Generative Structural Theory v1.66 — Budget-Two Reachability from Arbitrary Minimal 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 v1.65 determined the exact deletion budgets from bidirected tree starts on the cofinally complete growth sequence E of v1.63 Example Φ2. On E, complete stages alternate with stages that attach one new vertex to vertex 0 in both directions. From a bidirected tree start, budget 1 permits only the preserving continuation that attaches every later vertex to 0 as a leaf. The least budget to a prescribed bidirected tree is therefore 0 for that continuation's limit and 2 for every other target tree. Version v1.66 extends budget-two reachability to every inclusion-minimal initial structure. The main finite result shows that any inclusion-minimal strongly connected relation on a finite set W with at least two vertices can be transformed into a bidirected tree on W in at most |W| − 1 steps, deleting at most one ordered pair per step and preserving inclusion-minimal strong connectivity throughout. The proof constructs a directed ear decomposition and processes its ears in reverse order, replacing each ear by a bidirected branch while maintaining a forest attached to the remaining strongly connected part at single vertices. On E, this finite transformation can be performed from any starting stage, with vertices outside the fixed initial core attached to 0 as bidirected leaves. Once a bidirected tree is reached, the edge-exchange construction of v1.65 applies. Thus, for every starting stage s_0, every y ∈ Ms_0, and every tree T with vertex set N_0, there is a selection sequence with uniform deletion budget 2 that stabilizes with L = U = P_T, where P_T is the bidirected relation of T. No local finiteness assumption on T is required. For an initial structure that is not a bidirected tree, budget 0 cannot reach any P_T, so the least budget d_min(y; s_0, T) belongs to {1, 2}. An explicit directed-triangle example attains the value 1. The least budget sufficient uniformly for all starting stages, all minimal starts, and all target trees on E is exactly 2; its lower bound follows from the bidirected tree starts treated in v1.65. The classification of the values 1 and 2 for starts that are not bidirected trees, including whether the value 2 occurs for such a start, remains open here. Reachability is defined by L = P_T; the constructions in this version establish the stronger conclusion L = U = P_T. The reachability results concern the specific growth sequence E. The supplement contains supporting records, consistency checks, scripts, and finite-case computations; the finite computations do not replace the general proofs.