We report the first measurement of $\Lambda$ and $\overline{\Lambda}$ transverse polarization inside jets
in unpolarized proton-proton collisions at $\sqrt{s}$ = 200 GeV. Polarization was measured as a function of the jet transverse
momentum ($p_T$), the fraction of the jet momentum carried by $\Lambda$($\overline{\Lambda}$) ($z$), and the transverse
momentum of $\Lambda(\overline{\Lambda})$ relative to the jet axis ($j_T$). Significant polarization was observed, with positive
values for $\Lambda$ and negative values for $\overline{\Lambda}$, showing clear dependencies on the jet $p_T$ and the momentum
fraction $z$. These results provide new insights into the mechanism of $\Lambda$ spontaneous polarization and offer crucial
constraints for the extraction of polarizing fragmentation functions.