= Supercompact Sigma-two downward reflection
{title2=$V\models\sigma\Longrightarrow V_\kappa\models\sigma\quad(\sigma\in\Sigma_2)$}
If $\kappa$ is a <supercompact cardinal>, each true $\Sigma_2$ statement, allowing parameters in $V_\kappa$, holds in $V_\kappa$. Write it as an existential witness followed by a $\Pi_1$ assertion. Choose a supercompact <elementary embedding> whose transitive target contains the witness in $V_{j(\kappa)}$. Downward absoluteness of the universal bounded-matrix assertion puts the witness in that rank segment, and elementarity reflects this to $V_\kappa$.
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