Generative Structural Theory v1.67 — Unit-Budget Tree Reachability without Bidirected Tree Terms on the Cofinally Complete Growth Sequence E

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 v1.66 established that, on the cofinally complete growth sequence E of v1.63, deletion budget 2 reaches every prescribed bidirected tree from every inclusion-minimal initial structure. For an initial structure that is not a bidirected tree, the least budget is therefore 1 or 2. It left open whether a selection sequence with budget 1 that never has a bidirected tree as a term can nevertheless have a bidirected tree as its lower limit. Version v1.67 answers this existence question affirmatively. Starting at stage 0 from the directed triangle y = {(0,1), (1,2), (2,0)}, an explicit selection sequence moves a directed core to successively larger vertices and leaves the processed vertices as permanent bidirected leaves at 0. Every term is an inclusion-minimal strongly connected spanning sub-relation of its stage, and no term is a bidirected tree. The initial transition deletes no pair; every subsequent transition deletes exactly one pair. The sequence stabilizes with L = U = PT_0, where T_0 is the infinite star centred at 0 on the vertex set N_0. Explicit bounds show that both (0,v) and (v,0) remain present for all n ≥ 2v + 1, while an ordered pair (u,w) with u,w ≥ 1 is absent for all n > 2 min(u,w). Since budget 0 would retain the initial pair (1,2), which does not belong to PT_0, the least deletion budget for this initial structure and target is exactly 1. The same star can also be reached with budget 1 through a finite bidirected tree. The result established here is the existence of a stabilizing route that never passes through a bidirected tree term. The classification of trees reachable with budget 1 from non-bidirected-tree initial structures, and whether least budget 2 occurs for such an initial structure on E, remain open within this series.