Generative Structural Theory v1.70 — Exact Classification of Unit-Budget Reachability through Bidirected Trees for Consecutive Finite Path Branches 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 v1.69 established unit-budget reachability on the cofinally complete growth sequence E for the trees T_c with consecutive finite path branches. For a strictly increasing sequence of integers c = (c_j) with c_0 = 0, T_c has node set N_0 and edges {u, p(u)}, where p(u) = 0 if u is one of c_1, c_2, ... and p(u) = u + 1 otherwise. Its rooted branches are the finite paths 0 – c_j – (c_j − 1) – ... – (cj−1 + 1), of length c_j − cj−1, counting the edge to 0. Every such tree is reached from the directed triangle y = {(0, 1), (1, 2), (2, 0)} at stage 0 by a selection sequence with deletion budget 1 that stabilizes with L = U = PT_c and has no bidirected tree term. Version v1.70 classifies exactly when the same target can also be reached through a bidirected tree. On E, from y at stage 0, the following five conditions are equivalent: (i) a selection sequence with deletion budget 1 and PT_c ⊆ L has a bidirected tree term; (ii) a selection sequence with deletion budget 1 stabilizes with L = U = PT_c, has only bidirected tree terms from some stage onward, and deletes no pair from that stage onward; (iii) a selection sequence with deletion budget 1 and L = PT_c deletes pairs at only finitely many transitions; (iv) T_c has finitely many edges not incident with 0; and (v) only finitely many rooted branches have length at least 2. Condition (i) requires only containment in the lower limit, which is weaker than reaching the target in the sense L = PT_c; the equivalence therefore also covers that exact reachability condition. For the constructive direction, choose k such that p(u) = 0 for every u ≥ k + 1. Follow the construction of v1.69 through stage 2k + 1, then replace its directed triangle core at the complete stage 2k + 2 by two bidirected edges incident with 0. This transition deletes exactly one ordered pair and produces P_{T_c[V₂ₖ₊₂]}. Thereafter, each new node is added as a bidirected leaf at 0 without further deletion, yielding L = U = PT_c. Thus, the minimum deletion budget from y is 1 for every T_c. If only finitely many rooted branches have length at least 2, both kinds of unit-budget reaching sequences exist: those that meet a bidirected tree and those that never do. If infinitely many rooted branches have length at least 2, no unit-budget selection sequence whose lower limit contains PT_c can meet a bidirected tree. More generally, on E, any unit-budget selection sequence with lower limit P_T for a tree T having infinitely many edges not incident with 0 must delete a pair at infinitely many transitions, regardless of its starting stage or initial minimal structure, whenever such a sequence exists. The reachability classification concerns the specific growth sequence E, the specified initial triangle y, and the family T_c. Reachability for other trees with finite branches at 0 and for other initial structures remains outside this classification. The supplement contains the supporting records, scripts, and finite checks; the general results rest on the proofs.