Sharp logarithmic Sobolev inequalities on free groups by variational comparison

We give a variational comparison proof of the sharp logarithmic Sobolev inequality with constant $2$ for the word-length Poisson semigroup on free group von Neumann algebras. Our approach extends the Lagrange method for cyclic groups in [Yao26] to a framework in finite tracial von Neumann algebras. The inequality was recently established by Xie and Zhang in [XZ26] using a completely different approach.