Generative Structural Theory v1.44 — Unit-Budget Reachability of Bidirected Two-Ended Trees with Arbitrary Finite Branches 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-v1.43 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, ...}, the bidirected tree of any two-ended caterpillar and that of any two-ended tree with branches of depth at most two. Here the target is the bidirected tree P_T of a two-ended tree T with finite branches with node set {0, 1, 2, ...}: a two-way infinite path, the spine, with a finite tree rooted at each of its nodes, with no common bound on the sizes or on the heights 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_G); a nonempty finite node set induces a connected subtree iff its spine nodes form a nonempty interval of the spine and each of its other nodes has its parent in the set, or it has no spine node and lies in D(v_0), the set of one of its nodes v_0 and the descendants of v_0, each of its nodes other than v_0 having its parent in the set (Lemma C_G); with branches of depth three or more such a set need not be a star (Example G*). For every such tree, 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 G); the least budget is therefore 0 or 1, as Corollary Z_G decides (Corollary G5). The main tool is a recursive procedure, SETTLE(v), that takes the descendants of a node v out of the working relation, from the leaves towards v, each by the node-out of v1.40 carrying a settled subtree as in v1.43 Lemma NC, so that at its end the settled piece on v and its descendants meets the working relation only at v (Lemma S_G, by induction on the height); a further node-out then takes v itself out to its parent (Corollary L_G). In these steps every stage is minimal and at most one pair, of the working relation, is deleted at each step, also when the working relations grow between the steps in the ways allowed. The construction settles the branch of the root, separates a left and a right working relation beside it, and lets the two sides take turns, one turn each, starting with the left; a turn advances the end of its side by one spine node and then settles the whole branch of the new end, after all the nodes of that branch have entered, which takes finitely long as every branch is finite. Trees with an infinite branch, trees with three or more ends and one-ended trees other than caterpillars are not treated. No novelty is claimed; the proofs are given by hand, with no machine enumeration or computation.