Generative Structural Theory v1.58 — Unit-Budget Reachability of Bidirected Trees with a Locally Finite Core 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-v1.57 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), on one double ray (v1.56) or on a tripod core (v1.57). Here the set B of the nodes of infinite degree is infinite, and the least subtree H of T containing B, the core, is locally finite: every node of H has finitely many neighbours in H, with no bound on the number of branch points or of ends of H or on the degrees in H. 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 or on the order in which the nodes enter. The root c is the node of B of least label, and the settled part of the core is a finite subtree Q of H that contains c and the parent of each of its nodes other than c; each node f of its frontier in H has its core side, a relation that holds the whole untreated part below f, which may be finite in H and need not be a ray or hold infinitely many nodes of B; no core side is ever part of the locally finite side. A first turn at c, one separation of v1.48 with one moved part for each neighbour of c in H, makes the root side of c and its core sides (Lemma D_Θ). A core turn at a frontier node f extends the settled part by f, on a core tree that holds only the untreated part below f, while every other core side, each joined at its own end, the root sides and the locally finite side are protected and only receive new nodes; it splits the core side of f into one new core side for each child of f in H, handing every present node of the old untreated part outside the settled part over to exactly one new relation (Lemma K_Θ): at a node not in B by a turn of the procedure of v1.48, whose separation also passes the infinite branches outside the core to the locally finite side (Lemma C_Θ), and at a node of B by the advance part of such a turn followed by one separation, whose kept part becomes the root side of that node, or, at a leaf of H, by no separation, the core side then ending (Lemma J_Θ). Round ρ = 1, 2, ... has one core turn at each node of the frontier fixed at its start, the nodes of H at distance ρ from c, and then one root turn at each node of B settled before the round; core sides and root sides made during a round take part from the next round; 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 and at the introductions, at the node of least label of its frontier at that time (Lemma E_Θ). A node of H at distance l from c is settled by the end of the core turns of round l, and the root turn at b with index k falls in round dist(c, b) + k. The rounds are finite; a uniform bound on the number of core sides is not required (it is bounded on one test tree and unbounded on the other), and the number of root sides grows without bound. One turn runs at a time and at most one elementary step at a transition (Lemma U_Θ). A global invariant gives the covering, the joints and the minimality at the turn boundaries (Lemma V_Θ). Every wait is finite, every node is settled at a finite stage (Lemma F_Θ), and no settled pair is ever deleted (Claim Θ3). 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 Θ): 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-v1.56 this holds for every tree with node set N_0 whose nodes of infinite degree are finitely many, or infinitely many with a locally finite least connecting subtree (Corollary Θ6), which includes the trees of v1.55-v1.57, and the least uniform budget is 0 exactly under the condition of Proposition Z_K of v1.48 and 1 otherwise (Corollaries Θ5 and Θ6). The test trees are a comb core whose spine has no node of B, the nodes of B being at the tips of its teeth, and a binary core whose number of core sides at the start of round ρ is |S_ρ| = 2ρ-1 + 2ρ-2 for ρ ≥ 3, doubling at each next round from round 3 on. Unit-budget reachability is not established here for trees whose least connecting subtree of the nodes of infinite degree has a node with infinitely many neighbours in it. Online constructions and other growth sequences are not treated. No novelty is claimed.