Generative Structural Theory v1.61 — Unit-Budget Reachability of Arbitrary Bidirected Trees on Complete Growth Sequences

A growth sequence is a sequence of finite relations R_n on 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, and its deletion budget bounds the number of pairs of x_n that xₙ₊₁ drops; the number of pairs added is not bounded. Version 1.60 established unit-budget reachability of every bidirected tree on the original Z3 growth sequence. This work extends that result to the class Γ of complete growth sequences: each V_n is finite and non-empty, the sets V_n increase with union N_0, and R_n contains all ordered pairs of distinct nodes of V_n. The order of arrival is arbitrary, and each transition may introduce no nodes, one node or any finite number of nodes, with no uniform bound on the size of an arriving batch. For every growth sequence in Γ, every tree T with node set N_0, every starting stage s_0 and every initial structure y in Ms_0, there is a selection sequence with xs_0 = y, x_n in M_n, |x_n \ xₙ₊₁| ≤ 1 and L = U = P_T, where P_T is the bidirected relation of T. Thus every pair of P_T is present from some stage onward, and every other pair is absent from some stage onward, with the stage depending on the pair. The construction replaces the initialization used on Z3 and identifies the properties of the growth sequence needed by the subsequent operations. Explicit transition rules accommodate empty arrivals and arbitrary finite batches while preserving minimality and the deletion budget. Finite termination, fairness and stabilization are established without inserting stages or modifying the given growth sequence. The least uniform deletion budget is 0 exactly when y = P_T[Vs_0] and T[V_n] is connected for every n ≥ s_0; otherwise it is 1. Examples include one-node arrivals, growing batches, repeated stages and arbitrary enumerations of N_0. Growth sequences with non-complete edge sets, quantitative bounds on stabilization stages and online constructions remain outside the scope of this work.