Background. Let $X$ be an affine variety over a field $k$ of characteristic $0$. We can define then the loop space of $X$, denoted $\mathcal{L}X$, to be the functor taking a $k$-algebra $A$ to the set $X(A((t)))$. This is representable by an ind-scheme (of ind- infinite type). Intuitively this is the mapping space $\operatorname{Map}(D^{*},X)$, where $D^{*}$ is the formal punctured $k$-disk, $\operatorname{spec}k((t))$. This is not quite correct however, indeed for it to be correct we would need $\operatorname{spec}(A)\times D^{*}$ to be $\operatorname{spec}(A((t)))$, which it is not as soon as $A$ is infinite dimensional as a $k$ vector space. Nonetheless viewing it as a mapping space seems reasonable, in particular there is a well defined evaluation map from the universal correspondence. Note that for this to work we cannot naively take $\mathcal{L}X\times D^{*}$, again we have to complete the product. For example if $X=\mathbb{A}^{1}$ with coordinate $x$, then $\mathcal{O}(\mathcal{L}X\times D^{*})$ must contain the two way infinite Laurent series $x(t):=\sum_{i}x_{i}t^{i}$, where the $x_{i}$ are the natural coordinates on loop space. Notice that the Laurent tails tend to $0$ in $\mathcal{O}(\mathcal{L}X)$.
Question. Is there a natural algebro-geometric context in which $X$, $D^{*}$ and $\mathcal{L}X$ all live, so that we genuinely have $\operatorname{Map}(D^{*},X)\cong\mathcal{L}X$? Such a context must handle pro-discrete objects and adic ones in which a uniformiser is inverted. In particular can I use solid $k$-modules to build an appropriate context?
Naive guess. Thanks to Clausen-Scholze, we have a very well behaved symmetric monoidal category of solid $k$-modules. I believe that all the objects above sit naturally in this category, and that the products are as required. For example, if $A$ is a pro-discrete $k$-algebra, then $A\otimes_{\operatorname{solid}}k((t))$ is (as a module at least) the algebra of two way infinite Laurent series whose Laurent tails tend to $0$, according to Solid tensor product of pro-discrete space with Laurent series.
I cannot locate in the Clausen-Scholze work a definition of space modeled on the symmetric monoidal category $\operatorname{Solid}_{k}$, which leads me to believe that either it is a very special case of the analytic rings constructions, or that it is not useful.