Let $\mathcal{L}=\{0,S,+,\cdot,=,<,X,R,S\}$ be the language of arithmetic with three additional predicate symbols $X(v)$, $R(v,u)$ and $S(v,u)$.

Let $\phi(x),\psi(x,y)$ and $\eta(x,y)$ be formulas in $\mathcal{L}$. Define the formula $\Theta_e(\phi,\psi,\eta)$ in $\mathcal{L}$ for some $e\in \mathbb{N}$ which says:

- $\phi(0)$
- $\psi$ is a partition of $\{x:\phi(x)\}$ into classes of size $e$.
- $\eta$ is a partition of $\{x:\phi(x)\land \neg x=0\}$ into classes of size $e$.

Theorem. For every fix prime number $p,q\in \mathbb{N}$, ${\bf I}\Delta_0(R,S,X)+\Theta_p(R,S,X)+\{\neg \Theta_q(R',S',X'):R',S',X'\in \Delta_0(R,S,X)\}\not \vdash \bot$

Proof. One of the proofs of this theorem can be found here.

Q1. Is there any similar result for counting modulo a composite number? For example is ${\bf I}\Delta_0(R,S,X)+\Theta_7(R,S,X)+\{\neg \Theta_6(R',S',X'):R',S',X'\in \Delta_0(R,S,X)\}$ consistent?

Q2. Suppose that the theory above is consistent. Does it imply separation of complexity classes like $AC_0[6]$ and $NP$?

Thanks in advance.