Generative Structural Theory v1.55 — Unit-Budget Reachability of Bidirected Trees with Infinitely Many Infinite-Degree Vertices Lying on a Single 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–v1.54 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 finitely many nodes of infinite degree (v1.49–v1.54). Here the set B of the nodes of infinite degree of T is infinite and all its nodes lie on one ray of T. The least subtree H of T containing B, the core, is then a ray from a node r 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. As H is infinite, the core phase of v1.54, which settles the whole core before the branches are handled, would not end. The construction here runs one sequence of turns in rounds. After a first separation at r (Lemma X_W), round ρ = 1, 2, ... has one core turn at the ρ-th node h_ρ of H after r, which extends the settled finite part of the core by one node (Lemma K_W): at a node not in B by a turn of the procedure of v1.48 on a core tree that is not locally finite (Lemma C_W), and at a node b of B by the advance part of such a turn followed by one separation, whose kept part, the branches at b not in H, becomes the new root side of b, while the untreated infinite rest of the core stays in one relation, the core side, which is never part of the locally finite side (Lemma J_W). The core turn is followed by one root turn at each node of B then settled, the root sides being counted after the core turn, so that a root side made by it is handled in the same round; the root turn at h_j with index k falls in round max{1, j} + k − 1. 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 made of the settled part of the core, the branches at its nodes not in B and the branches introduced at the nodes of B, a tree that grows at the core turns as well as at the introductions (Lemma E_W). One turn runs at a time and at most one elementary step at a transition (Lemma U_W). A global invariant gives the covering, the joints and the minimality at the turn boundaries (Lemma V_W). Every wait is finite and every round ends after finitely many stages, though the number of root sides grows without bound; 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_W); no settled pair is ever deleted (Claim W3). 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 W): 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–v1.54 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 (Corollary W6), and the least uniform budget is 0 exactly under the condition of Proposition Z_K of v1.48 and 1 otherwise (Corollaries W5 and W6). Trees with infinitely many nodes of infinite degree not all on one ray, online constructions and other growth sequences are not treated. The proofs of Theorem W and Corollaries W5 and W6 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.
Author: Okino, Koji
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223467448
Cites:
Cited by:
Related to: