Generative Structural Theory v1.38 — A Unit Deletion Budget Reaches the Bidirected Ray on the Original Z3 Growth Sequence

A growth sequence is a sequence of finite relations R_n on node sets V_n with V_n ⊆ V_{n+1}, R_n ⊆ R_{n+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 x_{n+1} drops; the number of pairs added is not bounded. On the growth sequence of v1.36 Example Z3, whose stage n has the nodes 0, ..., 2n and 2n + 2 and all pairs between them, the bidirected ray P, the relation with pairs in both directions between adjacent nodes of the sequence 0, 1, 2, ..., is a lower limit that no preserving selection sequence reaches; v1.36 reached it with a deletion budget of 2, and v1.37 left open whether a budget of 1 suffices on this growth sequence, showing only that such a sequence keeps a growing number of pairs outside P. Here the growth sequence is kept fixed, with no inserted stages, and a selection sequence with deletion budget 1 is given whose lower and upper limits are both P. At each stage the two new nodes enter in a directed triangle hung at the previous top node, at no deletion; the two outside pairs of each triangle are removed later, one per transition, from the bottom up, at half the speed of the growth. The number of pairs outside P at stage n is exactly n + 2, the least that v1.37 allows, and still every such pair leaves after finitely many stages. So the least uniform deletion budget for reaching P on this growth sequence is 1. No novelty is claimed. The proofs and the example 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.