= Solution
{c}
Use the cardinalhood formula $C$ from part (b), and let a finite flag choose between it and its negation:
$$
\boxed{\varphi_3(x,i):\quad(i=0\land C(x))\lor(i=1\land\neg C(x)).}
$$
Here $0,1$ abbreviate their parameter-free definitions as <Finite von Neumann ordinals>, and both belong to $M$. This is one <first-order formula> with two free variables.
For $\beta=(\omega_1)^M$, part (b) proved $M\models C(\beta)$ and $N\models\neg C(\beta)$. The shared parameters $(\beta,0)$ therefore make $\varphi_3$ true in $M$ and false in $N$, disproving upward <set-theoretic absoluteness>. The shared parameters $(\beta,1)$ make it false in $M$ and true in $N$, disproving downward <set-theoretic absoluteness>. Hence the same formula fails both directions, with explicit witnesses to each failure.
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