Generative Structural Theory v1.45 — Unit-Budget Reachability of Bidirected One-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-v1.44 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, that of any two-ended tree with branches of depth at most two and that of any two-ended tree with finite branches. Here the target is the bidirected tree P_T of a one-ended tree T with finite branches with node set {0, 1, 2, ...}: a one-way infinite path s_0, s_1, s_2, ..., 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_H); 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_H); a stage may consist of four off-spine nodes on a path (Example H*). 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, at a stage that depends on the pair (Theorem H); the least budget is therefore 0 or 1, as Corollary Z_H decides (Corollary H5). The finite-branch lemmas of v1.44 are restated for one-ended trees and proved in full: the procedure SETTLE(v) takes the nodes of D(v) other than v out of the working relation, from the leaves towards v, so that the settled piece meets the working relation only at v, and a further node-out then takes v to its parent; their proofs use of the tree only the parent relation and the finiteness of the branches. The construction waits until the whole branch of the root has entered, settles it and keeps one working relation; it then repeats turns, each of which waits until the whole branch of the next spine node has entered, advances the end by one spine node, changing the settled part and starting the settling of the new branch at the same transition, and settles that branch. Every transition of the main phase, including waiting transitions, preserves the invariant (Lemma T_H); every turn ends after finitely many stages, with no bound common to the branches; every node joins the settled part at a finite stage, and from the stage at which both nodes of a pair have joined it the pair is selected iff it is in P_T. Trees with an infinite branch and trees with three or more ends are not treated. The target class is defined in §2; its equivalence with the class of all locally finite one-ended trees is not proved here. No novelty is claimed; the proofs are given by hand, with no machine enumeration or computation.