[proofplan]
We prove the weak $CD(K,N)$ inequality by reducing it to the weighted Jacobian comparison along optimal transport geodesics. For bounded-support absolutely continuous endpoints, optimal transport is represented almost everywhere by a map $T$ and the interpolating maps $F_t$ move points along minimizing geodesics from $x$ to $T(x)$. The weighted change-of-variables formula expresses the density $\rho_t$ through the weighted Jacobian of $F_t$. The curvature bound $\operatorname{Ric}_{V,N}\ge Kg$ enters exactly through the standard weighted transport-Jacobian comparison theorem, whose scalar inequality is then integrated against the source measure.
[/proofplan]
[step:Choose an optimal transport map and its interpolating geodesic maps]
Let $\mu_0=\rho_0m$ and $\mu_1=\rho_1m$ be bounded-support Borel probability measures on $M$ that are absolutely continuous with respect to $m$. Since $m=e^{-V}\operatorname{vol}_g$ with $e^{-V}>0$ smooth, $\mu_0$ is also absolutely continuous with respect to $\operatorname{vol}_g$.
We use the standard optimal-map theorem on complete Riemannian manifolds for absolutely continuous source measures (citing a result not yet in the wiki: existence and uniqueness almost everywhere of the quadratic-cost optimal transport map on complete Riemannian manifolds). It gives a Borel map
\begin{align*}
T:M\to M
\end{align*}
such that $T_{\#}\mu_0=\mu_1$. Let $\operatorname{id}_M:M\to M$, $x\mapsto x$, denote the identity map on $M$, and define the plan
\begin{align*}
\pi:=(\operatorname{id}_M,T)_{\#}\mu_0
\end{align*}
which is optimal for the cost $d_g^2$.
For $\mu_0$-a.e. $x\in M$, choose the unique constant-speed minimizing geodesic
\begin{align*}
\gamma_x:[0,1]\to M
\end{align*}
with $\gamma_x(0)=x$ and $\gamma_x(1)=T(x)$. Define the interpolating maps
\begin{align*}
F_t:M\to M,\qquad F_t(x):=\gamma_x(t).
\end{align*}
For every $t\in[0,1]$, define
\begin{align*}
\mu_t:=(F_t)_{\#}\mu_0.
\end{align*}
By construction, $(\mu_t)_{t\in[0,1]}$ is the $W_2$-geodesic induced by the optimal dynamical plan concentrated on the geodesics $\gamma_x$.
[/step]
[step:Express the intermediate densities using weighted Jacobians]
For each $t\in[0,1]$, write $\mu_t=\rho_t m$ for the density of $\mu_t$ with respect to $m$. The standard transport change-of-variables theorem applies on the full $\mu_0$-measure set where $F_t$ is differentiable and the relevant transport Jacobians exist (citing a result not yet in the wiki: a.e. Jacobian equation for Riemannian optimal transport maps).
For such an $x$, define the Riemannian volume Jacobian map
\begin{align*}
J_t:M \to [0,\infty), \qquad x \mapsto \det d(F_t)_x
\end{align*}
where $d(F_t)_x:T_xM\to T_{F_t(x)}M$ is the differential of $F_t$ at $x$ and the determinant is computed using $g_x$ and $g_{F_t(x)}$. Define the weighted Jacobian map
\begin{align*}
J_t^m:M \to [0,\infty), \qquad x \mapsto e^{-V(F_t(x))+V(x)}J_t(x)
\end{align*}.
The weighted change-of-variables formula gives, for $\mu_0$-a.e. $x$ and every $t\in[0,1]$,
\begin{align*}
\rho_t(F_t(x))J_t^m(x)=\rho_0(x).
\end{align*}
In particular, using $F_1=T$,
\begin{align*}
\rho_1(T(x))J_1^m(x)=\rho_0(x)
\end{align*}
for $\mu_0$-a.e. $x$.
[guided]
The reason for introducing $J_t^m$ rather than only the Riemannian Jacobian $J_t$ is that the densities $\rho_t$ are taken with respect to the weighted measure $m$, not with respect to $\operatorname{vol}_g$. Since
\begin{align*}
d m=e^{-V}\,d\operatorname{vol}_g,
\end{align*}
a change of variables through $F_t$ picks up the usual volume distortion $J_t(x)$ and also the ratio between the weights at $x$ and at $F_t(x)$. This is exactly
\begin{align*}
J_t^m(x)=e^{-V(F_t(x))+V(x)}J_t(x).
\end{align*}
The transport change-of-variables theorem says that mass is conserved along the map $F_t$. Written with respect to $m$, this conservation law is
\begin{align*}
\rho_t(F_t(x))J_t^m(x)=\rho_0(x)
\end{align*}
for $\mu_0$-a.e. $x$. At time $t=1$, the map $F_1$ is $T$, so the same identity gives
\begin{align*}
\rho_1(T(x))J_1^m(x)=\rho_0(x).
\end{align*}
These two identities are the algebraic bridge from pointwise Jacobian comparison to entropy convexity. Once we control $J_t^m(x)$ from below in terms of $J_1^m(x)$, we can rewrite that lower bound as an inequality involving $\rho_t(F_t(x))^{-\frac1N}$, $\rho_0(x)^{-\frac1N}$, and $\rho_1(T(x))^{-\frac1N}$.
[/guided]
[/step]
[step:Apply weighted Jacobian comparison along transport geodesics]
For $\mu_0$-a.e. $x\in M$, define the transport length map
\begin{align*}
\ell:M \to [0,\infty), \qquad x \mapsto d_g(x,T(x))
\end{align*}
We use the standard weighted transport-Jacobian comparison theorem under the lower bound $\operatorname{Ric}_{V,N}\ge Kg$ (citing a result not yet in the wiki: weighted Jacobian comparison under finite-dimensional Bakry-Emery Ricci lower bounds). Its hypotheses are satisfied because $(M,g)$ is complete, $V$ is smooth, $N>n$, and the tensor inequality
\begin{align*}
\operatorname{Ric}_{V,N}\ge Kg
\end{align*}
holds everywhere by assumption.
The comparison theorem states that, for $\mu_0$-a.e. $x\in M$ and every $t\in[0,1]$,
\begin{align*}
(J_t^m(x))^{\frac1N}\ge \tau_{K,N}^{(1-t)}(\ell(x))+\tau_{K,N}^{(t)}(\ell(x))(J_1^m(x))^{\frac1N}.
\end{align*}
Here the term $\operatorname{Hess}_gV$ and the term $-(N-n)^{-1}dV\otimes dV$ arise by differentiating the weighted factor $e^{-V(F_t(x))}$ along $\gamma_x$ and applying the arithmetic-geometric inequality splitting the geometric dimension $n$ from the artificial dimension $N-n$.
[/step]
[step:Convert the Jacobian comparison into the pointwise density inequality]
Fix $t\in[0,1]$. On the full $\mu_0$-measure set where the previous identities hold, the change-of-variables formula gives
\begin{align*}
\rho_t(F_t(x))^{-\frac1N}=\rho_0(x)^{-\frac1N}(J_t^m(x))^{\frac1N}.
\end{align*}
Multiplying the weighted Jacobian comparison by $\rho_0(x)^{-1/N}$ gives
\begin{align*}
\rho_t(F_t(x))^{-\frac1N}\ge \tau_{K,N}^{(1-t)}(\ell(x))\rho_0(x)^{-\frac1N}+\tau_{K,N}^{(t)}(\ell(x))\rho_0(x)^{-\frac1N}(J_1^m(x))^{\frac1N}.
\end{align*}
Using the endpoint identity $\rho_1(T(x))J_1^m(x)=\rho_0(x)$, we obtain
\begin{align*}
\rho_0(x)^{-\frac1N}(J_1^m(x))^{\frac1N}=\rho_1(T(x))^{-\frac1N}.
\end{align*}
Therefore, for $\mu_0$-a.e. $x\in M$,
\begin{align*}
\rho_t(F_t(x))^{-\frac1N}\ge \tau_{K,N}^{(1-t)}(d_g(x,T(x)))\rho_0(x)^{-\frac1N}+\tau_{K,N}^{(t)}(d_g(x,T(x)))\rho_1(T(x))^{-\frac1N}.
\end{align*}
[/step]
[step:Integrate the pointwise inequality to obtain the weak $CD(K,N)$ condition]
Since $\mu_t=(F_t)_{\#}\mu_0$ and $\mu_t=\rho_t m$, the change-of-variables formula for pushforwards gives
\begin{align*}
\int_M \rho_t(z)^{1-\frac1N}\,d m(z)=\int_M \rho_t(F_t(x))^{-\frac1N}\,d\mu_0(x).
\end{align*}
Integrating the pointwise density inequality with respect to $\mu_0$ yields
\begin{align*}
\int_M \rho_t(z)^{1-\frac1N}\,d m(z)\ge \int_M \left[\tau_{K,N}^{(1-t)}(d_g(x,T(x)))\rho_0(x)^{-\frac1N}+\tau_{K,N}^{(t)}(d_g(x,T(x)))\rho_1(T(x))^{-\frac1N}\right]\,d\mu_0(x).
\end{align*}
Because $\pi=(\operatorname{id}_M,T)_{\#}\mu_0$, the last integral is exactly
\begin{align*}
\int_{M\times M}\left[\tau_{K,N}^{(1-t)}(d_g(x,y))\rho_0(x)^{-\frac1N}+\tau_{K,N}^{(t)}(d_g(x,y))\rho_1(y)^{-\frac1N}\right]\,d\pi(x,y).
\end{align*}
Multiplying by $-1$ gives
\begin{align*}
-\int_M \rho_t(z)^{1-\frac1N}\,d m(z)\le -\int_{M\times M}\left[\tau_{K,N}^{(1-t)}(d_g(x,y))\rho_0(x)^{-\frac1N}+\tau_{K,N}^{(t)}(d_g(x,y))\rho_1(y)^{-\frac1N}\right]\,d\pi(x,y).
\end{align*}
This is precisely the weak Lott-Sturm-Villani $CD(K,N)$ inequality for the bounded-support absolutely continuous endpoint measures $\mu_0$ and $\mu_1$. Since the endpoint measures were arbitrary subject to those hypotheses, $(M,d_g,m)$ satisfies the claimed weak $CD(K,N)$ condition.
[/step]
