Generative Structural Theory v1.82 — Beyond the Detour Procedure: Structural Conditions for Further Leaf Reductions

(Q9) of v1.81 asks for conditions under which the 4 cases that the procedure P of v1.81 leaves among the 44 left by v1.79 (1299, 1548, 1861 and 2036) reach a relation with a leaf of the cases (i)-(iii) of leaf detachment, and more generally the residual instances at which P ends without one. This text does not settle (Q1) of v1.76, the two-deletion chain question; it gives partial results. Lemma F_Ϯ moves an open ear of a minimal strongly connected relation h (an ear whose removal leaves h strongly connected on the other nodes) to new ends u' and v', keeping its inner pairs: the result is minimal, with at most two deletions, exactly under a condition (F3) on the pair (u', v'). Theorem F_Ϯ gives, at a leaf l of the tree S outside the ear, the condition (F4) under which l is of case (i) after the move; then, if (Q1) holds on one node less, a chain from h to P_S inside h ∪ P_S with deletion bound two follows. Corollary Ϯ1: if l is a transit node of h without the ear, with in-neighbour a and out-neighbour b distinct, moving the ear to run from a to b, along two edges of S, is such a step, (F3) and (F4) holding by themselves; in the new relation l is a transit node opened by the ear. Corollary Ϯ2 moves the ear to have both ends at one node t when h without the ear and l is strongly connected. Corollary Ϯ3: the procedure P_F, which is P with the rule F (Theorem F_Ϯ at a leaf) after its six rules, gives such a chain whenever it ends at a leaf of the cases (i)-(iii). 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 P_F makes the steps of P wherever P ends at such a leaf, and that it reaches a leaf of case (i) in each of the 4 cases by one step of the rule F, of the form of Corollary Ϯ1, with no auxiliary relation k: from h in 1299 and 1861 (one step of P_F), and after the two steps of P in 1548 and 2036 (three steps of P_F). A step of Theorem F_Ϯ from h itself to a leaf of case (i) also exists in 1548 and 2036, which P_F does not take; so a one-step move exists for all 4, while P_F reaches 1548 and 2036 in three steps. Among the 21,192 residual instances of this six-node population, P_F reaches a relation with a leaf of cases (i)-(iii) in 18,434 instances, compared with 13,530 for P. These counts are findings of the six-node enumeration, not general statements.