PRH | Essay | 7.17 • From Hadele–Hidele Systems to Toric Frobenioids

We give a clean way to pass from an analytic Hadele-Hidele system (a unitary model of the adelic $a x+b$ action with a torus/Floquet closure of the log-scale) to a purely combinatorial, torus-graded Frobenioid that keeps only degrees, the placement of prime powers on the $u$-circle, and the circle holonomy. In short: we forget addition and remember the valuation/monodromy layer. We construct a small graded category $\mathbf{T F}(\mathcal{O})$ and a canonical functor

$$
\mathrm{T}: \mathcal{O} \longmapsto\left(\mathbf{T F}(\mathcal{O}), \operatorname{deg}, \mathcal{L}_\alpha\right),
$$

functorial under unitary equivalences and degree-preserving intertwiners, compatible with Hecke twists, and-over function fields with circle length $L=\log q$-an equivalence with the Frobenioid of effective divisors. No Fourier/Poisson identity is used; the transfer is extracted from operators and torus holonomy.