The coordinate biring of $\mathbf{S}\mathbf{p}\mathbf{e}\mathbf{c}(\mathbb{Z})\mathrm{/}{\mathbb{F}}_1$

We propose to define ${\mathbb{F}}_1$-algebras as integral bi-rings with the co-ring structure being the descent data from $\mathbb{Z}$ to ${\mathbb{F}}_1$. The coordinate bi-ring of $\mathbf{S}\mathbf{p}\mathbf{e}\mathbf{c}(\mathbb{Z})\mathrm{/}{\mathbb{F}}_1$ is then the co-ring of integral linear recursive sequences equipped with the Hadamard product. We associate a noncommutative moduli space to this setting, show that it is defined over ${\mathbb{F}}_1$, and has motive $\prod_{n \geq 0} \frac{s-n}{2 \pi}$.