Böhm-out lemma

For a finite path in a Böhm tree, choose $q$ at least the head arity at every preceding node. Replace relevant variables by $U_q=\lambda z_1\cdots z_{q}w.w{z}_1\cdots z_q$. After exposing each node, supply its initial bound variables with $U_q$, pad its $m$ arguments to $q$, and apply the projection $\lambda z_1\cdots z_q.z_i$ selecting the next child. Each step returns that child with accumulated substitutions, so iteration reaches a substitution instance of the chosen subtree. Tupling preserves all children until selection, even when the same variable occurs at several nodes.

Two distinct closed lambda terms in normal form for both beta reduction and eta conversion admit a separating context that maps them respectively to the two Church Booleans. Finite Böhm trees and the Böhm-out lemma isolate their first genuine head disagreement. The qualification about eta conversion matters: distinct beta-normal forms that are eta-equivalent cannot be separated this way.