Generative Structural Theory v1.64 — Continuation without Prescribed Tree Reachability on Cofinally Complete Growth Sequences

A growth sequence is a sequence of finite relations R_n on non-empty 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. Its deletion budget bounds the number of ordered pairs of x_n that xₙ₊₁ drops; the number of pairs added is not bounded. Version v1.63 established, for every growth sequence, that an infinite selection sequence with uniform deletion budget d exists if and only if b_∞ ≤ d, where b_∞ is the supremum of the least budgets required on finite initial intervals. On eventually complete growth sequences, continuation within a positive budget is equivalent to reaching any prescribed bidirected spanning tree as the lower limit, and also to reaching it with stabilization. Whether this implication extends to cofinally complete growth sequences, whose complete stages occur infinitely often, was left open. Version v1.64 gives a negative answer. It constructs a growth sequence J that is cofinally complete but not eventually complete. At each odd stage, a new vertex z_m is added to the preceding complete relation with only the pairs (0, z_m) and (z_m, 1); the next even stage is complete. The union of the stage relations is K(ℕ₀), the relation of all ordered pairs of distinct non-negative integers. The exclusion lemma shows that, when a new vertex is entered only from a and left only to a distinct vertex b, no inclusion-minimal strongly connected spanning sub-relation contains (a, b). More generally, it cannot contain the first forward pair of a simple a–b path together with the reverse pairs of the remaining edges. In particular, it cannot contain a bidirected a–b path. On J, the relations F_n = {(1, 0)} ∪ {(0, v), (v, 1) : v ∈ V_n ∖ {0, 1}} form a preserving selection sequence with deletion budget 0. From every starting stage s_0 and every initial minimal structure y, continuation is possible within a finite uniform budget, with b_∞(y; s_0) ≤ |y ∖ Fs_0+1|. Nevertheless, for every starting stage, every initial minimal structure, every selection sequence without any restriction on its deletions, and every tree T with vertex set ℕ₀, its bidirected relation P_T is not contained in the lower limit L. The finite path of T between 0 and 1 cannot be held in both directions by a minimal structure at any sufficiently late odd stage. This obstruction persists although every ordered pair between distinct vertices of any fixed finite set is available at all sufficiently late stages. Thus, continuation within a budget does not in general guarantee prescribed tree reachability on cofinally complete growth sequences. The result answers the general question (Q1) of v1.63, while leaving open the separate unit-budget reachability question for the growth sequence E of v1.63 Example Φ2.