Generative Structural Theory v1.43 — Unit-Budget Reachability of Bidirected Two-Ended Trees with Branches of Depth at Most Two on the Original Z3 Growth Sequence

A growth sequence is a sequence of finite relations R_n on node sets V_n with V_n ⊆ Vₙ₊₁, R_n ⊆ Rₙ₊₁ and every stage strongly connected; a selection sequence chooses at every stage an inclusion-minimal strongly connected spanning sub-relation x_n of R_n, and its deletion budget bounds the number of pairs of x_n that xₙ₊₁ drops; the number of pairs added is not bounded. On the growth sequence of v1.36 Example Z3, whose stage n has the nodes 0, ..., 2n and 2n + 2 and all pairs between them, v1.40, v1.41 and v1.42 showed that from every starting stage and every inclusion-minimal strongly connected initial structure, deletion budget 1 reaches the bidirected tree of any one-ended caterpillar, the bidirected double ray along any bijection from the integers onto {0, 1, 2, ...} and the bidirected tree of any two-ended caterpillar. Here the target is the bidirected tree P_T of a two-ended tree T of depth at most two with node set {0, 1, 2, ...}: a two-way infinite path, the spine, with a finite tree of height at most two rooted at each of its nodes, with no common bound on the sizes of these trees and no condition on the labels or on the order in which the nodes enter. Budget 0 reaches P_T from a given start exactly when the initial structure is P_T restricted to the nodes of that stage and the subtree induced by the nodes of every later stage is connected (Corollary Z_2); at depth two such a node set may have no spine node, being a star at a node of depth one (Lemma C_2, Example F*), so the spine-interval test of v1.42 is sufficient here but not necessary. For every such tree, budget 1 reaches P_T from every starting stage and every inclusion-minimal strongly connected initial structure, with lower and upper limits both P_T, so that the membership of every pair settles (Theorem F); the least budget is therefore 0 or 1, as Corollary Z_2 decides (Corollary F5). The main tool is a node-out that carries a settled subtree (Lemma NC): the node-out of v1.40 runs on a working relation while a settled piece meets it only at the node being moved, with every stage minimal and at most one pair deleted at each step, also when the working relations grow between the steps in the ways allowed; with it a node of depth one is taken out together with its children, the children first (Corollary U). The construction settles the branch of the root, separates a left and a right working relation beside it, and lets the two sides advance in turn; before a branch is settled it waits until all the nodes of that branch have entered, which takes finitely long as every branch is finite. Trees with branches of depth three or more and trees with three or more ends are not treated. No novelty is claimed; the proofs are given by hand, with no machine enumeration or computation.