Frobenius Factorization Loci for Gauss Maps of Smooth Plane Curves

We construct canonical closed subschemes representing relative Frobenius factorization of a morphism from a smooth projective curve family, including over nonreduced bases. For Gauss maps of smooth plane curves of degree d in characteristic p, we identify these subschemes with explicit linear coefficient loci. For q = p^e > 2, the eth locus is empty unless q divides d - 1; otherwise it is the smooth open in the space of equations sum_i X_i R_i(X_0^q, X_1^q, X_2^q), of dimension 3 binomial((d-1)/q + 2, 2) - 1. The classification on geometric points is due to Pardini and Homma. Our scheme-level argument uses the pulled-back Euler sequence, Serre duality and a regular-sequence calculation to rule out infinitesimal thickenings. We obtain dimensions and geometric irreducibility of exact-height strata, handle the characteristic-two exception, and give explicit smooth families with height jumps. These are complete results for the specified relative factorization and smooth-plane scopes associated with AIM-ARITHMETIC_GEOMETRY-0050. They do not construct characteristic-only components of space-curve Hilbert schemes or close the full AIM source question. The preprint is AI-assisted, self-audited and unrefereed. Independent review and proof-assistant formalization are not claimed; novelty and absolute priority remain undetermined.