Corollary 32 ([Distance to the affine class]). Fix δ as above, a function ε(r)≥0 tending to zero, a nonempty compact set Z⊂ U_δ , and η>0 . Uniformly over the fixed family of map pairs, all x∈G , and all deterministic k_0∈Br((1+δ) C,e^…

For a continuous function on Z , its distance in the supremum norm to the set of restrictions of Aπ+b is a continuous, 1 -Lipschitz function of that continuous function. The affine restriction set is closed: take a subsequence with fixed A , and evaluate at one point of Z to see that the offsets converge. If failed, there would be a sequence of maps, basepoints, and deterministic allowed suffixes for which the displayed probability is bounded below by a positive constant. Profile compactness supplies a subsequential law limit. Proposition 31 ([Affine profile limits]) supports that limit on the affine class. The event whose probability is bounded below is closed, so the Portmanteau inequality is a contradiction. For an independent random suffix, condition on its value. The bad probability is an average of deterministic bad probabilities, and hence is bounded by the same supremum, which tends to zero. For a compact exhaustion of U_δ , the zero sets of the continuous affine-distance functions are closed. For each compact in the exhaustion and each η>0 , the Portmanteau inequality for the open event that this distance exceeds η shows that its limiting probability is zero. Thus every limiting law is supported on functions affine on all these compact sets. Use a constant-label subsequence and evaluate at a fixed point to see that their labels and offsets are compatible across the exhaustion; equivalently, the class of functions affine on the entire domain is closed in C_ loc ( U_δ,W) . The function ε(r) was fixed before taking the limit. This assertion is not a supremum over all possible subexponential envelopes at one finite scale. Nor does it assert concentration near one particular pair (A,b) : different affine profiles may still occur in the limiting law.