Is there a family $\mathcal{P}$ of integral polytopes such that for every $n\in\mathbb N_{>1}$ $\exists p\in\mathcal{P}:vol(p)=n$ and

>1. $\forall q\in\mathcal{P}\backslash\{p\}\quad vol(q)\neq n$

>3. coordinates of $p$ are $O(\log\log n)$ in bit length

>3. dimension of $p$ is $O(\frac{\log n}{\log\log n})$

>4. closed in the sense if $vol(p)=ab$ such that $a,b>1\in\mathbb N_{>1}$ then $p=p_1\star p_2$ for a polytope product $\star$ where $vol(p_1)=a$ and $vol(p_2)=b$ and so if $p\in\mathcal P$ represents a prime by volume it cannot be decomposed?

<s>Is there any way to describe such a family so that given $n\in\mathbb N_{>1}$

>4. $p$ **cannot** be described in polynomial time from $n$?</s>