Generative Structural Theory v1.63 — Cofinal Completeness and Unbounded Deletion Requirements on 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 v1.62 treated eventually complete growth sequences, whose stages contain all ordered pairs of distinct nodes from some finite stage onward. This version considers the larger class Φ of cofinally complete growth sequences: for every m there is a stage n ≥ m with R_n = K(V_n), where K(W) denotes all ordered pairs of distinct nodes of W. The node sets have union ℕ₀. Eventual completeness implies cofinal completeness, and cofinal completeness implies that the union of the relations is K(ℕ₀); neither converse holds. A forcing lemma describes a bidirected path of length k attached at one node of an arbitrary finite loopless strongly connected relation. Adding a new node that closes the path into a directed cycle forces every selection to discard all k reverse pairs of the path. Repeating this construction with path lengths m + 1 produces a single cofinally complete growth sequence D that is not eventually complete. On D, from every starting stage and every initial minimal structure, no infinite selection sequence has a finite uniform deletion budget. Thus infinitely many complete stages, and even a complete union of the relations, do not guarantee bounded-budget continuation. For an arbitrary growth sequence, a starting stage s₀, and an initial minimal structure y, let b_N be the least possible maximum deletion count over a selection from y through stage N. These finite-interval values are non-decreasing. Writing b_∞ = sup_N b_N, König’s lemma gives an exact existence criterion: an infinite selection sequence with uniform deletion budget d exists if and only if b_∞ ≤ d. When b_∞ is finite, it is the least uniform budget for continuation and is attained by a selection sequence. These results also apply to growth sequences with no complete stage. Continuation within a budget is distinguished from reaching a prescribed bidirected tree P_T as a limit. On eventually complete growth sequences, b_∞ equals the finite-prefix value b_* of v1.62. For every integer d ≥ 1 and every tree T on ℕ₀, continuation with budget d, reachability with lower limit P_T, and reachability with lower and upper limits L = U = P_T are equivalent. No such equivalence is established here for cofinally complete sequences that are not eventually complete. Examples illustrate both unbounded forced deletions and zero-budget continuation, while target reachability beyond eventual completeness remains a question for further study.