 Supercompact Sigma-two downward reflection

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 $V_{\kappa }$.