Generative Structural Theory v1.65 — Exact Deletion Budgets from Bidirected Tree Starts 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 v1.64 separated continuation from prescribed tree reachability on cofinally complete growth sequences and left open the budget-1 reachability question for the growth sequence E introduced in v1.63. On E, each odd stage adds one new node joined in both directions only to node 0, and the following even stage makes the relation complete without changing the node set. Thus complete stages occur arbitrarily late, and deletion budget 0 permits indefinite continuation from every initial minimal structure. This version determines the exact least deletion budget for reaching any prescribed bidirected tree from a bidirected tree initial structure on E. The main finite lemma establishes rigidity under one deletion: if S is a tree on a finite node set W and q is an inclusion-minimal strongly connected relation on W, then |P_S ∖ q| ≤ 1 implies q = P_S, with no bound on the number of added pairs. Consequently, from any starting stage s_0 and any bidirected tree P_S in Ms_0, the selection sequence with deletion budget at most 1 is unique and preserving. Its limit is PS*, where S* is obtained from S by attaching every node outside Vs_0 to node 0 as a leaf. Hence no other tree can be reached with budget 1; in particular, no locally finite tree, including the ray, can be reached from such a start with that budget. Deletion budget 2 reaches every tree T with node set N_0. At complete stages, the construction adds a missing target edge and removes a non-target edge from the resulting cycle. Each target edge is eventually retained permanently, and every non-target edge is eventually absent, giving L = U = P_T. No local finiteness assumption on T is required. The exact least budget is therefore d_min(P_S; s_0, T) = 0 when T = S*, and 2 otherwise. This least budget is defined using reachability as L = P_T, and in both cases the construction achieves the stronger conclusion L = U = P_T. Although b_∞ = 0 on E, the least-budget formula for eventually complete growth sequences does not extend to all cofinally complete growth sequences. Initial minimal structures that are not bidirected trees remain outside the classification established here.