Formal geometry of free algebras

Let $X\in Sp$, $Free:Sp\to CAlg$ the free algebra functor

On the free algebra $Free(X)$, two notions coincide.

• A. Based loop space at the origin (co-algebraic Koszul dual). ${\Omega }_{0}Spec(Free(X))\simeq Spec(\mathbb{S}{\otimes }_{Free(X)}\mathbb{S})\simeq Free(\Sigma X)$ (Aside: observe this is a way to construct the additive structure on the group-stacks Super-${\mathbb{G}}_a$ (smooth version))
• B. Functions on free loop space (THH) $THH(Free(X))\simeq Free(S^1\otimes X)\simeq Free(\Sigma X)$
  Observe the structure A uses the augmentation, while B does not. So these notions certainly diverge when $Free(X)$ is replaced by a general ring $R$.