Resolution of Singularities in Positive Characteristic: Frobenius-Hasse Towers and Exceptional-History Descent

Abstract

Let k be a perfect field of characteristic p>0. We introduce an object-level construction for canonical strong embedded resolution and principalization over k. The construction program replaces monotonicity of pointwise numerical invariants by a well-founded history of addressed comparison factors. Starting from the differential-integral saturation of a marked Rees algebra, we construct total-Hasse activity packets, filtered coefficient cubes, semilinear Frobenius-Hasse sources, and literal transform data for ordinary permissible blowups. A six-row defect calculus routes local problems to certified surface, toroidal-monomial, binomial, and additive-type procedures, after which clean centre portfolios are serialized and descended on a global nerve. The central structure is a global replacement certificate transporting successor addresses, paid quotients, typed traces, displayed parents, reopening data, and terminal truth across macroblocks. We prove that a complete state equipped with this certificate admits a strict multiset replacement in a single dependent well-founded order; hence the iteration terminates and the exhausted state reconstructs a regular strict transform having normal crossings with the ordered boundary. We further formulate an object-level realization of the certificate through rigid generation, cross-generation no-reset, complete wild-capacity control, a centre-or-typed-exit alternative, structured cofibres, displayed-parent allocation, and literal terminal truth. The program page is also available at website https://sites.google.com/view/positive-char-resolution