Generative Structural Theory v1.77 — Two-Deletion Chains: Leaf-Detachment Reduction and Sufficient Conditions for Budget-Two Reachability

v1.76 left open the two-deletion chain question (Q1): for every finite set V, every minimal strongly connected relation h on V and every tree S with node set V, is there a chain of minimal strongly connected relations from h to the bidirected tree P_S, each contained in h ∪ P_S, with at most two pairs deleted at each step? A positive answer would give budget 2 from every start under the condition (AT) on the class Φ_I of v1.76. This text does not settle (Q1); it gives partial results. A minimal relation that holds both pairs of an edge splits at that edge into two minimal relations (Lemma B_Ϥ); so no step from a directed Hamiltonian cycle h reaches a minimal relation x holding both pairs of an edge {u, v} with x minus h ⊆ Pu,v (Proposition H_Ϥ), while a step adding pairs of several edges of S is not excluded. (Q1) is reduced to the detachment of a leaf of S, a chain that makes the leaf a pendant node at its neighbour (Proposition R_Ϥ); over all finite node sets it is equivalent to the detachment of some leaf and to that of any given leaf (Proposition Q_Ϥ), so that the reduction is another form of the question. A leaf is detached in one step when h without it is strongly connected (Proposition D_Ϥ) and, given a chain on the other nodes, when its only pair in comes from its neighbour or its only pair out goes to it (Proposition V_Ϥ, through Lemma L_Ϥ, which moves the pairs out of a node to a new node), the minimality, the allowed pairs, the deletions and the end of every intermediate relation being proved. So, if (Q1) holds on m - 1 nodes, it holds for every instance on m nodes with a leaf of one of these kinds (Theorem I_Ϥ), and it holds, with no search, on a class K defined by recursion (Corollary Ϥ3), with budget 2 on the growth sequences for the starts whose instance is in K. A finite check of the drafting side finds (Q1) and the detachment of every leaf in every case with at most six nodes and in random samples with seven and eight nodes; at six nodes 21192 of its 166896 cases have no leaf of the kinds of Theorem I_Ϥ.