Can multiple Boolean variables and equations be converted to a single integer variable and multiple modulo equations? e.g. let $x,y,z \in \mathbb{B}$ (Boolean)  and $w \in \mathbb{Z}$ (integers)  and $p,q,r \in \mathbb{P}$ (primes)
For $x$ let $(0,1)$ be represented by integers $(\overline{a},a)$ mod p
For $y$ let $(0,1)$ be represented by integers $(\overline{b},b)$ mod q
For $z$ let $(0,1)$ be represented by integers $(\overline{c},c)$ mod r
The Boolean relationships could be translated as:
$(w-\overline{a})(w-a) \equiv 0$ mod $p$
$(w-\overline{b})(w-b) \equiv 0$ mod $q$
$(w-\overline{c})(w-c) \equiv 0$ mod $r$
A logical AND equation could be translated as:
$x \wedge y = True \Longrightarrow (w-a)(w-b) \equiv 0$ mod $pq$
Is there a way to represent an OR equation wrt to $w$?
$x \vee y = True \Longrightarrow ?$
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A little experimenting:
$y = \overline{x} \Longrightarrow  (w-a)(w-\overline{b}) + (w-\overline{a})(w-b) \equiv (\overline{a}-a)(b-\overline{b}) $ mod $pq$
This is not a sufficient condition but it is a necessary condition.
An inverter is a key building block.
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Thanks to @Rob Arthan and @user326210 I think I have a working solution.
The objective is to transform multiple Boolean variables and multiple Boolean equations
into one integer variable and multiple modulo equations (focusing on 3 SAT).
Let the Boolean variables be $x_k \in \mathbb{B}$.
The equation $x_k^2 = x_k$ ensures that $x_k = 0 \ or \ 1$
For each $x_k$ let there be a variable $a_k$ and unique prime $p_k$such that:
$x_k \equiv a_k$ mod $p_k$
Let $w \in \mathbb{Z}$. be an integer such that:
$a_k \equiv w$ mod $p_k$
The Boolean equation $x_k^2 = x_k$ is transformed to:
$w^2 \equiv w$ mod $p_k$
The NOT operation is given by:
$\overline{a_k} \equiv 1-a_k \equiv 1-w$ mod $p_k$
Translating an  AND equation $x_1 \wedge x_2 \wedge \overline{x_3}$ :
Consider the term $p_2 p_3 w + p_1 p_3w + p_1 p_2 (1-w)$ mod $p_1 p_2 p_3$
When a variable does not solve the AND equation its term drops out ($\equiv 0$ mod $p_1 p_2 p_3$).
When it is a solution it leaves a residue e.g. $p_2 p_3,  p_1 p_3, p_1 p_2$.
$p_2 p_3 w + p_1 p_3w + p_1 p_2 (1-w) \equiv p_2 p_3  + p_1 p_3 + p_1 p_2 $ mod $p_1 p_2 p_3$
Variables are translated to $w$ , inverted variables $\overline{x_k}$ are translated to $(1-w)$.
Translating an OR equations $x_1 \vee x_2 \vee \overline{x_3}$:
Consider the term $p_2 p_3 w + p_1 p_3w + p_1 p_2 (1-w)$ mod $p_1 p_2 p_3$
When a variable does not solve the OR equation its term drops out ($\equiv 0$ mod $p_1 p_2 p_3$).
When it is a solution it leaves a residue e.g. $p_2 p_3,  p_1 p_3, p_1 p_2$.
There are 7 combinations of residues to have one or more solutions $r_1 \dots r_7$.
Let $t = \alpha w + \beta $  represent the term ...
The OR equation is translated to:
$(t-r_1) \dots (t-r_7) \equiv 0$ mod $p_1 p_2 p_3$
 A: I am not sure exactly what you're doing with this notation, but perhaps the following boolean identity will help:
$$a \vee b = \overline{\overline{a}\wedge \overline{b}} $$

Here is a related question which might also help: can you start with a collection of boolean variables $x_1, \ldots, x_k$ and a boolean formula $\varphi(x_1, x_2, \ldots, x_k)$, then convert the formula into an equivalent formula $\psi(n_1, n_2, \ldots, n_k)$ whose solutions are analogous (with the obvious correspondence that $0\leftrightarrow \mathtt{true}, 1\leftrightarrow \mathtt{false}$)?
For example, the boolean formula might be $\varphi(x_1, x_2, x_3) = (x_1\wedge x_2) \vee \overline{x_3}$ and you want an integer formula $\psi$ which has the same solutions.
To convert a boolean formula $\varphi$ into an integer equation $\psi$, you only need to perform the following substitutions:
$$\begin{align*}
a \wedge b \quad\longrightarrow\quad& a\cdot b\\
a \vee b \quad\longrightarrow\quad& a + b - a\cdot b\\
\overline{a} \quad\longrightarrow\quad& (1-a)\\
\end{align*}$$
(This is called arithmetization.)  Finally, if you want to ensure that there are no "extra" solutions (that is, if you want to force all of the integer variables to take on values of 0 and 1), you can also require the equations:
$$x_i^2 = x_i$$
for each of the variables $x_i$.
