As the idea of an absolute geometry over the field with one element $\mathbb{F}_1$ becomes more clear, two approaches seem to have crystallized, being based on different assumptions and going into different directions: on the one hand, there is Borger's, which interprets an $\mathbb{F}_1$-algebra $R$ to be a torsion-free $\lambda$-ring, i.e. a ring $R_0$ equipped with commuting operators $\phi_p$ for each prime number $p$, such that $\phi_p$ restricts to the Frobenius endomorphism on the tensor product $R_0\times \mathbb{F}_p$. On the other hand, there is an ongoing collaboration between Connes and Consani that basically builds on the approach of understanding an $\mathbb{F}_1$-algebra to be a generalization of a pointed monoid, and that has ended up using $\Gamma$-spaces, covariant functors from finite pointed sets to pointed sets, so that an $\mathbb{F}_1$-algebra in this sense is a monoid object in the topos of such functors. Naturally, the question arises as to how those two approaches are related. In https://arxiv.org/abs/2004.08879, Connes and Consani somewhat treat this question, highlighting the fact that each $\lambda$-ring carries an action of the multiplicative semigroup of natural numbers $\mathbb{N}^\times$ and can thus be understood as a ring-valued sheaf on the delooping $B\mathbb{N}^\times$ of that semigroup, or, equivalently, a ring object in the presheaf topos $PSh(B\mathbb{N}^\times)$. However, afterwards they go into some reflections based on cyclic homology whose significance and connection to the current question is not clear to me. Thus, I am asking here:
What exactly is the connection between Borger's and Connes' and Consani's approaches? In particular, can it be rightfully said that Connes' and Consani's approach can subsume Borger's? Can furthermore Connes' and Consani's approach be translated into Borger's?
Furthermore, in Section 9 of https://arxiv.org/abs/1304.6532, the spectrum of the integers as a plain ring is over $\mathbb{F}_1$ identified with the spectrum of the "witty ring" created by the integers, where the "wittification" functor is the right adjoint of the functor that forgets the $\lambda$-structure of $\lambda$-rings, which is understood as base exchange from $\mathbb{F}_1$ to $\mathbb{Z}$, and it is found that no such wittification can have any geometric points, indicating that they behave "very unlike $\mathbb{F}_1$-geometric objects of finite dimension", though they still have points valued in other wittifications. Is there some parallel to this in Connes' and Consani's theory, indicating that the spectrum of the integers should not be thought of as a finite-dimensional object over $\mathbb{F}_1$?
