Generative Structural Theory v1.51 — Unit-Budget Reachability of Bidirected Trees with One Infinite-Degree Vertex and Finitely 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 pairs between them, v1.40-v1.50 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 one node of infinite degree whose branches at that node are all finite (v1.49), and every such tree with exactly one infinite branch at that node, locally finite (v1.50). Here the node r of infinite degree has finitely many infinite branches, m >= 1 of them, each 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 branches together with r form a tree T_inf, which is locally finite, as r has finitely many children in it; so the turns of v1.48 can be run on T_inf as a whole, and the run reduces to two sides taken in turn, the finite branches and T_inf, as in v1.50, without running a separate procedure for each infinite branch: one procedure of v1.48 is run on all the infinite branches together. At the start one run of the separation lemma of v1.48 at r, with m + 1 parts, keeps the union B_fin of the finite branches, which is infinite, in the working relation and moves the nodes of the m infinite branches into m stars at r, the nodes arriving meanwhile included (Lemmas S_P for m = 2 and R_P for every m); its output is a state of the procedure of v1.48 for T_inf at the boundary of a turn, with the settled set {r} and the frontier made of the m children of r in the infinite branches, each star fitting one of the forms allowed there, with no leaf, one leaf or more (Lemma J_P). The early phase that makes a hub of an arbitrary initial structure is that of v1.40, on the original Z3; the separation, the turns and the arguments on them use of the growth sequence only the properties recorded in v1.50 (Lemma G_P, which excludes that phase and the explicit stages of the example). 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_inf at its frontier node of least label; meanwhile the relations of the other side wait and gain only entering nodes hung at their hubs or stars. A global invariant gives the covering, the joints at r and the minimality in every state (Lemma V_P); suspension and resumption leave no operation half done (Lemma H_P); every transition deletes at most one pair, of the relation in which its step runs, a star losing pairs only when it has become the working relation of a turn (Lemma U_P); every wait is finite and both sides progress (Lemma F_P); the settled pairs are disturbed by neither side (Claim P3). 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 P); the least uniform budget is 0 exactly under the condition of Proposition Z_K of v1.48 and 1 otherwise (Corollary P5). Trees with infinitely many 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 P and Corollary P5 were adopted by the review, in an AI-assisted reading that takes the cited basic lemmas of earlier versions as known. No novelty is claimed. The proofs are presented as AI-assisted symbolic arguments, without machine enumeration or numerical computation on the relations studied. AI disclosure: The manuscript was drafted and reviewed with AI assistance. These activities do not constitute independent human peer review or formal machine-checked verification. Scripts were used for document processing and consistency checks; those checks do not verify the mathematical proofs.