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

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

>2. coordinates of $p$ are $b$ bits in length with $b\leq C\log\log n$ in bit length at a fixed $C>0$

>3. dimension $p$ satisfies upper bound $p\leq\frac{C'{\log n}}b$ at a fixed $C'>0$

>4. closed in the sense if $vol(p)=ab$ and $a,b\in\mathbb N_{>1}$ then $p=p_1\star p_2$ 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>