Generative Structural Theory v1.46 — Unit-Budget Reachability of Bidirected Three-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-v1.45 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, of any two-ended tree with finite branches and of any one-ended tree with finite branches. Here the target is the bidirected tree P_T of a three-ended tree T with finite branches with node set {0, 1, 2, ...}: a centre c and three one-way infinite paths from c that meet one another only at c, the skeleton, with a finite tree rooted at each skeleton node, 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_I, with both directions written out); a nonempty finite node set induces a connected subtree iff its skeleton nodes form an interval of one direction or c with an initial part of each direction, and each of its other nodes has its parent in the set, or it has no skeleton 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_I); a stage may hold no skeleton node, and a later one skeleton nodes of all three directions (Example I*). 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 I); the least budget is therefore 0 or 1, as Corollary Z_I decides (Corollary I5). The finite-branch lemmas of v1.45 are restated for three-ended trees and proved in full; their proofs use of the tree only the parent relation, the sets D(v) and the finiteness of the branches, and they treat one branch inside one working relation, the other relations being further relations of a list meeting one node at a time. The construction settles the branch of the centre with one working relation, then separates the three directions: the nodes of two directions present at that moment, a finite set fixed then, are taken out one at a time by a node-out to c and joined to c in two stars, while every node that enters meanwhile is put at once into the relation of its direction; every stage of this separation is minimal and deletes at most one pair (Lemma W_I, with an invariant for every stage of it). The three 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 two working relations only grow. Every transition of the main phase preserves the invariant, the hanging of entering nodes in all three directions included (Lemma T_I); every turn ends after finitely many stages, with no bound common to the turns, so 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. By the argument of König's infinity lemma, written out here, the one-ended trees with finite branches of v1.45 are exactly the locally finite one-ended trees, so that v1.45 Theorem H holds for every locally finite one-ended tree (Lemma E_I, Corollary O_I). Trees with an infinite branch and trees with four or more ends are not treated, and the equivalence of the three-ended class of §2 with the class of all locally finite trees with exactly three ends is not proved here. No novelty is claimed; the proofs are given by hand, with no machine enumeration or computation.