Generative Structural Theory v1.47 — Unit-Budget Reachability of Bidirected Locally Finite Trees with Finitely Many Ends 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-v1.46 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, of the double ray along any bijection from the integers onto {0, 1, 2, ...}, of any two-ended caterpillar, of any two-ended tree with branches of depth at most two, and of any two-ended, one-ended or three-ended tree with finite branches. Here the target is the bidirected tree P_T of any locally finite tree T with node set {0, 1, 2, ...} and finitely many ends, with no condition on the labels or on the order in which the nodes enter. The ends are taken by the standard definition, and for the rays of a tree being separated by no finite set of nodes is shown to be the same as having a common tail (Fact N_J). A tree with node set {0, 1, 2, ...} is locally finite with exactly k >= 1 ends iff it has a skeleton with a finite core and k rays all of whose branches are finite: a finite set C of nodes inducing a connected subtree, the core, and k one-way infinite paths, each beginning at a node of C, its base, and otherwise disjoint from C and from one another, the bases not necessarily distinct (Lemma D_J, with König's argument for the finiteness of the branches, Lemma K_J); a single centre may not suffice for four or more ends. Hence the classes of v1.45, v1.44 and v1.46 are exactly the locally finite trees with one, two and three ends, and the reachability theorems of those versions hold for all such trees (Corollary E_J). 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_J, with a connectivity test for finite node sets, Lemma C_J, and a four-ended example with two branching nodes, Example J*). For every locally finite tree with finitely many ends, 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 J with Lemma D_J); the least uniform budget is therefore 0 for the starts that satisfy the condition of Corollary Z_J and 1 for all others (Corollary J5). The finite-branch lemmas of v1.46 are restated for several attachment points: a working relation may meet settled pieces in single nodes under the condition (H) of v1.43, and two relations of a list that share a node may be merged (Lemmas P_J and U_J); the procedure SETTLE with Lemma S_J and Corollaries L_J and B_J is given with statements and proofs in full under the relaxed hypothesis (S2*). The construction waits until the finite central part has entered, separates the directions 2, ..., k into stars at their bases while all the bases are still nodes of the one working relation, and then settles the central part from the core nodes farthest from the base of the direction 1 inwards, the stars staying attached at their bases; every stage of both phases is minimal and deletes at most one pair (Lemmas W_J and Q_J, with an invariant for every stage). The k directions then take turns in a fixed cyclic order, each turn waiting until the whole branch of the next skeleton node of its direction has entered, advancing that direction by one node and settling that branch, while the other working relations only grow. Every transition of the main phase preserves the invariant, the hanging of entering nodes in all the directions included (Lemma T_J); every turn ends after finitely many stages, the number of steps on a branch being shown finite apart from the waits for new nodes; every direction advances without bound and every node joins the settled part at a finite stage; from then on a pair with both nodes settled is selected iff it is in P_T, and no other pair with both nodes settled is introduced again. Trees with infinitely many ends and trees that are not locally finite are not treated. No novelty is claimed; the proofs are given by hand, with no machine enumeration or computation.