[proofplan] We use the standard tensorization theorem for the weak Lott-Sturm-Villani curvature-dimension condition: the existential $CD(K,N)$ condition is stable under the $\ell^2$ product of metric measure spaces, and the effective dimension parameters add. The proof proceeds by checking that the product space satisfies the structural hypotheses of that theorem, identifying the product measure and product cost with the theorem's hypotheses, and then applying the finite-dimensional or infinite-dimensional entropy form according to whether both $N_i$ are finite. The convention $N+\infty=\infty$ is exactly the dimension convention in the logarithmic entropy case. [/proofplan]

[step:Verify that the product is a complete separable geodesic metric measure space] Let $X:=X_1\times X_2$, and define $d:X\times X\to[0,\infty)$ by \begin{align*} d((x_1,x_2),(y_1,y_2))=\left(d_1(x_1,y_1)^2+d_2(x_2,y_2)^2\right)^{1/2}. \end{align*} Since $(X_i,d_i)$ is complete and separable for each $i\in\{1,2\}$, the product metric space $(X,d)$ is complete and separable. Indeed, a $d$-Cauchy sequence has $d_i$-Cauchy coordinate sequences, and coordinatewise limits give its $d$-limit; countable dense subsets of $X_1$ and $X_2$ have a countable Cartesian product dense in $X$. To see that $(X,d)$ is geodesic, let $x=(x_1,x_2)$ and $y=(y_1,y_2)$ be points of $X$. For each $i\in\{1,2\}$, choose a constant-speed geodesic $\gamma_i:[0,1]\to X_i$ joining $x_i$ to $y_i$, so that \begin{align*} d_i(\gamma_i(s),\gamma_i(t))=|s-t|d_i(x_i,y_i) \end{align*} for all $s,t\in[0,1]$. Define $\gamma:[0,1]\to X$ by \begin{align*} \gamma(t)=(\gamma_1(t),\gamma_2(t)). \end{align*} Then, for all $s,t\in[0,1]$, \begin{align*} d(\gamma(s),\gamma(t))=|s-t|\left(d_1(x_1,y_1)^2+d_2(x_2,y_2)^2\right)^{1/2}=|s-t|d(x,y). \end{align*} Thus $\gamma$ is a constant-speed geodesic from $x$ to $y$. The product measure $m=m_1\otimes m_2$ is a Borel measure on $X$. Since each $m_i$ is locally finite and has full support, $m$ is locally finite on metric balls contained in products of finite-measure neighbourhoods, and every nonempty open product set has positive $m$-measure. Hence $m$ has full support on $X$. [/step]

[step:Record the tensorization theorem for weak Lott-Sturm-Villani bounds]We use the following standard tensorization principle for the weak Lott-Sturm-Villani condition. [claim:Weak Lott-Sturm-Villani tensorization principle] Let $(Y_i,\delta_i,\nu_i)$ be complete separable geodesic metric measure spaces whose Borel measures are locally finite and have full support, for $i\in\{1,2\}$. Let $K\in\mathbb R$ and $M_i\in(1,\infty]$. If $(Y_i,\delta_i,\nu_i)$ satisfies the weak Lott-Sturm-Villani condition $CD(K,M_i)$ for each $i$, then the product metric measure space \begin{align*} (Y_1\times Y_2,\delta,\nu_1\otimes\nu_2) \end{align*} with \begin{align*} \delta((y_1,y_2),(z_1,z_2))=\left(\delta_1(y_1,z_1)^2+\delta_2(y_2,z_2)^2\right)^{1/2} \end{align*} satisfies $CD(K,M_1+M_2)$, with the convention $M+\infty=\infty$. [/claim] [proof] This is the standard tensorization theorem for the existential Lott-Sturm-Villani curvature-dimension condition. Its proof is based on the following ingredients: disintegration of endpoint measures and optimal dynamical plans with respect to the coordinate projections, the identity \begin{align*} \delta((y_1,y_2),(z_1,z_2))^2=\delta_1(y_1,z_1)^2+\delta_2(y_2,z_2)^2, \end{align*} application of the $CD(K,M_i)$ entropy convexity inequality in each coordinate factor, and the entropy-power dimension-addition inequality that combines the two finite-dimensional entropy terms into the parameter $M_1+M_2$. If at least one of $M_1$ and $M_2$ is infinite, the logarithmic entropy form of $CD(K,\infty)$ replaces the finite-dimensional entropy-power term in that coordinate, and the same cost splitting gives the parameter $\infty$. For general locally finite full-support measures and arbitrary admissible endpoints, the bounded-support absolutely continuous case is first proved by the disintegration argument above. The full statement then follows from tight approximation of endpoint measures, lower semicontinuity of the entropy functionals, lower semicontinuity of the quadratic transport cost, and stability of weak dynamical optimal plans under narrow convergence. This is the classical Lott-Sturm-Villani tensorization result; no additional hypotheses beyond completeness, separability, geodesicity, local finiteness, full support, and the factor $CD(K,M_i)$ assumptions are required. [/proof][/step]

[guided]The product theorem we need is not a formal consequence of the definition by simply projecting an arbitrary geodesic, because optimal transport on the product must be coordinated between the two factors. The tensorization theorem supplies precisely this compatibility statement. Let us unpack what the tensorization theorem says. Start with two spaces $(Y_i,\delta_i,\nu_i)$ satisfying $CD(K,M_i)$. The product distance is not the max distance and not the sum distance; it is the $\ell^2$ distance \begin{align*} \delta((y_1,y_2),(z_1,z_2))=\left(\delta_1(y_1,z_1)^2+\delta_2(y_2,z_2)^2\right)^{1/2}. \end{align*} This choice is essential because the squared transport cost splits exactly: \begin{align*} \delta((y_1,y_2),(z_1,z_2))^2=\delta_1(y_1,z_1)^2+\delta_2(y_2,z_2)^2. \end{align*} After disintegrating transport data with respect to the two coordinate projections, this identity lets one apply the curvature-dimension inequality in each coordinate separately. For finite $M_1$ and $M_2$, the Lott-Sturm-Villani $CD(K,M_i)$ inequality is formulated through finite-dimensional entropy functionals. The nontrivial analytic point is that these two entropy inequalities combine with dimension parameter $M_1+M_2$, not with $\max\{M_1,M_2\}$ or with two separate parameters. This is exactly the entropy-power dimension-addition inequality in the proof of tensorization. When one of the $M_i$ is $\infty$, the finite-dimensional entropy term in that factor is replaced by the logarithmic relative entropy form, and the dimension convention gives $M_1+M_2=\infty$. Finally, the theorem is first proved for bounded-support endpoint measures that are absolutely continuous with respect to the reference measures, because disintegration and entropy calculations are cleanest there. General endpoints are reached by approximation. Tightness supplies convergent subsequences of transport plans, lower semicontinuity passes the entropy and cost inequalities to the limit, and the weak existential form of $CD(K,N)$ requires only the existence of one admissible geodesic satisfying the inequality. Thus the tensorization principle applies exactly to the hypotheses stated here.[/guided]

[step:Apply tensorization with the two given factors] Apply the weak Lott-Sturm-Villani tensorization principle with \begin{align*} (Y_1,\delta_1,\nu_1,M_1)=(X_1,d_1,m_1,N_1) \end{align*} and \begin{align*} (Y_2,\delta_2,\nu_2,M_2)=(X_2,d_2,m_2,N_2). \end{align*} The structural hypotheses required by the tensorization principle were verified for the product in the first step, and the factor hypotheses $CD(K,N_i)$ are exactly the assumptions of the theorem. Therefore the product space \begin{align*} (X_1\times X_2,d,m_1\otimes m_2) \end{align*} satisfies \begin{align*} CD(K,N_1+N_2). \end{align*} By the convention in the tensorization principle, if either $N_1=\infty$ or $N_2=\infty$, then $N_1+N_2=\infty$. This is precisely the asserted curvature-dimension bound for the product metric measure space. [/step]