Generative Structural Theory v1.62 — Deletion Budgets and Finite-Prefix Reachability on Eventually Complete Growth Sequences

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 v1.61 established that deletion budget 1 reaches the bidirected relation P_T of every tree T with node set N_0 on every complete growth sequence, from every starting stage and every initial minimal structure, with lower and upper limits L = U = P_T. This paper extends the setting to the class Λ of eventually complete growth sequences: the node sets increase to N_0, and there is a finite stage N_c such that R_n contains every ordered pair of distinct nodes of V_n for all n ≥ N_c. Before that stage, the relations need not be complete, and pairs between existing nodes may be added later. Eventual completeness is stronger than requiring only that the union of the relations be complete. There is no common finite deletion budget for the whole class Λ. For every integer k ≥ 1, an explicit growth sequence forces exactly k deletions at its first transition. On this sequence, the least budget for reaching P_T is k for every tree T with node set N_0. For a starting stage s_0, an initial structure y, and N ≥ max{s_0, N_c}, the finite interval from s_0 to N is represented by a layered directed graph. Its vertices are stage–structure pairs (n, x), and each transition has weight |x \ x′|. The bottleneck value b_* is the minimum, over passages starting at y, of the largest transition weight. Extending a minimal structure by attaching new vertices as bidirected leaves shows that b_* is independent of the choice of N. For every integer d ≥ 1, a selection sequence from y with deletion budget d and lower limit P_T exists if and only if b_* ≤ d. This is also equivalent to the existence of such a sequence with L = U = P_T. Thus, once a stage beyond completion is fixed, reachability with a positive budget reduces to a finite passage problem. The budget-0 case is characterized for every growth sequence with node union N_0: it is possible exactly when y = P_T[Vs_0] and, for every n ≥ s_0, the induced tree T[V_n] is connected and P_T[V_n] ⊆ R_n. On Λ, the least budget is therefore 0 under these conditions and max{1, b_*} otherwise. The containment condition is automatic after completion, but connectedness is still required throughout the tail. The existence of a completion stage is assumed; no procedure for detecting eventual completeness from finitely many observed stages is claimed.