[proofplan]
The result is exactly the smooth Riemannian specialization of the von Renesse--Sturm characterization of lower Ricci curvature bounds by convexity of Boltzmann entropy on quadratic Wasserstein space. The proof therefore records the precise external theorem being used, translates its notation into the present notation, and verifies each of its hypotheses. Once the translation is made, the forward and reverse implications are the two directions of that theorem with the same coefficient convention.
[/proofplan]
[step:State the external von Renesse--Sturm theorem in the needed form]
We use the following standard external theorem of von Renesse and Sturm, in the form proved for smooth complete connected finite-dimensional Riemannian manifolds without boundary in von Renesse and Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature, Communications on Pure and Applied Mathematics, 2005.
Let $(N,h)$ be a complete connected finite-dimensional smooth Riemannian manifold without boundary. Let $d_h$ be its Riemannian distance and let $m_h$ be its Riemannian volume measure. For $K\in\mathbb R$, the following are equivalent.
First, $\operatorname{Ric}^h_q(w,w)\ge K h_q(w,w)$ for every $q\in N$ and every $w\in T_qN$.
Second, for every pair $\nu_0,\nu_1\in\mathcal P_2(N)$ with finite entropy relative to $m_h$, there exists a constant-speed geodesic $(\nu_t)_{t\in[0,1]}$ in the quadratic Wasserstein space $(\mathcal P_2(N),W_{2,h})$ such that, for every $t\in[0,1]$,
\begin{align*}
\operatorname{Ent}_{m_h}(\nu_t)\le (1-t)\operatorname{Ent}_{m_h}(\nu_0)+t\operatorname{Ent}_{m_h}(\nu_1)-\frac K2 t(1-t)W_{2,h}(\nu_0,\nu_1)^2.
\end{align*}
Here $W_{2,h}$ is defined from $d_h$ by the quadratic optimal-transport formula, and $\operatorname{Ent}_{m_h}$ is the extended relative entropy, equal to $\int_N \sigma\log\sigma\,dm_h(q)$ when $\nu=\sigma m_h$ and $\sigma\log\sigma\in L^1(N,\mathcal B(N),m_h)$, and equal to $+\infty$ otherwise.
[guided]
The proof uses a deep external theorem, so the first task is to state exactly what is being imported. The imported result is not a heuristic localization principle; it is the von Renesse--Sturm equivalence theorem for smooth Riemannian manifolds. In the notation used here, it says the following.
Let $(N,h)$ be a complete connected finite-dimensional smooth Riemannian manifold without boundary. Let $d_h:N\times N\to[0,\infty)$ be the distance induced by $h$, let $m_h$ be the Riemannian volume measure, and let $W_{2,h}$ be the quadratic Wasserstein distance constructed from $d_h$. Define the extended entropy $\operatorname{Ent}_{m_h}$ by the rule that, if $\nu=\sigma m_h$ and $\sigma\log\sigma\in L^1(N,\mathcal B(N),m_h)$, then
\begin{align*}
\operatorname{Ent}_{m_h}(\nu)=\int_N \sigma\log\sigma\,dm_h(q),
\end{align*}
and otherwise $\operatorname{Ent}_{m_h}(\nu)=+\infty$.
The theorem of von Renesse and Sturm states that, for a real number $K$, the tensor inequality
\begin{align*}
\operatorname{Ric}^h_q(w,w)\ge K h_q(w,w)
\end{align*}
for every $q\in N$ and every $w\in T_qN$ is equivalent to the following entropy convexity property: whenever $\nu_0,\nu_1\in\mathcal P_2(N)$ have finite entropy, there is a constant-speed $W_{2,h}$-geodesic $(\nu_t)_{t\in[0,1]}$ joining them such that, for every $t\in[0,1]$,
\begin{align*}
\operatorname{Ent}_{m_h}(\nu_t)\le (1-t)\operatorname{Ent}_{m_h}(\nu_0)+t\operatorname{Ent}_{m_h}(\nu_1)-\frac K2 t(1-t)W_{2,h}(\nu_0,\nu_1)^2.
\end{align*}
This is the exact coefficient convention used in the present theorem. The theorem is quoted as an external standard theorem because its proof is the original von Renesse--Sturm localization and volume-distortion argument; reproving that argument here would duplicate the external characterization rather than prove a separate elementary lemma.
[/guided]
[/step]
[step:Verify that the present objects match the theorem]
Apply the external theorem with $N:=M$ and $h:=g$. The hypotheses required there are satisfied because the present statement assumes that $(M,g)$ is complete, connected, finite-dimensional, smooth, and without boundary. Under this substitution, $d_h=d_g$, $m_h=\operatorname{vol}_g$, $W_{2,h}=W_2$ by the defining infimum over couplings, and $\operatorname{Ent}_{m_h}=\operatorname{Ent}_{\operatorname{vol}_g}$ by the displayed definition of the extended entropy. Therefore the two conditions in the external theorem are exactly the two conditions listed in the present statement.
[/step]
[step:Obtain the implication from Ricci lower bound to entropy convexity]
Assume that
\begin{align*}
\operatorname{Ric}_p(v,v)\ge K g_p(v,v)
\end{align*}
for every $p\in M$ and every $v\in T_pM$. By the verification in the preceding step, this is precisely the Ricci lower bound hypothesis in the von Renesse--Sturm theorem with $N=M$ and $h=g$. Hence, for every $\mu_0,\mu_1\in\mathcal P_2(M)$ satisfying $\operatorname{Ent}_{\operatorname{vol}_g}(\mu_0)<\infty$ and $\operatorname{Ent}_{\operatorname{vol}_g}(\mu_1)<\infty$, the theorem supplies a constant-speed $W_2$-geodesic $(\mu_t)_{t\in[0,1]}$ from $\mu_0$ to $\mu_1$ such that, for every $t\in[0,1]$,
\begin{align*}
\operatorname{Ent}_{\operatorname{vol}_g}(\mu_t)\le (1-t)\operatorname{Ent}_{\operatorname{vol}_g}(\mu_0)+t\operatorname{Ent}_{\operatorname{vol}_g}(\mu_1)-\frac K2 t(1-t)W_2(\mu_0,\mu_1)^2.
\end{align*}
This proves the displacement convexity condition in the statement.
[/step]
[step:Obtain the implication from entropy convexity to Ricci lower bound]
Assume conversely that the stated displacement convexity condition holds. By the identification of notation above, this is exactly the entropy convexity hypothesis in the von Renesse--Sturm theorem for the manifold $(M,g)$, the distance $d_g$, and the measure $\operatorname{vol}_g$. The reverse implication of that theorem gives
\begin{align*}
\operatorname{Ric}_p(v,v)\ge K g_p(v,v)
\end{align*}
for every $p\in M$ and every $v\in T_pM$. This proves the Ricci lower bound condition in the statement.
[guided]
Now we start from the entropy side. The assumption is not merely an informal convexity statement; it has the same endpoints, same geodesic existence quantifier, and same coefficient as the von Renesse--Sturm theorem stated above. More explicitly, for every finite-entropy pair $\mu_0,\mu_1\in\mathcal P_2(M)$, the hypothesis provides a constant-speed geodesic $(\mu_t)_{t\in[0,1]}$ in $(\mathcal P_2(M),W_2)$ satisfying
\begin{align*}
\operatorname{Ent}_{\operatorname{vol}_g}(\mu_t)\le (1-t)\operatorname{Ent}_{\operatorname{vol}_g}(\mu_0)+t\operatorname{Ent}_{\operatorname{vol}_g}(\mu_1)-\frac K2 t(1-t)W_2(\mu_0,\mu_1)^2
\end{align*}
for every $t\in[0,1]$.
The external theorem requires exactly this hypothesis after the substitution $N=M$, $h=g$, and $m_h=\operatorname{vol}_g$. The preceding verification shows that $W_{2,h}$ becomes $W_2$ and that $\operatorname{Ent}_{m_h}$ becomes $\operatorname{Ent}_{\operatorname{vol}_g}$. Therefore all hypotheses of the reverse direction of the von Renesse--Sturm theorem are satisfied. Its conclusion is the pointwise tensor inequality
\begin{align*}
\operatorname{Ric}_p(v,v)\ge K g_p(v,v)
\end{align*}
for every $p\in M$ and every $v\in T_pM$. This is exactly the Ricci lower bound condition in the present theorem.
[/guided]
[/step]
[step:Combine the two directions]
The forward direction proves that the pointwise Ricci tensor lower bound implies the stated weak $K$-displacement convexity of $\operatorname{Ent}_{\operatorname{vol}_g}$. The reverse direction proves that the same weak $K$-displacement convexity implies the pointwise Ricci tensor lower bound. Hence the two conditions are equivalent.
[/step]
