Finite Free Models and Taylor–Wiles–Kisin Conditions for Framed Deformation Rings — E8 Intelligence Research

FINDING: Deformation rings of Galois representations admit finite free models over local coefficient rings; the Taylor–Wiles–Kisin prime condition controls when the patching argument produces a power series ring (defect zero) versus a more singular framed deformation ring. | MATH: Let R^□ be the framed universal deformation ring of a residual representation ρ̄: G_{F_S} → GL_n(k), k finite of char p. Taylor–Wiles–Kisin condition: for each place v | p, the local deformation problem is "unobstructed" and the local ring R^□_v is a power series ring over O = W(k) in n² + [F_v:Q_p]·n(n−1)/2 variables; global patching yields R^□ ≅ O[[x₁,…,x_g]] / (Taylor–Wiles ideal) with g = dim of the Selmer-type tangent space. Finite free module condition: R^□ is a finite free O-module of rank = |GL_n(k)|-orbit count when R^□ is O-flat and the Kisin variety (the mod-p fiber of the local deformation ring) has the expected dimension. Key numerical invariant: the "defect" δ = dim R^□ − dim expected; δ = 0 ⇔ T Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com