Generative Structural Theory v1.36 — Revisable Structural Selection: Stabilization and Limit Preservation

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; M_n is the set of inclusion-minimal strongly connected spanning sub-relations of R_n. A selection sequence chooses x_n in M_n at every stage and may revise earlier choices: a pair selected at one stage may be dropped at the next, within a deletion budget. Its lower limit L (the pairs selected from some stage on) and upper limit U (the pairs selected infinitely often) are defined for every sequence, and the sequence stabilizes iff L = U. A deletion budget of 1, a stage structure that is the only inclusion-minimal one at every stage, and stabilization of every pair together do not make L strongly connected: in the example the return pair moves one stage further at each stage, L is a one-way infinite path, and the union of the parent relations has no inclusion-minimal strongly connected spanning sub-relation at all. L is strongly connected iff for every two nodes one finite path between them is selected at every stage from some stage on. If every stage is inclusion-minimal and L is strongly connected, then L is inclusion-minimal and the sequence stabilizes; inclusion-minimality at each stage alone guarantees neither. Further examples show a sequence that does not stabilize under a budget of 2, and a budget-2 sequence whose limit, a two-way infinite path, is on the same growth sequence the union of no sequence that keeps its earlier choices. A sufficient condition stated on the growth sequence and the rule alone, and the case of budget 1 beyond the first example, are not treated. No novelty is claimed. The proofs and examples are presented as symbolic arguments, without machine enumeration or numerical computation on the relations studied. AI assistance was used in drafting and review, as detailed in the disclosures.