Generative Structural Theory v1.81 — Detours past Transit Nodes: Structural Conditions for Preparatory Steps

(Q7) of v1.80 asks for conditions on a minimal strongly connected relation h and a tree S under which steps of Lemma R_Ϫ or Lemma M_Ϫ lead to a relation with a leaf of the cases (i)-(iv) of leaf detachment. This text does not settle (Q1) of v1.76, the two-deletion chain question; it gives partial results. Lemma D_Ϭ gives three local detour constructions under explicit neighbourhood conditions at a transit node w with in-neighbour a, out-neighbour b distinct from a, and return pair (b, a). The basic construction moves an additional pair (b, z) to start at a when a is entered only from b; two variants carry this detour through a bidirected pair at a or at z under the corresponding hypotheses of the lemma. Each construction gives a minimal relation, with one or two pairs deleted, in which a reaches b without passing through w. Theorem T_Ϭ applies it at a leaf l of S: when the new pair lies in the bidirected tree P_S, the step stays inside h ∪ P_S and makes l a leaf of case (i); if (Q1) holds on one node less, a chain from h to P_S inside h ∪ P_S with deletion bound two follows. Proposition L_Ϭ closes a leaf onto its neighbour in S (case (iii)); Corollaries Ϭ1-Ϭ3 combine these steps with v1.80 Lemma R_Ϫ and v1.79 Lemma A_Ϩ (a step at a node entered only from the leaf, the satellites of the leaf re-hung first, and a detour at a node that is not a leaf followed by its re-hanging). Corollary Ϭ4: a procedure P that applies these rules and v1.80 Theorem K_Ϫ in a fixed order, each step decided by testing conditions on the current relation and S, gives such a chain whenever it ends at a leaf of the cases (i)-(iii), with no auxiliary relation k of case (iv). These statements hold for every finite node set. A finite check at six nodes (every labelled minimal relation with one tree of each shape) finds that, of the 44 cases left by v1.79, the conditions of Theorem T_Ϭ hold initially in exactly the 18 that v1.80 reached by one new step to the cases (i)-(iii), and P reaches a leaf of the cases (i)-(iii) in 40, so that 28 of the 32 cases not explained by Theorem K_Ϫ are reached by the new rules, with no k; no condition is given for the other 4 (1299, 1548, 1861 and 2036), which are not counterexamples to (Q1); on all 21192 residual instances at six nodes P ends at such a leaf in 13530. These coverage figures are findings of the six-node enumeration, not general statements.