Generative Structural Theory v1.42 — Unit-Budget Reachability of Bidirected Two-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, v1.40 and v1.41 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 and the bidirected double ray along any bijection from the integers onto {0, 1, 2, ...}. Here the target is the bidirected tree P_T of a two-ended caterpillar T with node set {0, 1, 2, ...}: a two-way infinite path, the spine, with finitely many leaves, possibly none, attached to each of its nodes, with no condition on the labels or on the order in which the nodes enter. Budget 0 reaches P_T from a start exactly when the initial structure is P_T restricted to the nodes of that stage and, at every later stage, the indices of the spine nodes present form an interval of integers and every leaf present has its spine node present (Corollary Z_E); Example E* gives a caterpillar with leaves for which this holds at every stage. For every two-ended caterpillar, 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 E); the least budget is therefore 0 or 1, as Corollary Z_E decides (Corollary E5). The construction takes a spine node as root and classifies each node as left, central or right according to its spine index; the central nodes are the root and its leaves. It first settles the leaves of the root then present, while one working relation holds all other nodes, and then separates a left and a right working relation that meet the settled part only at the root (Lemma I). The two sides then advance in turn: at each turn the end of that side moves by one spine node, and the leaves of the new end then present are taken out of the working relation by the node-out of v1.40 and settled; leaves entering later are joined at their entry, and at most one pair is deleted at each stage. Trees with branches of depth two 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.