Lemma 20 ([Uniform conditional test]). Let I and e be disjoint sets of fields with admissible union. For each ν , let h_I be a function of the labels on I with values in T_ x_ν X_ν , and suppose h_I≤B for a constant independent of ν and …

Fix t . Use the coupling in Proposition 16: it gives t matches agreeing on I , each branch marginal at total variation distance at most η_ ν,t from μ_ν , and their e -labels at distance at most η_ ν,t from independent uniform letters, where η_ ν,t →0 . Increasing the error if necessary allows one bound for all these comparisons. Applying the fixed inversion σ preserves this assertion. The vector h_I has the same value in every branch. Form the inductive mean m_t of the branch vectors v_ σ(e_j) . Equation gives, for every coupled tuple, 1t∑_ j=1 ^th_I v_ σ(e_j) _I m_t ≤Bm_t. The last step uses only the norm bound; it does not assume that h_I is independent of the leaves. Under independent uniform leaves, Equation applies by Equations and. Under the coupling, m_t≤M , by convexity of the ball of radius M . Consequently the total variation comparison changes the expected squared norm by at most 2M^2η_ ν,t . The branch comparisons change the expectation of each scalar-product test by at most 2BMη_ ν,t . It follows that 𝔼_ μ_ν h_I v_ σ(e) ≤B√ 1/t+2M^2η_ ν,t +2BMη_ ν,t . First take the upper limit in ν with t fixed, and then let t→∞ . All error bounds used only B and M , so the conclusion is uniform over the action-dependent functions in the statement.