As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a decomposition $K_i(R[T,T^{-1}])\cong K_i(R)\oplus K_{i-1}(R)$ for $R$ regular.

Now turn to the philosophical part, the field $\mathbb{F}_1$. For a root system $\Phi$, the natural candidate for the algebraic group of type $\Phi$ over $\mathbb{F}_1$ is the Weyl group $W(\Phi)$ - the formula for the group order is right as noted by Tits, see also several other MO questions on $\mathbb{F}_1$ such as this or this. Now I want to understand the loop group of this group over $\mathbb{F}_1$. The notation would probably be something like $G(\Phi,\mathbb{F}_1[T,T^{-1}])$ and I apologize if anyone is offended by this abuse of notation. I would expect the affine Weyl group for the affine root system $\tilde{\Phi}$ to be a reasonable candidate.

The above philosophical guessing has some consequences, and I would like to know if the following statements have been considered, and if they are true or false. In the following, I specialize to the case of root systems $A_n$ and $\tilde{A}_n$, just for definiteness. In that case $W(A_n)\cong S_{n+1}$ is the symmetric group, and the affine Weyl group $W(\tilde{A}_n)$ is an extension of $S_{n+1}$ by $\mathbb{Z}^{n}$, where $\mathbb{Z}^n$ is identified with the hyperplane $x_1+\cdots+x_{n+1}=0$ in $\mathbb{Z}^{n+1}$ with the permutation action of $S_{n+1}$. Now here are the questions:

Are there stabilization theorems for the homology of $W(\tilde{A}_n)$? How do these stabilization theorems compare to those for $S_{n+1}$, in particular is the stable range the same or one less? More generally, has the homology of $W(\tilde{A}_n)$ been computed by any chance (this is loosely related to this MO-question on cohomology of the action of $S_n$ on $T_n$).

Provided there are stabilization theorems, have people considered the group completion of $\bigsqcup_n BW(\tilde{A}_n)$ and compared it to the plus-construction space $BW(\tilde{A}_\infty)^+$? A stabilization theorem should imply that the two are weakly equivalent.

Does the fundamental theorem of algebraic K-theory hold in this situation? In other words, can we express the homotopy groups of $BW(\tilde{A}_\infty)^+$ in terms of stable homotopy groups of spheres? Maybe $BW(\tilde{A}_\infty)^+$ even splits as two copies of the sphere spectrum (one copy suitably shifted)?

Of course, similar questions can be formulated for the other classical series, and I would be happy to know the answers in these cases as well.