Generative Structural Theory v1.49 — Unit-Budget Reachability of Bidirected Trees with One Infinite-Degree Vertex and 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-v1.48 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). Here, as a first step beyond local finiteness, T is a tree with node set {0, 1, 2, ...} with one node r of infinite degree such that every connected component of T minus {r} is finite; no common bound is put on the sizes or the depths of these finite branches, and no condition on the labels or on the order in which the nodes enter; the infinite stars are the simplest examples. 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 M); 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 M5, with Proposition Z_K of v1.48, whose proof holds for every tree with node set {0, 1, 2, ...}). The frontier of a finite settled set containing r is infinite, so the construction of v1.48, with one working relation for each node of a finite frontier, is not applied as it is; instead the settled part, P_T on r and the branches settled so far, and one working relation meet only in r; at the boundary of a turn the working relation holds r and the nodes present of all the branches not yet settled, and while a branch is processed these nodes are held by the working relation and the pieces pending within the settling. The branches are taken one at a time, in the order of the labels of the children of r: each is awaited until all of its finitely many nodes have entered, settled by the recursive procedure SETTLE of v1.48, moved to r by a node-out, with a renewal of the hub when needed, and merged into the settled part. The lemmas of v1.48 for this procedure were stated for locally finite trees; they are restated and proved for a finite branch of a tree with no condition on the degrees, with the joint conditions kept and a supply condition added (Lemma S_M, Corollary L_M): the uses of local finiteness in the earlier proofs are listed and replaced, and the length of a node-out is bounded by degrees in the finite working relation, not in T. Every fresh node is a node entering at the transition of its step, from the infinitely many branches not yet settled, assigned to the step before the other entering nodes are hung. An invariant holding the pieces pending within the settling gives the covering of the node set, the joints and the minimality in every state (Lemma V_M); one branch is processed in finitely many stages, every transition deleting at most one pair (Lemma B_M); every wait is for a named node, one named finite branch or one further node of a named infinite set, and every node is settled at a finite stage (Lemma F_M). Trees with an infinite branch at a node of infinite degree, trees with several nodes of infinite degree, arbitrary 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. AI and document-processing disclosure: The manuscript, including its mathematical arguments, was drafted and reviewed with AI assistance at the author's request. Document assembly, copying, reference generation, and consistency checks were performed by scripts. The expression “by hand” above describes symbolic mathematical arguments without machine enumeration or numerical computation on the relations studied; it does not mean that the manuscript or its proofs were manually written by a human without AI assistance. The AI review and script-based consistency checks are not independent human peer review or formal machine-checked verification of the proofs. Details of the drafting process, review, retained records, and deleted records are disclosed in the manuscript and supplement.