Generative Structural Theory v1.41 — Unit-Budget Reachability of Bidirected Double Rays 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 ⊆ 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. 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, v1.40 showed that from every starting stage and every inclusion-minimal strongly connected initial structure, deletion budget 1 reaches the bidirected tree of any one-ended caterpillar with node set {0, 1, 2, ...}, with equal lower and upper limits. Here the target is the bidirected double ray P_σ along a bijection σ from the integers onto {0, 1, 2, ...}: the relation with a pair in each direction between σ(k) and σ(k + 1) for every integer k. Budget 0 reaches P_σ from a start exactly when the initial structure is P_σ restricted to the nodes of that stage and the positions under σ of the nodes of every later stage form an interval of integers (Corollary Z_D, from Proposition Z of v1.40); Example D* gives a σ for which those positions form an interval at every stage. For every σ, budget 1 reaches P_σ from every starting stage and every initial structure, with lower and upper limits both P_σ, so that the membership of every pair settles (Theorem D); for double rays the least budget is therefore 0 or 1, as Corollary Z_D decides (Corollary D5). The construction fixes the node 0 as a root and calls the other nodes left or right by their positions under σ. A separation phase moves the left nodes one at a time, by the node-out of v1.40, out of a single working relation into a second one that meets the first only at the root; it uses new nodes of the right side only, and so ends after finitely many stages (Lemma P). The settled part is then an interval of the double ray with a working relation at each end; each working relation is fed only by the new nodes of its own side, the two ends advance in turn, and at most one pair is deleted at each stage. This manuscript does not extend the reachability result beyond double rays; two-ended caterpillars with leaves and trees with three or more ends remain outside its scope. No novelty is claimed; the proofs are given by hand, with no machine enumeration or computation.