On Local Well-posedness of the Periodic Korteweg-de Vries Equation Below $H^{-\frac{1}{2}}(\mathbb{T})$

We utilize a modulation restricted normal form approach to establish local well-posedness of the periodic Korteweg-de Vries equation in $H^s(\mathbb{T})$ for $s>-\frac{2}{3}$. This work creates an analogue of the mKdV result by Nakanishi, Takaoka, and Tsutsumi for KdV, extending the currently best-known result of $s\ge -\frac{1}{2}$ without utilizing the theory of complete integrability.