Generative Structural Theory v1.56 — Unit-Budget Reachability of Bidirected Trees with Infinitely Many Infinite-Degree Vertices Lying on a Single Double Ray 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 pairs between them, v1.40–v1.55 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 finitely many nodes of infinite degree (v1.49–v1.54) and every tree whose infinitely many nodes of infinite degree all lie on one ray (v1.55). Here the set B of the nodes of infinite degree of T lies on one double ray (two-way infinite path) of T and goes on infinitely in both of its directions. The least subtree H of T containing B, the core, is then that double ray, numbered ... h_{-1}, h_0, h_1, ... from a node r = h_0 of B, and every component of T without the nodes of H is a locally finite tree, finite or infinite, joined to H by one edge, with no condition on the sizes, the depths or the labels of the components, on the gaps between the nodes of B along H or on the order in which the nodes enter. The core now has two ends to be extended. The settled part of the core is kept as a finite interval {h_{-l}, ..., h_m} of H containing r, and the two untreated infinite parts beyond its two ends are held by two relations, the core sides, neither of which is ever part of the locally finite side. A first separation at r with two moved parts makes the root side of r and the two core sides (Lemma D_Y). A core turn on either side extends the settled interval by one node at that end, on a core tree that holds only the untreated part on that side, while the core side of the other side, joined at the other end, the root sides and the locally finite side are protected and only receive new nodes (Lemma K_Y): at a node not in B by a turn of the procedure of v1.48 (Lemma C_Y), and at a node b of B by the advance part of such a turn followed by one separation, whose kept part becomes the new root side of b (Lemma J_Y). Round ρ = 1, 2, ... has the core turn at h_ρ, the core turn at h_{-ρ}, and then one root turn at each node of B in the settled interval {h_{-ρ}, ..., h_ρ}, the root sides being counted after the two core turns; each root turn settles a finite branch or introduces an infinite one by the separation lemma of v1.48 and is followed, whenever the locally finite side has a relation, by one turn of v1.48 on a locally finite tree that grows at the core turns on both sides and at the introductions (Lemma E_Y). The root turn at b with index k falls in round max{1, a(b)} + k − 1, a(b) being the round in which b is settled (a(r) = 0). One turn runs at a time and at most one elementary step at a transition (Lemma U_Y). A global invariant gives the covering, the joints and the minimality at the turn boundaries (Lemma V_Y). Every wait is finite and every round ends after finitely many stages, with at most 2ρ + 1 root turns; once the locally finite side has a relation, every later round has at least one turn of it under the least-label rule, and every node is settled at a finite stage (Lemma F_Y); no settled pair is ever deleted (Claim Y3). 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 Y): each pair of P_T is present, and each other pair absent, from some stage on that depends on the pair, and no stage common to all pairs is claimed. With v1.48–v1.55 this holds for every tree with node set N_0 whose nodes of infinite degree are finitely many, or infinitely many and all on one ray, or infinitely many and all on one double ray (Corollary Y6), and the least uniform budget is 0 exactly under the condition of Proposition Z_K of v1.48 and 1 otherwise (Corollaries Y5 and Y6). Unit-budget reachability is not established here for trees with infinitely many nodes of infinite degree whose least connecting subtree is neither a ray nor a double ray. Online constructions and other growth sequences are not treated. No novelty is claimed. The proofs are presented as symbolic arguments, without machine enumeration or numerical computation on the relations studied.
Author: Okino, Koji
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223449163
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