Generative Structural Theory v1.59 — Unit-Budget Reachability of Bidirected Trees Whose Core Has Exactly One Infinite-Degree Vertex 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-v1.58 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 (v1.48), every tree with finitely many nodes of infinite degree (v1.49-v1.54) and every tree whose infinitely many nodes of infinite degree have a locally finite least connecting subtree, the core (v1.58, after the ray, the double ray and the tripod of v1.55-v1.57). Here exactly one node c of the core H has infinitely many neighbours in H, and every other node of H has finitely many; each component of H without c is a locally finite tree, finite or infinite, and the components of T without the nodes of H are locally finite trees, with no condition on their sizes, depths or labels or on the order in which the nodes enter. The children of c in H, u_1 < u_2 < ..., one for each core branch D(u_j) at c, are infinitely many, so a separation at c with one moved part for each of them would not end; c may have any number of children outside H, none included. The core branches are therefore introduced one at a time. Those not yet introduced are held, with the branches of c outside the core not yet handled, in one relation joined at c, the reserve, minimal and locally essential with a hub; no step of a turn other than a turn at c touches them, and none of their nodes is settled or enters the locally finite side, a core side or another root side (Lemma P_Ξ). An introduction turn is one separation of v1.48 with one moved part, D(uₘ₊₁), whose kept part, the rest of the reserve, is infinite whatever the number of branches of c outside the core; it hands the present nodes of D(uₘ₊₁) over to a star at c, which becomes a new core side at the end of the turn, with budget 1 and every other side protected (Lemma W_Ξ). Inside an introduced branch the core turns, the splitting of core sides, the root sides and the locally finite side of v1.58 are used, with the one clause of v1.58 on the core nodes outside the settled part replaced, as the core branches not yet introduced are held by the reserve (Lemma K_Ξ). Round ρ = 1, 2, ... has one core turn at each node of the introduced frontier fixed at its start, then one introduction turn, at uρ+1, then one root turn at each node of B settled before the round that has a branch outside the core left; core sides and root sides made during a round, the one made by its introduction turn included, take part from the next round; an L-turn follows the introduction turn and each root turn whenever the locally finite side has a relation. With the weight η(h) = j + dist(u_j, h) for a node h in D(u_j) ∩ H, the settled part of the core at the start of round ρ is the finite set of the nodes of weight at most ρ - 1, and h is settled by the end of the core turns of round η(h), the round of introduction of its branch plus its distance from c; the distance alone does not bound it, as infinitely many nodes of H lie at distance 1 from c; every node outside H is also settled at a finite stage, by the cases of Lemma F_Ξ (iv), and the bound by η above is not claimed for an arbitrary node outside H. The rounds are finite, with no uniform bound required on their numbers of turns; one turn runs at a time and at most one elementary step at a transition (Lemma U_Ξ); every node is settled at a finite stage (Lemma F_Ξ) and no settled pair is ever deleted (Claim Ξ3). 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 (Theorem Ξ): each pair of P_T is present, and each other pair absent, from some stage on that depends on the pair, and no stage common to all pairs is claimed. With v1.56 and v1.58 this holds for every tree with node set N_0 whose nodes of infinite degree are finitely many, or infinitely many with at most one node having infinitely many neighbours in their least connecting subtree (Corollary Ξ6), and the least uniform budget is 0 exactly under the condition of Proposition Z_K of v1.48 and 1 otherwise (Corollaries Ξ5 and Ξ6). The test trees are an infinite star core, a core whose branches at c are finite stars and infinite combs in turn, and a core with infinitely many binary core branches together with infinitely many branches of c outside the core. Unit-budget reachability is not established here for trees whose least connecting subtree of the nodes of infinite degree has two or more nodes with infinitely many neighbours in it. Online constructions and other growth sequences are not treated. No novelty is claimed.