Generative Structural Theory v1.48 — Unit-Budget Reachability of Arbitrary Bidirected Locally Finite Trees 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-v1.47 showed that deletion budget 1 reaches, from every starting stage and every inclusion-minimal strongly connected initial structure, the bidirected tree P_T of a sequence of classes of trees, ending with every locally finite tree with finitely many ends (v1.47). Here the target is P_T for every locally finite tree T with node set {0, 1, 2, ...}, with no condition on the number of its ends, on the labels or on the order in which the nodes enter. 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 L); the least uniform budget is 0 for the starts in which 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, and 1 for all others (Corollary L5, with Proposition Z_K, whose proof holds for every tree with node set {0, 1, 2, ...}). The construction chooses a root and keeps a finite settled set closed under the parent; at the boundary of a turn there is one working relation for each node of its frontier, while within a turn the relation in progress may also hold the nodes of directions not yet separated and pending pieces may be present. Its main tool is a separation lemma: one working relation holding a node f and the nodes of finitely many disjoint sets below it is split, by node-outs to f, into the relation of one infinite set and stars at f for the others, every stage minimal and every transition deleting at most one pair of the relation separated, the joints with the other relations kept (Lemma X_K). Each turn takes the node of least label of the frontier at its start and runs three parts: an advance into that node; the settling of its finite child subtrees, after each has entered completely, by a recursive procedure whose lemma needs no skeleton and no condition on the ends (Lemma S_L, with Corollaries L_L and B_L); and the separation into its infinite children. The states within a turn are kept apart from its boundary, at which the settled set is complete and every frontier node is infinite, and the covering of the node set, the joints and the minimality are shown in every state, the pieces pending within the settling included (Lemma V_L). Every transition keeps the invariant and deletes at most one pair (Lemmas I_L and T_L); every wait is for a named node, a named finite set or one further node of a named infinite set, every turn ends after finitely many stages, and every node is settled at a finite stage, although the frontier grows (Lemma F_L). The rooted binary tree is treated first in full (Theorem K and Corollary K5). Trees that are not locally finite and online constructions are not treated. No novelty is claimed; the proofs are given by hand, with no machine enumeration or computation.