Properties of parametric Presburger families

Let $S_t$ be a parametric Presburger family, meaning a family of subsets of $\mathbb{N}^d$ definable over the natural numbers using quantifiers, boolean operations, and inequalities of the form

$$\vec a(t)\cdot \mathbf{x}\le b(t),$$

where $b\in\mathbb{Z}[t]$ and $\vec a\in\mathbb{Z}[t]^d$. Parametric Presburger-family properties conjecture. Properties 1, 2, 3, 3a, 3b, and 4 all hold for $S_t$. These properties respectively concern the eventual quasi-polynomial behavior of solution counts and selected solutions, eventual periodicity of relevant parameter sets, and rational generating functions; the conjecture unifies several quasi-polynomial and generating-function phenomena, while the paper establishes relationships and special cases rather than the full assertion.