Generative Structural Theory v1.50 — Unit-Budget Reachability of Bidirected Trees with One Infinite-Degree Vertex and One Infinite Locally Finite Branch 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-v1.49 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) and every tree with one node of infinite degree whose branches at that node are all finite (v1.49). Here exactly one branch at the node r of infinite degree is infinite and locally finite, and the other branches are finite, with no common bound on their sizes or depths and no condition on the labels or on the order in which the nodes enter. The infinite branch cannot be awaited as a whole, and the nodes of it, restricted to it, do not enter in the form of Z3. A lemma records that the finite operations of v1.48 and v1.49 use of the growth sequence only that the stages are finite and increasing, that every node enters at a finite stage, that an entering node is new, that all pairs of a stage are present, and that an infinite set has a further node entering after every stage (Lemma G_N); so they can be run on the infinite branch inside Z3. The construction separates, at r, the union B_fin of the finite branches, kept in one working relation as in v1.49, from the infinite branch, kept in the relations of v1.48 for the locally finite tree T' made of r and that branch (the separation lemma of v1.48 applied with B_fin, which is infinite, as the kept part). Turns of the two kinds alternate, each running whole: a turn settles the finite branch at the child of r of next label, as in v1.49, or is a turn of v1.48 for T'; meanwhile the relations of the other side wait at the boundary of their own turns, where they are complete working relations, and gain only entering nodes hung at their hubs or stars, which deletes nothing. So at every transition at most one elementary step runs in the whole relation, and the budget 1 is not a sum of two budgets. During a turn in the infinite branch the settled relation splits into P_T of the finite branches settled, which with the working relation of the finite branches is fixed apart from the hanging and meets the rest in r only, and PT' of the part of T' settled, on which v1.48 runs, the two parts being joined by the lemma of v1.43 on relations meeting in one node. A global invariant gives the covering of the node set, the joints and the minimality in every state (Lemma V_N); suspension and resumption leave no operation half done (Lemma P_N); every transition deletes at most one pair (Lemma U_N); every wait is finite and both sides progress (Lemma F_N); the settled pairs are disturbed by neither side (Claim N3). 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 N); the least uniform budget is 0 exactly under the condition of Proposition Z_K of v1.48 and 1 otherwise (Corollary N5). Trees with two or more infinite branches at a node, an infinite branch that is not locally finite, several nodes of infinite degree, and online constructions are not treated. The proofs of Theorem N and Corollary N5 were adopted in the AI-assisted review, in a reading that takes the cited basic lemmas of earlier versions as given. No novelty is claimed. AI disclosure: The text and mathematical arguments were prepared with AI assistance. The proofs are presented as symbolic mathematical arguments, without machine enumeration or numerical computation on the relations studied. This describes the method of argument, not manual authorship by a human without AI assistance. AI assistance does not constitute independent human peer review or formal machine-checked verification of the proofs.