Generative Structural Theory v1.39 — Unit-Budget Reachability and Stabilization from Arbitrary Initial Minimal Structures 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, v1.38 gave one selection sequence with deletion budget 1 whose lower and upper limits are both the bidirected ray P, the relation with pairs in both directions between adjacent nodes of the sequence 0, 1, 2, .... Here the starting point is arbitrary: for every starting stage s and every inclusion-minimal strongly connected structure y of stage s, a selection sequence with deletion budget 1 is built that starts from y and whose lower and upper limits are both P, so that the membership of every pair settles. The construction first subdivides pairs until each pair is the only pair out of its first node or the only pair into its second node, keeps a hub at which the entering nodes not used by the current repair step are hung at no deletion, and then advances a front node by node: the pairs at the front node are gathered into one pair in and one pair out by splitting the node, the node is taken out of the remaining relation and moved into the bidirected prefix, where it is joined to the next node by a pair in each direction; it is not deleted from the node set V_n. A lemma on ears keeps every intermediate stage inclusion-minimal. With budget 0 the pairs of y stay, and y has a pair outside P, so the least uniform deletion budget for reaching P is 1 from every start. A first attempt, to repair y into the forms of v1.38 Example Z7 on the node set of stage s, fails for some y, since with budget 1 the number of pairs cannot fall on a fixed node set; this is not a proof that P cannot be reached. No novelty is claimed. The proofs and the example are presented as symbolic arguments, without machine enumeration or numerical computation on the relations studied. AI assistance was used in drafting and review, as detailed in the disclosures.