```module main {
    // Define state variables for circuit C1
    var y1: boolean;
    var r1: boolean;

    // Define input and state variables for circuit C2
    input x2: boolean;
    var y2: boolean;
    var r2: boolean;

    // Initial state of the variables
    init {
        y1 = false;
        r1 = false;
        y2 = false;
        r2 = false;
    }

    // Transition functions for the circuits
    next(r1) = !r1; // r1 transitions to not r1
    next(y1) = r1; // y1 transitions to current value of r1

    next(r2) = x2 || r2; // r2 transitions to x2 OR current r2
    next(y2) = x2 || r2; // y2 becomes x2 OR current r2

    // Specification to be verified (example)
    specification g1 : assert always (y1 == !r1);
    specification g2 : assert always (y2 == (x2 || r2));

    // Control specification, ensuring correctness of the circuit synchrony and transition logic
    control {
        check;
    }
}
```
In the corrected code above, I have changed `==` to `=` in the `next` assignments, as UCLID5 uses `=` to define the value of a variable in the next state.