Generative Structural Theory v1.71 — Unit-Budget Reachability of Trees with Finitely Many Edges Not Incident with Zero on the Cofinally Complete Growth Sequence E

A growth sequence is a sequence of finite relations R_n on non-empty 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. Its deletion budget bounds the number of ordered pairs of x_n that xₙ₊₁ drops; the number of pairs added is not bounded. The lower limit L consists of the pairs present at every sufficiently large stage, and the upper limit U consists of the pairs present at infinitely many stages. Stabilization means L = U. Version v1.70 classified unit-budget reachability through bidirected trees for trees whose branches at zero are finite paths with consecutive labels. Version v1.71 removes both restrictions and establishes a classification for every tree T with node set N_0, on the cofinally complete growth sequence E of v1.63, from the directed triangle y = {(0, 1), (1, 2), (2, 0)} at stage 0. Here P_T denotes the relation containing both orientations of every edge of T. The following four conditions are equivalent: (i) some selection sequence from y with deletion budget 1 and P_T ⊆ L contains a bidirected tree term; (ii) some such sequence stabilizes with L = U = P_T, is eventually a sequence of bidirected trees, and eventually stops deleting pairs; (iii) some selection sequence from y with deletion budget 1 and L = P_T deletes pairs at only finitely many transitions; and (iv) T has finitely many edges not incident with zero. The sufficient direction is constructive. A finite assembly lemma starts from a triangle state X(H_0; p_0, q_0), where H_0 is the forest on V with edges {0, v} for v in V minus {0, p_0, q_0}. Two auxiliary leaves at zero support a sequence of pull and retire moves, each deleting exactly one ordered pair, followed by a closing move that produces the prescribed bidirected tree. On E, this construction reaches PT[V_N] at a finite stage N. Subsequent vertices are added as bidirected leaves at zero without further deletions. No earlier term is a bidirected tree. As an additional result, every tree with finitely many edges not incident with zero is also reached by a sequence that stabilizes with L = U = P_T without ever containing a bidirected tree term. Its least deletion budget from y is therefore exactly 1. The classification concerns reachability through a bidirected tree from the specified start y. It is not a complete classification of unit-budget reachability without bidirected tree terms, nor of reachability from arbitrary initial minimal structures. The supplement contains construction code, finite checks, and supporting records; the general results are established by the proofs.