MOSFET goes into saturation mode much later
You forgot about RdsOn * Ids.

Since Vds rises with Ids*RdsOn and Ids=Vds/RdsOn ,

I think linear Saturation is better defined as ;

\$V_{DS(sat)} &gt;= V_{GS}  - V_t  + I_{DS}*R_{dsON}\$

You may sub RdsOn with another equation and reduce more.

Then $$I_{DS(sat)}&lt;= k V_{DS}^2$$

 REF