Generative Structural Theory v1.60 — Unit-Budget Reachability of Arbitrary Bidirected Trees 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, earlier versions of this series 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 tree whose nodes of infinite degree are finitely many (v1.56), every tree whose infinitely many nodes of infinite degree have a locally finite least connecting subtree, the core H (v1.58), and every tree whose core has exactly one node of infinite degree in it (v1.59); v1.59 §1.1 gives the account of these versions. This text treats every tree T with node set N_0. If the set B of the nodes of infinite degree of T is finite, the empty set included, the result is that of v1.56 Corollary Y6. For B infinite a construction is made here, with no restriction on the number or the placement of the nodes of infinite degree in the core, the least subtree H of T containing B: there may be none, one, finitely many or infinitely many of them, adjacent or not, at any depth, and every node of the core may be one. The core is then taken as a tree rooted at the node c of least label in B. A node b of the core with infinitely many children in the core cannot be separated from all its child core branches at once. Each such node, once settled, therefore has its own reserve, one relation minimal and locally essential with a hub, joined at b only, which holds the child core branches of b not yet introduced and the branches of b outside the core not yet handled; the joint of b toward its parent is the settled pair on that edge, apart from the reserve. The core turn that reaches such a node b settles it by the advance of the core turn of v1.58 at a node of infinite degree and runs no separation: the relation left on b and the present nodes of the set of its descendants is already a reserve, as that set is exactly the reserve set of b (Lemma R_Π). The reserves of different nodes have disjoint node sets, the home of each child core branch not yet introduced is the reserve of its parent, and the reserves of ancestors and descendants do not meet (the invariant (I_Π) with (π6), (π7), Lemma P_Π). Each settled node of infinite degree in the core introduces one child core branch per round, by a separation of v1.48 with one moved part, as at c in v1.59 (Lemma W_Π); the targets of a round (the nodes of the introduced frontier, the settled nodes of infinite degree in the core, and the settled nodes of infinite degree with a branch outside the core left) are fixed at its start and finite; the core sides, reserves and root sides made during a round take part from the next round, while a relation of the locally finite side made during a round may be handled by a later L-turn of the same round; an L-turn follows each introduction turn and each root turn when the locally finite side has a relation. With the weight κ, which grows by 1 at a step from a node of finite degree in the core and by j + 1 at a step from a node of infinite degree in the core to its j-th child in the core, the settled part at the start of round ρ is the finite set of the core nodes of weight at most ρ - 1, and a node h of the core is settled by the end of the core turns of round κ(h): the settling is shown along the finite path from the root, the waits at the nodes of infinite degree on it counted by the introduction indices, with no bound by the distance alone; the rounds with no core turn are finitely many. A node outside the core is settled at a finite stage by the cases of Lemma F_Π (iv) (the finite branches in core turns or root turns, the infinite ones through the locally finite side), and the bound by κ is not claimed for it. One turn runs at a time and at most one elementary step at a transition (Lemma U_Π), and no settled pair is ever deleted (Claim Π3). For every tree with node set N_0, budget 1 reaches P_T from every starting stage and every initial structure, with lower and upper limits both P_T: 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; the least uniform budget is 0 exactly under the condition of Proposition Z_K of v1.48 and 1 otherwise (Theorem Π, Corollaries Π5 and Π6, the Main Theorem). The test trees are a core with two adjacent nodes of infinite degree in it, a ray of such nodes each with infinitely many core branches, and a tree all of whose nodes have infinite degree, on which the introduction is repeated at every depth. Online constructions and other growth sequences are not treated. No novelty is claimed.