds on all of $D_{perf}(\mathcal{O}_X)$. The same argument works for all other properties, except that for property (7)(b) and (7)(c) we also use that $P \mapsto P^\vee$ is a self equivalence of $D_{perf}(\mathcal{O}_X)$. Small detail omitted. \medskip\noindent We will prove the equivalent conditions (2) -- (7) imply (1) using the induction principle of Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}. \medskip\noindent First, we prove (2) -- (7) $\Rightarrow$ (1) if $X$ is affine. Let $P = \mathcal{O}_X[0]$. From (7) we obtain $H^i (X, E) = 0$ for $i \ll 0$. Hence (1) follows since $E$ is determined by $R\Gamma (X, E)$, see Lemma \ref{lemma-affine-compare-bounded}. \medskip\noindent Now assume $X = U \cup V$ with $U$ a quasi-compact open of $X$ and $V$ an affine open, and assume the implication (2) -- (7) $\Rightarrow$ (1) is known for the schemes $U$, $V$, and $U \cap V$. Suppose $E \in D_\QCoh(\mathcal{O}_X)$ satisfies (2) -- (7). By Lemma \ref{lemma-direct-summand-of-a-restriction} and Theorem \ref{theorem-bondal-van-den-Bergh} there exists a perfect complex $Q$ on $X$ such that $Q|_U$ generates $D_\QCoh (\mathcal{O}_U)$.  Let $f_1, \dots , f_r \in \Gamma (V, \mathcal{O}_V)$ be such that $V \setminus U = V(f_1, \dots , f_r)$ as subsets of $V$. Let $K \in D_{perf}(\mathcal{O}_V)$ be the object corresponding to the Koszul complex on $f_1, \dots , f_r$. Let $K' \in D_{perf}(\mathcal{O}_X)$ be \begin{equation} \label{equation-detecting-bounded-below} K' = R (V \to X)_* K = R (V \to X)_! K, \end{equation} see Cohomology, Lemmas \ref{cohomology-lemma-pushforward-restriction} and \ref{cohomology-lemma-pushforward-perfect}. This is a perfect complex on $X$ supported on the closed set $X \setminus U \subset V$ and isomorphic to $K$ on $V$. By assumption, we know $R\Hom_{\mathcal{O}_X}(Q, E)$ and $R\Hom_{\mathcal{O}_X}(K', E)$ are bounded below. \medskip\noindent By the second description of $K'$ in (\ref{equation-detecting-bounded-below}) we have $$ \Hom_{D(\mathcal{O}_V)}(K[-i], E|_V) = \Hom_{D(\mathcal{O}_X)}(K'[-i], E) = 0 $$ for $i \ll 0$. Therefore, we may apply Lemma \ref{lemma-orthogonal-koszul-second-variant} to $E|_V$ to obtain an integer $a$ such that $\tau_{\leq a}(E|_V) = \tau_{\leq a} R(U \cap V \to V)_*(E|_{U \cap V})$. Then $\tau_{\leq a} E = \tau_{\leq a} R(U \to X)_*(E|_U)$ (check that the canonical map is an isomorphism after restricting to $U$ and to $V$). Hence using Lemma \ref{lemma-bounded-below-truncation} twice we see that $$ \Hom_{D(\mathcal{O}_U)}(Q|_U [-i], E|_U) = \Hom_{D(\mathcal{O}_X)}(Q[-i], R (U \to X)_* (E|_U)) = 0 $$ for $i \ll 0$. Since the Proposition holds for $U$ and the generator $Q|_U$, we have $E|_U \in D^+_\QCoh (\mathcal{O}_U)$. But then since the functor $R(U \to X)_*$ preserves bounded below objects (see Cohomology, Section \ref{cohomology-section-derived-functors}) we get $\tau_{\leq a} E \in D^+_\QCoh(\mathcal{O}_X)$. Thus  $E \in D^+_\QCoh (\mathcal{O}_X)$.