[proofplan] We apply Talagrand's certifiable lower-tail inequality to the nonnegative integer-valued function $Z$ at the upper median level $m$. The finite product structure, the one-coordinate Lipschitz condition, and the explicit $r$-certificate hypothesis are precisely the hypotheses of that inequality. Since $\mathbb{P}(Z\ge m)\ge 1/2$, dividing Talagrand's two-sided product bound by this median probability gives the asserted one-sided lower-tail estimate. [/proofplan] [step:Record the finite product notation and the upper median event] The theorem is stated on the finite product probability space \begin{align*} (\Omega,\mathcal F,\mathbb P)=\prod_{i=1}^n(\Omega_i,\mathcal F_i,\mathbb P_i), \end{align*} where $n\in\mathbb N$ is the number of coordinates. Thus each point $x\in\Omega$ has coordinate representation $x=(x_1,\dots,x_n)$ with $x_i\in\Omega_i$. Define the measurable upper median event \begin{align*} A := \{x \in \Omega : Z(x) \ge m\}=Z^{-1}(\{m,m+1,m+2,\dots\}). \end{align*} Since $Z$ is measurable and integer-valued, $A\in\mathcal F$. By the upper median assumption, \begin{align*} \mathbb{P}(A) = \mathbb{P}(Z\ge m) \ge \frac12. \end{align*} [/step] [step:Apply Talagrand's certifiable lower-tail inequality] We use Talagrand's certifiable lower-tail inequality for product probability spaces as the external concentration input. In the form needed here, it states that if $Z:\Omega\to\mathbb Z_{\ge 0}$ is $1$-Lipschitz under one-coordinate changes and has certificates of size at most $rs$ for each level $\{Z\ge s\}$, then for every integer $b\ge 1$ and every $u\ge 0$ with $b-u\ge 0$, \begin{align*} \mathbb P(Z\le b-u)\,\mathbb P(Z\ge b)\le \exp\left(-\frac{u^2}{4rb}\right). \end{align*} We apply this inequality with $b:=m$ and $u:=t$. Its hypotheses are satisfied because the theorem assumes that $\Omega$ is a finite product probability space, that changing one coordinate changes $Z$ by at most $1$, and that the level set $\{Z\ge s\}$ has certificates of size at most $rs$ for every integer $s\ge 1$. Therefore \begin{align*} \mathbb P(Z\le m-t)\,\mathbb P(Z\ge m)\le \exp\left(-\frac{t^2}{4rm}\right). \end{align*} Since $\mathbb P(Z\ge m)\ge 1/2$, division by $\mathbb P(Z\ge m)$ gives \begin{align*} \mathbb P(Z\le m-t)\le 2\exp\left(-\frac{t^2}{4rm}\right). \end{align*} [guided] The concentration input is Talagrand's certifiable lower-tail inequality. The theorem applies to a nonnegative integer-valued function on a finite product probability space when two structural conditions hold: first, changing one coordinate changes the value by at most $1$; second, every upper level event $\{Z\ge s\}$ can be forced by revealing at most $rs$ coordinates. These are exactly the two assumptions stated for $Z$. We now choose the level and deviation parameters in Talagrand's inequality. Let $b:=m$ and let $u:=t$. Since $m\in\mathbb Z_{\ge 1}$ and $0\le t\le m$, the parameters are admissible. The inequality gives \begin{align*} \mathbb P(Z\le m-t)\,\mathbb P(Z\ge m)\le \exp\left(-\frac{t^2}{4rm}\right). \end{align*} The median information is used only through the upper median half of the median condition. Namely, the assumption $\mathbb P(Z\ge m)\ge \frac12$ implies \begin{align*} \frac{1}{\mathbb P(Z\ge m)}\le 2. \end{align*} Dividing the previous product bound by $\mathbb P(Z\ge m)$ therefore yields \begin{align*} \mathbb P(Z\le m-t)\le 2\exp\left(-\frac{t^2}{4rm}\right). \end{align*} This is the desired lower-tail estimate. [/guided] [/step] [step:Conclude the lower-tail estimate] The preceding step proves exactly that, for every $0\le t\le m$ with $m-t\ge 1$, \begin{align*} \mathbb{P}(Z \le m-t)\le 2\exp\left(-\frac{t^2}{4rm}\right). \end{align*} This is the asserted lower-tail bound. [/step]