Noncommutative Oort–Tate Geometry, Divided Frobenius, and Lagrange Defects

Grothendieck asked whether every finite locally free group scheme of order n is annihilated by its n-th power map. Recently, a counterexample of rank 2² was found by A. Mathew working with OpenAI’s Sol. In this note, we give a geometric construction, for every prime p, of a finite flat noncommutative group scheme of rank p² whose p²-power map is nontrivial but whose p³-power map is trivial. The main objective of the leisurely exposition is to present the family as a natural interaction between Oort–Tate geometry, noncommutative deformation theory, skew-primitivity, and the additive cross-effect of Frobenius rather than as a coincidental collection of formulas. We also characterize the base ring as the universal p-local parameter ring on which the prescribed divided-Frobenius equations are exactly skew primitive for the radial coproduct.