Generative Structural Theory v1.37 — Revisable Structural Selection under a Unit Deletion Budget

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. GST v1.36 showed, under a budget of 2, a selection sequence that does not stabilize and one whose lower limit, a two-way infinite path, no preserving sequence on the same growth sequence reaches. Here both are examined under a deletion budget of 1. On the growth sequence of v1.36 Example Z2, and on every growth sequence obtained from it by inserting stages, every selection sequence with deletion budget 1 stabilizes: on three nodes the inclusion-minimal strongly connected relations are two directed triangles and three two-way paths, and from a two-way path every other one needs two deletions. On another growth sequence, with a core of four nodes, a selection sequence with deletion budget 1 does not stabilize. Inserting one stage between consecutive stages of v1.36 Example Z3 gives a new growth sequence on which a selection sequence with deletion budget 1 reaches the two-way infinite path, while no preserving sequence on this new growth sequence does. So on these two growth sequences the least uniform deletion budget for each phenomenon is 1. On the original growth sequence of v1.36 Example Z3, every selection sequence with deletion budget 1 keeps a growing number of pairs outside the path; whether one of them reaches the path is not decided. 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.