Generative Structural Theory v1.69 — Unit-Budget Reachability of Trees with 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 ⊆ V_{n+1}, R_n ⊆ R_{n+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 x_{n+1} drops; the number of pairs added is not bounded. Version v1.68 established that, on the cofinally complete growth sequence E, a tree reachable with deletion budget 1 must have only finite components after removal of vertex 0. Version v1.69 proves unit-budget reachability for a family of trees satisfying this necessary condition and establishes the existence of target trees that cannot be reached with budget 1 by any selection sequence containing a bidirected tree term. For any strictly increasing sequence of integers 0 = c_0 < c_1 < c_2 < ..., define T_c on the non-negative integers by the paths 0 - c_j - (c_j - 1) - ... - (c_{j-1} + 1), for j ≥ 1. The components of T_c minus {0} are finite paths with consecutive labels. Each is attached to 0 by the edge {0, c_j}; the corresponding path including this attachment edge has length c_j - c_{j-1}. No common bound on these lengths is required. Examples include the star at 0, infinitely many paths of length 2 attached to 0, and paths of lengths 1, 2, 3, ... attached to 0. Starting at stage 0 from the directed triangle y = {(0, 1), (1, 2), (2, 0)}, an explicit selection sequence on E reaches the bidirected relation P_{T_c} with lower and upper limits L = U = P_{T_c}. Every term is inclusion-minimal and strongly connected, and no term is a bidirected tree. The first transition deletes no pair, and every subsequent transition deletes exactly one pair. Budget 0 cannot reach a tree from this initial triangle, so the least deletion budget is exactly 1. A further result applies to every tree T on the non-negative integers having infinitely many edges not incident with 0: any selection sequence on E with budget 1 and P_T ⊆ L, from any starting stage and any initial minimal structure, contains no bidirected tree term. For T_c, this applies precisely when c_j - c_{j-1} ≥ 2 for infinitely many j. These trees are therefore reachable with budget 1 from y, but only through sequences that never contain a bidirected tree. The reachability construction concerns E, the specified initial triangle, and the stated label arrangement. Unit-budget reachability for more general finite branches, other label arrangements, and other initial structures remains outside the classification established here.