Now asked at MO.
For $\mathcal{R}=(R_i)_{i\in\mathbb{N}}$ a sequence of countable commutative ordered rings such that $R_i$ is an sub-ordered ring of $R_{i+1}$ and $R_0=\mathbb{Z}$, let $$Poly_\mathcal{R}=\{\sum_{i<n}a_ix^i\in (\bigcup_{i\in\omega}R_i)[x]: a_i\in R_i\}$$ be the set of polynomials whose $i$th term comes from the $i$th ring in the sequence. $Poly_\mathcal{R}$ naturally inherits the structure of an ordered ring, namely by taking as the set of positive elements all those $p$ whose leading coefficient is positive in the sense of the appropriate $R_i$. For example, we'll have $x>1$ since the leading coefficient of $x-1$ is $1$ which is positive.
The set $Poly_\mathcal{R}^+$ of nonnegative elements of $Poly_\mathcal{R}$ is a nonstandard model of Robinson's $\mathsf{Q}$. Unsurprisingly the theory of $Poly^+_\mathcal{R}$ depends on $\mathcal{R}$. I'm interested in the "generic" behavior of such models.
To make this precise, let $\mathfrak{X}$ be the space of all such ordered ring sequences topologized by taking as basic open sets of those of the form $$[(R_i)_{i<n}]:=\{\mathcal{S}=(S_i)_{i\in\omega}: \forall i<n(S_i=R_i)\}$$ for each finite sequence of ordered rings $(R_i)_{i<n}$. (Fine, this $\mathfrak{X}$ is a proper-class-sized space; do any of the usual tricks to fix this.) Say that a sentence $\varphi$ in the language of Robinson arithmetic is generically true (resp. false) iff $\{\mathcal{R}: Poly_\mathcal{R}^+\models\varphi\}$ is comeager in $\mathfrak{X}$ (resp. meager).
For example, "Every number is either even or odd" is neither generically true nor generically false. On the other hand, the more complicated sentence
$(*)\quad$ "There is some $k$ such that for all $n>k$ there is some $m<k$ such that $n+m$ is even"
is generically true. Given a (code for a) basic open set $(R_i)_{i<n}$, consider the extension $(S_i)_{i<n+1}$ with $S_i=R_i$ for $i<n$ and $S_n$ be some ring containing $R_{n-1}$ as a subring in which $2x=1$ has a solution. Then every $\mathcal{R}\in[(S_i)_{i<n+1}]$ has $Poly_\mathcal{R}^+\models(*)$: just take $k=x^n$. (Note that this is nontrivial since $Poly_{(\mathbb{Z},\mathbb{Z},\mathbb{Z}, ...)}^+\models\neg(*)$.)
In general, many basic facts about modular arithmetic have "perturbed" versions a la $(*)$ above which are generically true. I'm interested in unpacking this more. Suppose we have a sentence of the form $$\varphi\equiv\forall x_1,...,x_n\exists y_1,...,y_k\theta(x_1,...,x_n,y_1,...,y_k)$$ with $\theta$ quantifier-free. Define the perturbed version of $\varphi$, which I'll denote by "$\varphi^\circ$," as: $$\exists u\forall x_1,...,x_n\exists x_1',...,x_n'\mbox{ such that }$$ $$\bigwedge_{i\le n}\lfloor{x_i\over u}\rfloor\le x_i'\le x_iu\quad\mbox{ and }$$ $$\quad\exists y_1,...,y_k(\theta(x_1', ..., x_n', y_1,...,y_k)).$$
The idea is that $\varphi^\circ$ doesn't require $\exists\overline{y}\theta$ to hold on every tuple of $x$s; we're allowed to perturb the inputs by up to a fixed factor $u$.
- Note that by "disjointifying" the variables involved we have in an appropriate sense that the perturbation of a conjunction is the conjunction of the perturbations. In the original version of this question I didn't notice that variable-disjointification did this, and since commutation with conjunctions seems an obviously desirable property I used a much nastier notion of perturbation which I now realize was silly.
My question is:
Suppose $\varphi$ is an $\forall\exists$-sentence in the language of Robinson arithmetic and $\mathbb{N}\models\varphi$. Is $\varphi^\circ$ generically true?
(We can also consider "perturbed versions" of higher-complexity sentences, but the $\forall\exists$-case seems already interesting. However, the particular question above is boring for higher-complexity sentences: "There is some $k$ such that for every $n$ there is some $m\in (\lfloor {n\over k}\rfloor, nk)$ such that $m$ is not even" is generically false.)