Generative Structural Theory v1.68 — Exact Budget Two for Trees with Infinite Branches at Zero 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 Example Φ2, deletion budget 2 reaches every bidirected tree P_T from every starting stage and every inclusion-minimal initial structure. For an initial structure that is not a bidirected tree, the least budget is therefore either 1 or 2. Version v1.67 exhibited a budget-1 sequence from a directed triangle that stabilizes to the bidirected star at 0 without ever passing through a bidirected tree term. Version v1.68 proves that the least budget is exactly 2 whenever T − {0} has an infinite connected component. Here T is a tree with vertex set ℕ₀, and its branches at zero are the connected components remaining after 0 is removed. An infinite branch at zero does not mean that 0 has infinite degree: the ray 0–1–2–3–⋯ has one infinite branch at zero, although 0 has degree 1. Two finite lemmas establish the lower bound. First, if an inclusion-minimal strongly connected relation contains both (a,b) and (b,a), deleting these two pairs separates its vertices into two strongly connected sides, with no other pair between them. Second, a transition that retains both pairs and deletes at most one old pair cannot move an existing vertex from one side to the other, even when new vertices and arbitrarily many pairs may be added. On E, each new vertex has pairs only with 0 at its introduction. Consequently, once both pairs of an edge {0,r} are permanently retained by a budget-1 selection sequence, the side containing r remains a fixed finite set. This prevents the lower limit L from containing P_T whenever the branch of T at zero containing r is infinite. Thus, for every starting stage s_0 and every y ∈ Ms_0, no budget-1 selection sequence has P_T ⊆ L, and the upper bound from v1.66 gives d_min(y; s_0, T) = 2. Budget 2 can attain the stronger stabilization L = U = P_T. It follows that every tree reached with budget 1 on E must have only finite branches at zero, and hence infinite degree at 0. Conversely, every tree on ℕ₀ in which 0 has finite degree, including every locally finite tree, has least deletion budget 2 from every minimal start. In particular, from the same directed triangle y = {(0,1), (1,2), (2,0)} at stage 0, the least budget is 1 for the star at 0 and 2 for the ray. The results concern the specific growth sequence E. Whether having only finite branches at zero is sufficient for budget-1 reachability from some start, and which such trees are reachable from which starts, remain open.