Generative Structural Theory v1.78 — Residual Cases and Preparatory Chains for Leaf Detachment

v1.77 reduced the two-deletion chain question (Q1) of v1.76 (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?) to the detachment of a leaf, and gave the chain, assuming (Q1) on one node less, for every instance with a leaf of one of three cases (v1.77 Theorem I_Ϥ). This text does not settle (Q1); it gives partial results. Re-pointing the pairs into a leaf l with neighbour p (Proposition P_Ϧ) replaces the pairs into l by the one pair (p, l) and the relation on the other nodes by a minimal relation k that holds the pairs (p, b) for the pairs (l, b) of h and is otherwise inside (h ∪ P_S)[V']; the result g is minimal, inside h ∪ P_S, and has l as a leaf of the second case of v1.77. So, assuming (Q1) on one node less, the chain exists for every instance with a leaf of a new case (iv), where this step deletes at most two pairs (Theorem E_Ϧ), and whenever a chain with bound 2 inside h ∪ P_S leads from h to a relation with a leaf of the cases (i)-(iv) (Corollary Ϧ1); on a class K+ defined by recursion, which contains the class K of v1.77, the chain is built from given witnesses with no search for a chain (Corollary Ϧ2). A finite check classifies the 21192 cases at six nodes with no leaf of the three cases of v1.77: 566 lie in other sufficient conditions of v1.77, 17948 of the others have a leaf of case (iv), and each of the 2678 left has a preparatory chain of one step (2676 cases) or two steps (2 cases) to a relation with a leaf of the cases (i)-(iv), the whole chains being checked step by step; in 24 of them every one-step preparation is a stall, a finding on one-step preparations only. K+ holds 121470 of the 166896 cases at six nodes.