Generative Structural Theory v1.76 — Deletion Budgets for Eventually Available Target Trees

v1.75 gave the least deletion budget for every stage, every start and every tree on the class Φ_J of the growth sequences in which every new node comes as a leaf at 0 and the node set of every finite stage is later joined inside. This text drops the joining condition and asks only that the pairs of the target tree T come, P_T ⊆ ∪ R_n (the condition (AT)), on the class Φ_I of the growth sequences with 0 in V_0 and the pairs at a new node only (0, v) and (v, 0). (AT) is necessary and sufficient for P_T to be reached with some finite uniform deletion budget (Proposition N_Ϣ, Proposition E_Ϣ). From a bidirected tree start P_S the least budget is 0 if T = S*, the tree S with every later node hung at 0, and 2 otherwise, as on Φ_J: the exchanges edge by edge of v1.75 need only the two pairs of each target edge (Proposition C_Ϣ), and budget 1 fixes the selection (Proposition U_Ϣ). From a start that is not a bidirected tree, budget 2 reaches P_T when, besides (AT), the pairs among the nodes of the start all come (the condition (AW)), or when a chain of minimal relations with at most two deletions per step leads from the start to a bidirected tree of available pairs (Theorem A_Ϣ); a positive answer to a finite question on one node set, the two-deletion chain question (Q1), would give such a chain at every stage. The lower bound for an infinite branch at zero needs no joining (Proposition L_Ϣ). The value 1 of the table of v1.75 does not carry over: on a sequence K of Φ_I and a tree T_K with (AT), all of whose branches at zero are finite, the least budget from the directed triangle at 0 is 2 (Proposition K_Ϣ). Corollary Ϣ1 collects what is settled; the bound 2 for a general start under (AT) alone and a characterization of budget 1 are left open.