Generative Structural Theory v1.54 — Unit-Budget Reachability of Bidirected Trees with Finitely Many Infinite-Degree Vertices 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-v1.53 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 exactly one node of infinite degree (v1.49-v1.52) and every tree with exactly two (v1.53). Here the set B of the nodes of infinite degree of T is finite and has at least two nodes. The least subtree H of T containing B, the core, is finite, 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 or on the order in which the nodes enter. The core may branch, and a node of B may have children in the core whose branches hold other nodes of B, so the path phase of v1.53 does not apply as it is, and copies of the procedures of v1.52 or v1.53 run side by side could delete two pairs at one transition. The construction runs one sequence of turns. A finite core phase first separates from the working relation of the node r of B of least label the branches at its children in the core (Lemma X_S), and then settles the other nodes of the core one at a time, each when its parent is settled, of least label: a node not in B by a turn of the procedure of v1.48 on a core tree that need not be locally finite at the nodes of B not yet reached (Lemma C_S), a node b of B by the advance part of such a turn followed, when b has children in the core, by one separation whose kept part, the branches at b not in the core, becomes the root side of b, while its moved parts, the branches at the children of b in the core, stay in the core (Lemma J_S). When b has no child in the core, the relation remaining after the advance part becomes its root side directly, without a separation. At its end each root side meets the settled relation on the core in its own node of B only, and the root sides do not meet one another (Lemma D_S). After it, root turns at the nodes of B follow one another in a fixed cycle, each settling a finite branch or introducing an infinite one by the separation lemma of v1.48, the branches of larger label being the kept part, and each followed, whenever the L-side has a relation, by one turn of v1.48 on one locally finite tree that holds the settled core, the branches at its nodes not in B and the branches introduced at the nodes of B (Lemma E_S). One turn runs at a time and at most one elementary step at a transition (Lemma U_S). During a root turn, the other root sides and the locally finite side only gain new nodes; during an L-turn, every root side only gains new nodes, at its hub (Lemma H_S). A global invariant gives the covering, the joints and the minimality in every state (Lemma V_S); every wait is finite and every node is settled at a finite stage under the least-label rule (Lemma F_S); no settled pair is ever deleted (Claim S3). 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 S): 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-v1.53 this holds for every tree with node set N_0 with finitely many nodes of infinite degree (Corollary S6), and the least uniform budget is 0 exactly under the condition of Proposition Z_K of v1.48 and 1 otherwise (Corollaries S5 and S6). Trees with infinitely many nodes of infinite degree, online constructions and other growth sequences are not treated.