Lemma 16 ([Bounds for fitted solutions]). For a member v∈_m,C₀ centered at (0,0) or (0,1) , K≥2 , z∈ B_K , and 0<ℓ≤K , one has_ℓ⁺v(z)≤C L(K)⁴, _ℓ^-v(z)≤C L(K)^4. Every optimizing displacement for_Kv(z) or_Kv(z) has length at most C …

By Lemma 15 ([Envelope slopes]), it suffices to bound the two slopes at length K . Suppose an optimizing displacement has length BK with B≥2 . Comparing its value with the competitor having no displacement, and using in a box of size (B+2)K , gives (B^2K)/(2) ≤C BK(L(K)+ B)^2. This estimate holds for the supremum and for the infimum. Consequently B≤C(L(K)+ B)^2≤2C L(K)^2+2C( B)^2 . Choose a fixed B_0 so large that 2C( B)^2≤B/2 for B≥B_0 . The last inequality then yields B≤4C L(K)^2 for B≥B_0 , and the same type of bound holds when B<B_0 . This proves the displacement claim. The optimizer and its time argument now lie in a box of size C K L(K)^2 . Applying there and observing that L(C K L(K)^2)≤C L(K) bounds the absolute values of v there by C K L(K)^4 . Dropping the nonpositive cost in the expressions for the two slopes gives .