math.LO — Logic
Bounded Presentations and Normalization in Arithmetic Reconstruction
We study bounded presentations of selected constructions in absolute anabelian geometry and Frobenioid theory. In bounded Zermelo set theory with Choice, we prove uniform recursion for a common state space and a uniformly supplied transition rule. Under explicit index bounds, membership-table codes place the states and their histories at the same finite type level. We construct coherent comparisons for representative quotients, fiberwise full skeletons, isofibrational base changes, and countable towers, and verify the lifting properties for model Frobenioids and angular presentations. For nonzero numerical observables transforming by positive scalars, compatible normalization exists exactly when the associated character is trivial on every isotropy group. A self-equivalence of the realified arithmetic Frobenioid over preserves degree and every object isomorphism class but does not lift to the unit-trivialized perfection. The structural results give a relative uniform interpretation theorem: the weak theory constructs the bounded presentations and comparison maps, and specified arithmetic recognition theorems identify the resulting structures.