Generative Structural Theory v1.84 — Beyond Rules F and G: Structural Conditions for Residual Leaf Reductions

(Q13) of v1.83 asks for structural conditions for the general steps that reach a leaf of the cases (i)-(iii) of leaf detachment in the instances that the procedure P_FG^first of v1.83 leaves at six nodes and in which no sequence of candidates of its eight rules F, G, K, T, L, L', S and D+A reaches such a leaf (2178 of the 21192 residual instances of the population checked in v1.82 and v1.83). This text does not settle (Q1) of v1.76, the two-deletion chain question; it gives a partial answer to (Q13). Lemma ϴ1: when a leaf l of the tree S, with tree neighbour p, is a closed transit node of a minimal strongly connected relation h, with in-neighbour a and out-neighbour b, a ≠ b, and {a, b} is an edge of S, the relation g that h becomes when the path a, l, b is written (a, b) is minimal on the other nodes, and hanging l again at p, by the 2-cycle of l and p or by one of its two pairs kept and the other turned to p under the condition (F3) of v1.82, gives a minimal relation inside h ∪ P_S with at most two deletions in which l is of case (i), and of case (ii) or (iii) for the forms that enter or leave l at p. Theorem ϴ2 and Corollary ϴ3: with (Q1) on one node less, a chain from h to P_S inside h ∪ P_S with deletion bound two follows; the procedure P_FGH^first, the rules F, G and H (Lemma ϴ1) before those of P, gives such a chain whenever it ends at a leaf of the cases (i)-(iii), and, by v1.83 Proposition Ͽ1, succeeds wherever P_FG^first does. These statements hold for every finite node set. Finite checks at six nodes find that 2066 of the 2178 have one least general step; their one-step solutions, counted as pairs (x, l) of a relation and a leaf, are 23718 pairs; steps of Lemma ϴ1 are among them in 578 of the 2066, and in the 440 others with a transit leaf in such a pair no transit leaf has its two neighbours joined by an edge of S. P_FGH^first reaches a leaf of the cases (i)-(iii) in 19570 of the 21192 (P_FG^first 18988; 578 more at the initial relation and 4 at a later turn) and leaves 1622; with the nine rules, some sequence of candidates reaches such a leaf in 26 of them (class I) and none in 1596 (class II), while general steps do in all. Two or more least general steps are needed in 112 (110 + 2) of the 2178 of class II with the eight rules, and in 116 (114 + 2) of all 1622 left, class I included. These numbers, among them the 578 instances with a step of Lemma ϴ1 and the 582 gained by P_FGH^first, are findings of the six-node enumeration, not general statements.