--------------------------------------------- The next subsections are essentially stolen from \cite{BHJ15a} and \cite{BBJ15}. A (real rank 1) valuation $v$ on (the function field of) $X$ is \emph{divisorial} if $v=c\ord_E$, where $c>0$ and $E$ is a prime divisor on a normal variety $Y$ mapping birationally to $X$. The \emph{log discrepancy} of $v$ is then $A_X(v)=c(1+\ord_E(K_{Y/X}))$, where $K_{Y/X}$ is the relative canonical divisor. The projection $X\times\bC\to X$ takes divisorial valuations to divisorial (or trivial) valuations. Further, every divisorial valuation on $X$ has a unique preimage $w$ on $X\times\bC$ that is $\bC^*$-invariant (under the action on the second factor) satisfies $v(\tau)=1$, with $\tau$ denoting the coordinate on $\bC$; then $A_{X\times\bC}(w)=A_X(v)+1$. A non-Archimedean metric on $A$ will be viewed as a certain type of function $\varphi$ on the Berkovich analytification of $X$ with respect to the trivial norm on $\bC$. In particular, $\varphi$ restricts to a function on the space of divisorial valuations on $X$, and is in fact uniquely determined by this restriction. Every normal test configuration $(\cX,\cA)$ of $(X,A)$ defines a non-Archimedean metric $\varphi=\varphi_{\cX,\cA}$ as follows. Pick a normal variety $\cY$ with $\bC^*$-equivariant birational morphisms $\pi:\cY\to X\times\bC$ and $\rho:\cY\to\cX$, and let $p_1:X\times\bC\to X$ be the projection. Then $\rho^*\cA-\pi^*p_1^*L=\cO_\cY(D)$ for a unique $\bQ$-divisor $D$ supported on the central fiber. One has $\varphi(c\ord_E)=c\ord_E(D)$ for every irreducible component $E$ of the central fiber of $\cY$. Varying $\cY$ determines $\varphi$ completely. One denotes by $\cP_A^{\NA}$ the set of non-Archimedean metrics defined by normal (merely semiample) test configurations. \medskip \noindent \textbf{4.E. Non-Archimedean functionals.} In~\cite{BHJ15a}, natural non-Archimedean versions $E^{\NA}$, $J^{\NA}$, and $M^{\NA}$ of the basic functionals on $\cP_A$ are introduced. They are defined as functionals $\cP_A^{\NA}\to\bR$, with $$ J^{\NA}(\varphi)=\sup\varphi-E^{\NA}(\varphi) $$ for all $\varphi\in\cP_A^{\NA}$. The functional $M^{\NA}$ is closely related (but not equal) to the Donaldson-Futaki invariant $\DF$. More precisely, we have \begin{equation*} M^{\NA}(\cX,\cA)=\DF(\cX,\cA)+V^{-1}\left((\cX_{0,\red}-\cX_0)\cdot\cA^n\right), \end{equation*} so that $M^{\NA}(\cX,\cA)\le\DF(\cX,\cA)$ with equality when $\cX_0$ is reduced. Uniform K-stability is equivalent to an estimate $M^{\NA}\ge\delta J^{\NA}$ on $\cP_A^{\NA}$. In addition, let us introduce $L^{\NA}$ and $D^{\NA}:=L^{\NA}-E^{\NA}$ by the formula \begin{equation*} L^{\NA}(\varphi)=\inf_v(A_X(v)+\varphi(v)), \end{equation*} where $v$ ranges over divisorial valuations on $X$ (compare~\cite[Proposition 3.8]{Berm15}). Note that \begin{equation*} D^{\NA}\le M^{\NA} \quad\text{and}\quad D^{\NA}\le J^{\NA}. \end{equation*} One says that $X$ is \emph{uniformly Ding-stable} if $D^{\NA}\ge\delta J^{\NA}$ on $\cP_A^{\NA}$ for some $\delta>0$. This trivially implies uniform K-stability, and the converse is also true, as we shall see in Theorem~\ref{thm:Ding}. \medskip \noindent \textbf{4.F. From uniform K-stability to uniform Ding stability.} As a first step towards the proof of Theorem~0.7, one shows: \setref{Theorem ???} \begin{claimref}\label{thm:Ding} Let $X$ be a smooth Fano manifold and $\delta\ge 0$. Then $M^{\NA}\ge\delta J^{\NA}$ holds on $\cP_A^{\NA}$ if and only if $D^{\NA}\ge\delta J^{\NA}$ on $\cP_A^{\NA}$ \end{claimref} \begin{proof} One direction is clear since $M^{\NA}\ge D^{\NA}$. Conversely, suppose $M^{\NA}\ge \delta J^{\NA}$ on $\cP_A^{\NA}$. First observe that $D^{\NA}$ and $M^{\NA}$ coincide on any test configuration $(\cX,\cA)$ which is lc (in the sense that the pair $(\cX,\cX_0)$ has log canonical singularities) and such that $\cA\equiv-K_\cX$ (numerical equivalence). In order to establish $D^{\NA}(\cX,\cA)\ge\delta J^{\NA}(\cX,\cA)$ for an arbitrary normal, semiample test configuration, we may assume wlog that $(\cX,\cA)$ is semistable (and hence lc), since both sides are homogeneous under finite base change. Following the procedure of~\cite{LX}, we then use the Minimal Model Program to produce a continuous path $(\varphi_t)_{t\in[0,T]}$ in $\cP_A^{\NA}$ with $\varphi_0=\varphi_{(\cX,\cA)}$, $\varphi_T$ associated to an lc test configuration $(\cX',\cA')$ with $\cA'\equiv-K_{\cX'}$, and such that $t\mapsto D^{\NA}(\varphi_t)/J^{\NA}(\varphi_t)$ is non-increasing. One will then get as desired \begin{equation*} \frac{D^{\NA}(\varphi)}{J^{\NA}(\varphi)}\ge \frac{D^{\NA}(\varphi_T)}{J^{\NA}(\varphi_T)} =\frac{M^{\NA}(\varphi_T)}{J^{\NA}(\varphi_T)}\ge\delta. \end{equation*} More precisely, we run an MMP with scaling by $\cH:=r\cA-K_\cX$, $r\gg 1$, as in~\cite[\S~5.1]{LX}. This yields a sequence of rational numbers $1=:\lambda_0>\dots>\lambda_k>\lambda_{k+1}:=\frac{1}{r+1}$, a sequence of $\bC^*$-equivariant divisorial contractions and flips \begin{equation*} \cX=:\cX^0\dashrightarrow\cX^1\dashrightarrow\dots\dashrightarrow\cX^k \end{equation*} and, for each $j$, an affine path of lc, semiample test configurations $(\cX^j,\cA^j_\lambda)$, $\lambda\in[\lambda_{j+1},\lambda_j]$, such that \begin{equation}\label{equ:KRF} K_{\cX^j}+\cA^j_\lambda=\frac{r\lambda}{(r+1)\lambda-1}(K_{\cX^j}+\cA^j), \end{equation} with $\cA^j$ the push-forward of $\cA$ to $\cX^j$. Reparametrizing by $e^t=\frac{r\lambda}{(r+1)\lambda-1}$ then yields the desired path $(\varphi_t)$ in $\cP_A^{\NA}$. Indeed using (\ref{equ:KRF}) we prove that \begin{equation*} \frac{d}{dt}\left(D^{\NA}(\varphi_t)-J^{\NA}(\varphi_t)\right)=D^{\NA}(\varphi_t)-J^{\NA}(\varphi_t)\le 0 \end{equation*} and \begin{equation*} \frac{d}{dt}J^{\NA}(\varphi_t)=J^{\NA}(\varphi_t)-M^{\NA}(\varphi_t). \end{equation*} It follows that \begin{equation*} \left(\frac{d}{dt}D^{\NA}(\varphi_t)\right)J^{\NA}(\varphi_t)-D^{\NA}(\varphi_t)\left(\frac{d}{dt}J^{\NA}(\varphi_t)\right) =\left(D^{\NA}(\varphi_t)-J^{\NA}(\varphi_t)\right)M^{\NA}(\varphi_t)\le 0, \end{equation*} since $M^{\NA}\ge\delta J^{\NA}\ge 0$ by assumption, and we conclude as desired that $D^{\NA}/J^{\NA}$ is non-increasing along the path $(\varphi_t)$. \end{proof} \begin{claimref} The path $(\varphi_t)$ can be seen as a non-Archimedean version of the K\"ahler-Ricci flow. \end{claimref} \begin{claimref}\label{rmk:ss} The condition $M^{\NA}\ge 0$ is equivalent to K-semistability, cf.~\cite{BHJ15a}. The case $\delta=0$ of Theorem~\ref{thm:Ding} therefore says that K-semistability is also equivalent to Ding semistability, thereby showing that the main result of~\cite{Fuj} is also valid assuming only $X$ to be K-semistable. \end{claimref} \noindent \textbf{4.G. From uniform Ding stability to coercivity of the K-energy.} Suppose $D^{\NA}\ge\delta J^{\NA}$ on $\cP_A^{\NA}$, where $\delta\in{}]0,1[$, and pick $\delta'\in{}]0,\delta[$. The point is to show that $M\ge\delta' J-C$ on $\cP_A$ for some constant $C>0$. By Theorem 2.1, this proves Theorem~0.7. Arguing by contradiction, one can assume that there exists a sequence $(\phi_j)_1^\infty$ in $\cP_A$ such that $$ M(\phi_j)\le\delta' J(\phi_j)-j. $$ Since $M$ and $J$ are translation invariant, one can assume that $\phi_j$ is normalized so that $\sup(\phi_j-\phi_0)=0$. The inequality $M\ge-nJ$, which always holds, then implies $J(\phi_j)\ge\frac{j}{n+\delta'}\to+\infty$, and hence $E(\phi_j)\le-J(\phi_j)\to-\infty$. \medskip \noindent \textbf{Step 1: construction of a geodesic ray in $\cE^1$.} The technique follows Darvas and He~\cite{DH14}. Denote by $d_1$ the distance introduced by Darvas~\cite{Dar14} on the space $\cE^1$ of finite energy, so that $d_1$ defines the topology of convergence in energy. For each $j$ we let $(\phi_{j,t})_{0\le t\le T_j}$ be the geodesic segment connecting $\phi_0$ to $\phi_j$, parametrized so that $T_j=-E(\phi_j)$. By~\cite[Proposition~6.2]{BBGZ13} and~\cite[Theorem~1]{Dar13} we have $E(\phi_{j,t})=-t$ and $\sup(\phi_{j,t}-\phi_0)=0$, respectively. Thus $J(\phi_{j,t})\le\sup(\phi_{j,t}-\phi_0)-E(\phi_{j,t})=t$. In particular, $J(\phi_j)\le T_j$, so $M(\phi_j)\le\delta'T_j-j<\delta'T_j$ for all $j$. By~\cite{BeBe14}, $M$ is convex along geodesic segments, so $M(\phi_{j,t})\le\frac{t}{T_j}M(\phi_j)\le\delta't$ for $t\le T_j$. Since $M\ge H-nJ$, this yields $H(\phi_{j,t})\le(\delta'+n)t$ for $t\le T_j$, so for fixed $T>0$ and $t\le T$, the metrics $\phi_{j,t}$ lie in the set $$ \mathcal{K}_T:=\{\phi\in\cE^1\mid\sup(\phi-\phi_0)=0\text{ and }H(\phi)\le(\delta'+n)T\}, $$ which is a compact subset of the metric space $(\cE^1,d_1)$ by~\cite{BBEGZ15}. By the geodesic property, we also have \begin{equation*} d_1(\phi_{j,t},\phi_{j,s}) =d_1(\phi_{j,1},\phi_0)|t-s| \le C(J(\phi_{j,1})+1)|t-s| \le 2C|t-s|, \end{equation*} which shows that $t\mapsto\phi_{j,t}$ restrict to uniformly Lipschitz continuous maps $[0,T]\to\mathcal{K}_T$. Combining Ascoli's Theorem with a diagonal argument we may assume, after passing to a subsequence, that $\phi_{j,t}$ converges to a geodesic ray $(\phi_t)_{t\ge0}$ in $\cE^1$, uniformly for each compact time interval. Note in particular that $E(\phi_t)=-t$ for all $t$. The geodesic ray $(\phi_t)$ defines an $S^1$-invariant metric $\Phi$ on $p_1^*L$ over $X\times\bD^*$, such that the restriction of $\Phi$ to $X\times\{\tau\}$ is equal to $\phi_{\log|\tau|^{-1}}$. We claim that $\Phi$ is psh. To see this, note that the geodesic segment $(\phi_{j,t})$ defines an $S^1$-invariant psh metric $\Phi_j$ on $p_1^*L$ over $X\times A_j$, where $A_j=\{e^{-T_j}<|\tau|<1\}\subset\bC$. The locally uniform convergence of $\phi_{j,t}$ to $\phi_t$ implies that $\Phi_j$ converges to $\Phi$ locally uniformly in the $L^1$-topology; hence $\Phi$ is psh. Now $M$ is lsc on $\mathcal{K}_T$, so $M(\phi_t)\le\liminf_jM(\phi_{j,t})\le\delta't$. Hence $D(\phi_t)\le M(\phi_t)\le\delta't$ for $t\ge0$. \medskip \noindent \textbf{Step~2: Approximation by test configurations.} Since $\sup(\phi_t-\phi_0)=0$ for all $t$, $\Phi$ extends to a psh metric on $p_1^*L$ over $X\times\bD$. It has zero Lelong numbers at all points on $X\times\bD^*$ since $\phi_t\in\cE^1$ for all $t$. The generic Lelong number along $X\times\{0\}$ is also zero. As a consequence, the multiplier ideals $\cJ(m\Phi)$ are cosupported on proper subvarieties of the central fiber $X\times\{0\}$, and they are of course $\bC^*$-invariant. We claim that there exists $m_0$ large enough such that the sheaf $\cO((m+m_0) p_1^*L)\otimes\cJ(m\Phi)$ is generated by its global sections on $X\times\bD$ for each $m\ge 1$. More precisely, given a very ample line bundle $H$ on $X$, it is enough to choose $m_0$ such that $A:=m_0L-K_X-(n+1)H$ is ample on $X$. Indeed, as first observed in~\cite[Corollary 1.5]{DEL00}, this is a consequence of the Nadel vanishing theorem and the Castelnuovo-Mumford criterion for global generation. More precisely, since $\bD$ is Stein, it is enough to show that $\cO((m+m_0) p_1^*L)\otimes\cJ(m\Phi)$ is $p_2$-globally generated, with $p_2:X\times\bD\to\bD$ denoting the second projection. By the relative version of the Castelnuovo-Mumford criterion, this will be the case as soon as $$ R^j (p_2)_*\left(\cO((m+m_0)p_1^*L-jp_1^*H)\otimes\cJ(m\Phi)\right)=0 $$ for all $j\ge 1$, which is a consequence of Nadel's vanishing (compare~\cite[Theorem B.8]{BFJ12}). Denote by $\mu_m\colon\cX_m\to X\times\bC$ the normalized blow-up of $X\times\bC$ along $\cJ(m\Phi)$, with exceptional divisor $E_m$, and set $\cA_m:=\mu_m^*p_1^*L-\frac{1}{m+m_0} E_m$. Then $(\cX_m,\cA_m)$ is a normal, semiample test configuration for $(X,A)$, inducing a non-Archimedean metric $\varphi_m\in\cP_A^{\NA}$ with $\sup\varphi_m=0$, given by $$ \varphi_m(v)=\tfrac{1}{m+m_0}v(\cJ(m\Phi)) $$ for each $\bC^*$-invariant divisorial valuation $v$ on $X\times\bC$. For each $m\ge 1$, pick any $S^1$-invariant smooth psh metric $\Psi_m$ on the $\bQ$-line bundle $\cA_m$. By the results of~\cite{Berm15,BHJ15b} mentioned in the introduction, the corresponding subgeodesic ray $(\phi_{m,t})$ satisfies \begin{equation*} \lim_{t\to+\infty}\tfrac1{t}L(\phi_{m,t}) =L^{\NA}(\varphi_m) \quad\text{and}\quad \lim_{t\to+\infty}\tfrac1{t}E(\phi_{m,t}) =E^{\NA}(\varphi_m) =-J^{\NA}(\varphi_m). \end{equation*} The psh metric $\Phi_m$ on $p_1^*L$ over $X\times\bC$ induced by $\Psi_m$ has analytic singularities of type $\cJ(m\Phi)^{1/(m+m_0)}$. As a consequence of Demailly's local regularization theorem~\cite[Proposition 3.1]{Dem92}, $\Phi_m$ is therefore less singular than $\Phi$. More precisely, for each $r<1$, we can find a constant $C_{r,m}>0$ such that $\Phi_m\ge\Phi-C_{m,r}$ over $X\times\bD_r$. By monotonicity of $E$, the corresponding subgeodesic rays satisfy $$ E(\phi_{m,t})\ge E(\phi_t)-C_{m,r}=-t-C_{m,r} $$ for $t>-\log r$, and we infer \begin{equation}\label{e102} E^{\NA}(\varphi_m) =\lim_{t\to\infty}\tfrac1{t}E(\phi_{m,t}) \ge -1. \end{equation} \medskip \textbf{Step~3: asymptotics of $L$ along the geodesic ray.}\label{S102} We now relate the slope at infinity of $L(\phi_t)$ to the asymptotics of the non-Archimedean functional $L^{\NA}(\varphi_m)$ as $m\to\infty$. We shall prove that \begin{equation}\label{e101} \lim_{m\to+\infty}L^{\NA}(\varphi_m) =\lim_{t\to+\infty}\tfrac1{t}L(\phi_t). \end{equation} \setref{Theorem ???} \begin{claimref}\label{T101} Let $(\phi_t)$ be a subgeodesic ray in $\cE^1$ normalized by $\sup(\phi_t-\phi_0)=0$, and let $\Phi$ be the corresponding $S^1$-invariant psh metric on the pull-back of $-K_X$ to $X\times\bD$. Then: \begin{equation}\label{e105} \lim_{t\to+\infty}\tfrac1{t}L(\phi_t) =\inf_w\left(A_{X\times\bC}(w)-1-w(\Phi)\right), \end{equation} where $w$ ranges over $\bC^*$-invariant divisorial valuations on $X\times\bC$ such that $w(\tau)=1$, and $A_{X\times\bC}(w)$ is the log discrepancy of such a valuation. \end{claimref} Here $w(\Phi)$ is to be interpreted as a generic Lelong number on a suitable blowup, see~\cite{BFJ08}. Let us deduce~\eqref{e101} from Theorem~\ref{T101}. By~\cite{BHJ15a}, the projection map $X\times\bC\to X$ induces a bijection between $\bC^*$-invariant divisorial valuations $w$ on $X\times\bC$ satisfying $w(\tau)=1$ and divisorial (or trivial) valuations $v$ on $X$, and we have $A_{X\times\bC}(w)=A_X(v)+1$. As a result, we get \begin{equation*} L^{\NA}(\varphi_m) =\inf_v(A_X(v)+\varphi_m(v)) =\inf_w(A_{X\times\bC}(w)-1-\tfrac{1}{m+m_0}w(\cJ(m\Phi))), \end{equation*} and we are left showing that \begin{equation*} \inf_w(A_{X\times\bC}(w)-w(\Phi)) =\lim_{m\to\infty}\inf_w(A_{X\times\bC}(w)-\tfrac{1}{m+m_0}w(\cJ(m\Phi))). \end{equation*} But this follows formally from the inequalities \begin{equation*} w(\cJ(m\Phi)) \le m\,w(\Phi) \le w(\cJ(m\Phi))+A_{X\times\bC}(w). \end{equation*} \begin{proof}[Proof of Theorem~\ref{T101}] Since the metric $\Phi$ is psh on $X\times\bD$, the function $l(\tau):=L(\phi_{\log|\tau|^{-1}})$ is subharmonic on $\Delta$~\cite{Bern09}, and its Lelong number $\nu$ at the origin coincides with the negative of the left hand side of~\eqref{e105}. We must therefore show that $\nu$ is equal to $s:=\sup_w(w(\Phi)+1-A_{X\times\bC}(w))$. As in the proof of~\cite[Proposition~3.8]{Berm15} we use that $\nu$ is the infimum of all $c\ge0$ such that \begin{equation}\label{e103} \int_Ue^{-2(l(\tau)+(1-c)\log|\tau|)}id\tau\wedge d\bar{\tau} =\int_{X\times U}e^{-2(\Phi+(1-c)\log|\tau|)}id\tau\wedge d\bar{\tau}<+\infty, \end{equation} for some small neighborhood $U\subset\bD$ of the origin. Near each point of $X\times\{0\}$ we have $e^{-2\Phi}id\tau\wedge d\bar{\tau}=e^{-2\varphi}dV$, with $\varphi$ psh and $dV$ a smooth volume form. Setting $p:=\lfloor c\rfloor$ and $r=c-p\in[0,1[$ we get \begin{equation*} e^{-2(\Phi+(1-c)\log|\tau|)}id\tau\wedge d\bar\tau =|\tau|^{2p}e^{-2(\varphi+(1-r)\log|\tau|)}dV. \end{equation*} Since $\tau^{p}$ is holomorphic, it follows from~\cite[Theorem~5.5]{BFJ08} that \begin{equation*} \int_{X\times U}e^{-2(\Phi+(1-c)\log|\tau|)}id\tau\wedge d\bar{\tau}<+\infty \quad\Longrightarrow\quad \sup_w\frac{w(\Phi)+(1-r)w(\tau)}{pw(\tau)+A_{X\times\bC}(w)}\le 1, \end{equation*} where $w$ ranges over all divisorial valuations on $X\times\bC$. By homogeneity and by the $S^1$-invariance of $\Phi$, it suffices to consider $w$ that are $\bC^*$-invariant and normalized by $w(\tau)=1$. We then get $w(\Phi)+1\le p+r+A_{X\times\bC}(w)=c+A_{X\times\bC}(w)$, and hence $s\le\nu$. Conversely,~\cite[Theorem~5.5]{BFJ08} shows that \begin{equation*} \sup_w\frac{w(\Phi)+(1-r)}{p+A_{X\times\bC}(w)}<1 \quad\Longrightarrow\quad \int_{X\times U}e^{-2(\Phi+(1-c)\log|\tau|)}id\tau\wedge d\bar{\tau}<+\infty. \end{equation*} Since $c$, and hence $p$ is bounded above, say by $\nu+1$, and $A_{X\times\bC}(w)\ge1$, the left-hand condition is equivalent to the existence of $\varepsilon>0$ such that $w(\Phi)\le(1-\varepsilon)A_{X\times\bC}(w)-1+c$ for all $w$. Given $\rho>0$, we must therefore show that there exists $\varepsilon>0$ such that \begin{equation*} w(\Phi)\le(1-\varepsilon)A_{X\times\bC}(w)+1+s+\rho \end{equation*} for all $w$ as above. Arguing by contradiction, we get a sequence $w_j$ of $\bC^*$-invariant divisorial valuations with $w_j(\tau)=1$ such that \begin{equation*} w_j(\Phi)\ge(1-\tfrac{1}{j})A_{X\times\bC}(w_j)-1+s+\rho. \end{equation*} Let $W$ be the subset of the Berkovich analytification $(X\times\bC)^\an$ consisting of semivaluations $w$ that are $\bC^*$-invariant and satisfy $w(\tau)=1$, see~\cite{BerkBook}. Then $W$ is compact and even sequentially compact~\cite{poineau}, so after passing to a subsequence we may assume that $w_j\to w_\infty\in W$. For each $m\ge1$, the multiplier ideal sheaf $\cJ(m\Phi)$ satisfies \begin{equation*} \tfrac{1}{m}w_j(\cJ(m\Phi)) \ge w_j(\Phi)-\tfrac{1}{m}A_{X\times\bC}(w_j) \ge(1-\tfrac{1}{j}-\tfrac{1}{m})A_{X\times\bC}(w_j)-1+s+\rho \end{equation*} for all $j\ge 1$. Since $w\to w(\cJ(m\Phi))$ is continuous and $w\to A_{X\times\bC}(w)$ is lsc on $W$, this implies \begin{equation*} \tfrac{1}{m}w_\infty(\cJ(m\Phi)) \ge(1-\tfrac{1}{m})A_{X\times\bC}(w_\infty)-1+s+\rho \end{equation*} for all $m\ge1$. In particular, $A_{X\times\bC}(w_\infty)<\infty$. On the other hand we have, by definition of $s$, \begin{equation*} \tfrac 1m w(\cJ(m\Phi))\le w(\Phi)\le A_{X\times\bC}(w)-1+s \end{equation*} for all divisorial valuations $w$ in $W$. By density, the inequality $\frac1mw(\cJ(m\Phi))\le A_{X\times\bC}(w)-1+s$ is also valid for all quasimonomial valuations $w\in W$, and hence for all $w\in W$ by approximation via retraction to dual complexes~\cite{BFJ12,JM12}. Comparing these inequalities we infer \begin{equation*} A_{X\times\bC}(w_\infty)-1+s\ge (1-\tfrac 1 m)A_{X\times\bC}(w_\infty)-1+s+\rho, \end{equation*} and we reach a contradiction by letting $m\to\infty$. \end{proof} \noindent \textbf{Step 4: concluding the proof of Theorem 0.7.} By~\eqref{e101} we have \begin{equation*} \lim_{m\to\infty}L^{\NA}(\varphi_m) =\lim_{t\to\infty}\tfrac1{t}L(\phi_t) =\lim_{t\to\infty}\tfrac1{t}(D(\phi_t)+E(\phi_t)) \le-1+\delta'. \end{equation*} On the other hand, our assumption of uniform Ding stability yields $$ L^{\NA}(\varphi_m)-E^{\NA}(\varphi_m)=D^{\NA}(\varphi_m)\ge\delta J^{\NA}(\varphi_m)=-\delta E^{\NA}(\varphi_m), $$ and hence $$ L^{\NA}(\varphi_m)\ge(1-\delta)E^{\NA}(\varphi_m)\ge -1+\delta $$ by~\eqref{e102}, which contradicts $\delta'<\delta$. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \medskip \textbf{4.H. The semistable case.} In this final subsection, the proof of Theorem~0.8 will be breiefly sketched. The solvability of the twisted K\"ahler-Einstein equation \begin{equation}\label{equ:twist} \Ricci(\omega)=(1-\gamma)\omega+\gamma\omega_0 \end{equation} with $\gamma\in{}]0,1[$ is implied by the coercivity of the twisted Mabuchi K-energy $$ M_\gamma:=H-(1-\gamma) J^*=M+\gamma J^*. $$ By Theorem~\ref{thm:Ding}, K-semistability implies $D^{\NA}\ge 0$ on $\cP_A^{\NA}$. Arguing as in Steps 2, 3 and 4 above, we infer from this that $\lim_{t\to+\infty}D(\phi_t)/t\ge 0$ for each subgeodesic ray $(\phi_t)$ in $\cE^1$. Since $M_\gamma\ge D+\gamma J^*$, this implies $$ \lim_{t\to+\infty}\frac{M_\gamma\phi_t)}{t}\ge\gamma\lim_{t\to+\infty}\frac{J^*(\phi_t)}{t}. $$ Since $J^*$ and $J$ are comparable up to multiplicative constants, we obtain an estimate $$ \lim_{t\to+\infty}\frac{M_\gamma\phi_t)}{t}\ge\delta\lim_{t\to+\infty}\frac{J(\phi_t)}{t} $$ for some uniform constant $\delta>0$. Arguing as in Step 1 above, using the geodesic convexity of $M_\gamma$ established in~\cite{BeBe14}, we deduce from this the desired coercivity estimate $M_\gamma\ge\delta'J-C$, which proves the solvability of (\ref{equ:twist}).