Generative Structural Theory v1.53 — Unit-Budget Reachability of Bidirected Trees with Exactly Two Infinite-Degree Vertices on the Original Z3 Growth Sequence
Year: 2026
Type: preprint
Abstract: A growth sequence is a sequence of finite relations R_n on node sets V_n with V_n ⊆ V_{n+1}, R_n ⊆ R_{n+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 x_{n+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–v1.52 established unit-budget reachability for a sequence of classes of bidirected trees, including every locally finite tree (v1.48) and every tree with exactly one vertex of infinite degree (v1.49–v1.52). This version treats every tree T with vertex set ℕ₀ and exactly two vertices r and s of infinite degree. The unique path P between them is finite. Every component obtained by removing the vertices of P is a locally finite tree, finite or infinite, attached to exactly one vertex of P by one edge. No additional restrictions are imposed on the sizes, depths or labels of these components, and infinite branches may occur at internal vertices of P. Procedure R begins with a finite path phase. It separates the branch containing s from the working relation at r, settles the internal vertices of P and their finite branches, and converts the working relation at s into a second root side after its advance operation. The settled relation containing P connects the two root sides. In the main phase, root turns at r and s alternate. Each settles a finite branch or introduces an infinite branch, and each is followed, whenever the locally finite side has a working relation, by one turn on that side. Only one turn runs at a time, and each transition performs at most one elementary operation. During a root turn, the other root side and the locally finite side only gain new vertices; during a locally finite turn, both root sides only gain new vertices at their hubs. A global invariant preserves spanning, strong connectivity and inclusion-minimality at every stage. Every wait and every turn ends after finitely many stages. A least-label rule ensures that every vertex is eventually settled, while settled pairs are never deleted. Theorem R proves that, for every such tree T, every starting stage s₀ and every initial structure y ∈ M_{s₀}, there exists a selection sequence with x_{s₀} = y, x_n ∈ M_n, |x_n ∖ x_{n+1}| ≤ 1 and L = U = P_T, where P_T is the bidirected relation of T. The construction uses T in advance and keeps the original Z3 growth sequence unchanged. Corollary R5 gives the least uniform deletion budget for a selection sequence from y with lower limit P_T. It is 0 exactly when y = P_T[V_{s₀}] and T[V_n] is connected for every n ≥ s₀, and 1 otherwise. Trees with three or more vertices of infinite degree, online constructions and other growth sequences are outside the scope of this version.
Author: Okino, Koji
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223462293
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