Generative Structural Theory v1.40 — Unit-Budget Reachability of Bidirected One-Ended Caterpillars 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, v1.39 showed that from every starting stage and every inclusion-minimal strongly connected initial structure, deletion budget 1 reaches the bidirected ray on 0, 1, 2, ..., with equal lower and upper limits. Here the target is the bidirected tree P_T of a tree T with node set {0, 1, 2, ...}, every node on finitely many edges: the relation with a pair in each direction along every edge of T. For every such tree, budget 0 reaches P_T from a start exactly when the initial structure is P_T restricted to the nodes of that stage and T restricted to the nodes of every later stage is connected (Proposition Z). For one-ended caterpillars, the trees made of a one-way infinite path, the spine, and leaves attached to its nodes, finitely many at each, budget 1 reaches P_T from every starting stage and every initial structure, with lower and upper limits both P_T, so that the membership of every pair settles (Theorem C); for these trees the least budget is therefore 0 or 1, as Proposition Z decides. The construction keeps a settled bidirected subtree and one remaining relation that meets it in a single node, the front; it renews a hub of the remaining relation with one new node per step, takes out of the remaining relation first the leaves of the front and then the front itself, each joined to the settled part by a pair in each direction, and deletes at most one pair at each stage. On the same finite node set a bidirected tree cannot be turned under budget at most 1 into another inclusion-minimal strongly connected relation (Lemma O). Trees with two or more ends, or with branches of depth two or more, are not treated. No novelty is claimed; the proofs are given by hand, with no machine enumeration or computation.