Generative Structural Theory v1.35 — Toward Infinite Structural Selection: Minimal Structures, Admissible Extensions, and Compatibility

A growth sequence is a sequence of finite parent relations R_n on node sets V_n with V_n ⊆ Vₙ₊₁, R_n ⊆ Rₙ₊₁ and every stage strongly connected. At each stage three sets of spanning sub-relations are kept apart: the strongly connected ones (A_n), the inclusion-minimal ones (M_n) and those of least size (C_n), with C_n ⊆ M_n ⊆ A_n. A selection x in M_n passes to y in Mₙ₊₁ by a preserving transition (x ⊆ y) or under a deletion budget (|x minus y| <= k); for minimal members x ⊆ y holds iff y restricted to the old nodes is x, and a stage that adds no node never stops a preserving selection, so collisions need new nodes. For a fixed rule, continuation to every finite depth is equivalent to an infinite admissible sequence (an application of König's infinity lemma, in ZFC), and the infinite continuation from x is unique iff x is infinitely continuable and every infinitely continuable state reached from x has exactly one infinitely continuable successor. When a stage attaches a finite relation at one node, its minimal members are the unions of the old ones with those of the attached part: preserving successors exist iff the attached part is strongly connected, and their number is the number of minimal members of the attached part, so existence and uniqueness have different conditions. The union of an infinite preserving sequence is strongly connected and inclusion-minimal on the union of the stages; the converse fails (a counterexample with complete stages and a two-way path on the non-negative integers), and a set of pairs is such a union from a given stage iff each of its restrictions to the later stages is an inclusion-minimal member there. Examples with five successors and a unique infinite continuation, with collisions, and with branching at every stage are given. A complete classification of existence and uniqueness for general growth sequences, and the case of rules that allow deletions, are not solved in this manuscript. No novelty is claimed; the proofs and examples are given by hand, with no machine enumeration or computation.