Generative Structural Theory v1.52 — Unit-Budget Reachability of Bidirected Trees with One Infinite-Degree Vertex and Infinitely Many Infinite Locally 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 ordered pairs of distinct nodes between them, v1.40–v1.51 established unit-budget reachability from every starting stage and every inclusion-minimal strongly connected initial structure for successive classes of bidirected trees. These include every locally finite tree (v1.48) and trees with one node of infinite degree whose branches at that node are all finite (v1.49), or of which exactly one (v1.50) or finitely many (v1.51) are infinite, each locally finite. This version treats trees with node set N_0 in which a node r of infinite degree has countably infinitely many infinite branches, each locally finite, and every other branch is finite. No restriction is imposed on the number, sizes or depths of the finite branches, on the labels, or on their order of entry. The union B_fin of the finite branches may be infinite, finite or empty. The tree consisting of r and all the infinite branches is no longer locally finite, so the procedure of v1.48 and the single initial separation of v1.51 do not apply unchanged. Procedure Q handles the branches at r one at a time, in increasing order of the labels of its children. A root turn settles a finite branch as in v1.49, or introduces an infinite branch by the separation procedure of v1.48 with two parts. The part retained in the common working relation is the union of the branches at children of larger label; it remains infinite because r has infinitely many children. The introduced branch becomes a new frontier relation for the locally finite tree T_J consisting of r and the finitely many infinite branches introduced so far. Once an infinite branch has been introduced, each root turn is followed by one turn of the v1.48 procedure on T_J, at the frontier node of least label. Although T_J grows as further branches are introduced, this rule is fair: only finitely many nodes have a smaller label than any fixed node. Every wait and every turn ends after finitely many stages, every node is eventually settled, and later introductions preserve the settled relations. A global invariant maintains vertex coverage, the required intersections of the component relations, inclusion-minimal strong connectivity and the deletion budget of at most one pair per transition. Theorem Q establishes that, for every tree T in this class, every starting stage s and every y in M_s, there is a selection sequence with x_s = y, x_n in M_n, |x_n \ xₙ₊₁| ≤ 1 and L = U = P_T, where P_T is the bidirected relation of T. Thus every pair of P_T is eventually always present and every other pair is eventually always absent. Corollary Q5 gives the least uniform deletion budget: it is 0 exactly when y = P_T[V_s] and T[V_n] is connected for every n ≥ s, and 1 otherwise. Together with v1.49–v1.51, this covers every tree with node set N_0 having exactly one node of infinite degree. The construction uses the target tree in advance. Trees with several nodes of infinite degree and online constructions are not treated. The supplement contains the drafting and review records, recorded corrections, document-processing scripts and consistency-check outputs.