Generative Structural Theory v1.80 — Structural Analysis of the Remaining Preparatory Cases

v1.79 left 44 cases at six nodes, among the residual instances of (Q5) of v1.78, that have no preparation of one or two steps of its Lemmas A_Ϩ and B_Ϩ to a relation with a leaf of one of the cases (i)-(iv) and lie outside its Theorem H_Ϩ; each has a one-step preparation found by a search, so none is a counterexample to the two-deletion chain question (Q1) of v1.76. This text does not settle (Q1); it gives partial results. Under their stated hypotheses, three local operations preserve inclusion-minimal strong connectivity, each deleting a number of pairs fixed by its data (at most two in the steps used here), and stay inside h ∪ P_S when the pairs they add are pairs of the bidirected tree P_S. Lemma R_Ϫ moves pairs between a node w that is entered from one node c only and that node c, provided w has at least two pairs out or c does not reach the one out-neighbour of w without passing through w. Lemma M_Ϫ carries the part of the relation beyond a bidirected pair {c, s} onto a new pair: a step of one deletion on the other part that adds a pair (c, d) becomes a step on all nodes in which s lies on the path c, s, d. Lemma E_Ϫ re-hangs an ear w_1, ..., w_r whose ends are joined without it as the bidirected path w_1, ..., w_r hung at one node, deleting at most the two end pairs of the ear. Theorem K_Ϫ gives a construction from conditions on h and the tree S: if an ear is closed at a leaf l of S, its path lies in S, one of its nodes has a neighbour in S outside it and other than l, and apart from the ear l has exactly one pair in, (a, l), and one pair out, (l, b), with a = b or a reaching b without passing through the ear or l, then re-hanging the ear is one step to a relation in which l is of case (i); if (Q1) holds for every instance on one node less, a chain from h to P_S inside h ∪ P_S with deletion bound two follows. Corollary Ϫ1 states the same for any steps of these lemmas and of v1.79 that lead to a relation with a leaf of the cases (i)-(iv), the existence of the steps, and in case (iv) of the auxiliary relation k of v1.78, being hypotheses. The lemmas and the conditional theorem hold for arbitrary finite node sets. A finite check of the drafting side at six nodes finds that, in the 44 cases, steps of Lemmas A_Ϩ and B_Ϩ of any number reach a relation with a leaf of the cases (i)-(iv) in 12, each first at three steps, and never in 32, all reachable relations being examined; Theorem K_Ϫ applies to 12, one step of the new lemmas leads to a leaf of the cases (i)-(iii) in 18 more and to case (iv) in 12 more, and two steps in the last 2, the whole chains checked step by step. The coverage findings for the 44 cases are restricted to the six-node enumeration; no universal coverage by the proposed operations is claimed.