Generative Structural Theory v1.85 — Beyond Tree-Edge Reattachment: Tree-Path Conditions for Residual Leaf Reductions

(Q15) of v1.84 asks for structural conditions for the one-step solutions of the instances that the procedure P_FGH^first of v1.84 leaves at six nodes, in particular at a closed transit leaf whose two neighbours are not joined by an edge of the tree S, where the re-hanging across a tree edge of v1.84 (the rule H) does not apply. This text does not settle (Q1) of v1.76, the two-deletion chain question; it gives a partial answer to (Q15). Lemma ϒ1: let a leaf l of S, with tree neighbour p, be a closed transit node of a minimal strongly connected relation h, with in-neighbour a and out-neighbour b, a ≠ b, and let A be the set of the pairs of the path of S from a to b, directed from a to b, and h0 the relation h without the two pairs at l. Then h0 ∪ A is strongly connected on the other nodes; under the condition (C) that every pair of h0 is essential in h0 ∪ A, taking out only redundant pairs of the path gives a minimal relation g that keeps all of h0, and hanging l again at p by the 2-cycle of l and p gives a minimal relation inside h ∪ P_S with at most two deletions in which l is of the cases (i), (ii) and (iii). When {a, b} is an edge of S, (C) holds and the step is that of v1.84 Lemma ϴ1. (C) is a sufficient condition, not a necessary one. Theorem ϒ2 and Corollary ϒ3: 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_FGHJ^first, the rules F, G, H and J (Lemma ϒ1 with (C)) 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, ends at such a leaf wherever P_FGH^first does. These statements hold for every finite node set. Finite checks at six nodes find that, of the 1484 instances of class II with one least general step that P_FGH^first leaves, 482 have a closed transit leaf whose neighbours are not joined by an edge of S, and (C) holds at one of them in 98. P_FGHJ^first ends at a leaf of the cases (i)-(iii) in 19668 of the 21192 residual instances (P_FGH^first in 19570; the 98 gained all by one step of the rule J at the initial relation) and leaves 1524, 26 of class I and 1498 of class II. On all minimal relations with six nodes, at the leaves of this kind where (C) fails, the same construction gives a relation that is not minimal in 62608 of 65708 steps and a minimal one in the other 3100. A success counted here is an arrival at a leaf of the cases (i)-(iii), not an instance for which (Q1) is proved. These numbers are findings of the six-node enumeration, not general statements.