Generative Structural Theory v1.87 — Beyond a Single Tree Path: Full-Tree Leaf Reattachment under Budget Two

This version studies bidirected reattachment of a transit leaf using pairs from the whole target tree, extending the tree-path approach of v1.85 and v1.86. It establishes the deletion-budget obstruction and an exact existence condition for this form of reattachment. It does not settle the general two-deletion chain question (Q1) of v1.76. Let h be an inclusion-minimal strongly connected relation on a finite node set V, and let l be a leaf of a tree S on V. Suppose that l is a transit node of h with distinct in-neighbour a and out-neighbour b, and let p be its tree neighbour. Write h0 for h with the two pairs at l removed. For a relation x whose only pairs at l are (p,l) and (l,p), write g for the remaining relation on V′ = V ∖ {l}. When p is distinct from a and b, Lemma Ͻ1 gives |h ∖ x| = 2 + |h0 ∖ g|. Thus, under deletion budget two, a reattachment of this form must retain every pair of h0. Moreover, x is inclusion-minimal strongly connected on V if and only if g is inclusion-minimal strongly connected on V′. Consequently, when p is distinct from a and b, such a step inside h ∪ P_S exists under budget two exactly when condition (C'') holds: there is a set B ⊆ PS-l ∖ h0 such that h0 ∪ B is inclusion-minimal strongly connected on V′. Lemma Ͻ2 characterises this condition by cut sets, extending the cut characterisation to an arbitrary set of candidate pairs. Theorem Ͻ3 applies it to leaf reduction. Corollary Ͻ4 introduces the terminal rule X and proves that P_FGHJNX^first preserves the successes of P_FGHJN^first. A chain to P_S follows under the additional assumption that (Q1) holds on one fewer node. Finite checks at six nodes find that, among the 1484 instances of the population of v1.85, 482 have at least one closed transit leaf whose distinct neighbours are not joined by an edge of S. Of these 482 instances, 372 satisfy (C') at none of those leaves, with 542 such leaves in total. Condition (C'') holds at 8 of these leaves, in 8 instances, each with a unique set B consisting of one tree pair outside the path. On the fixed population of 21192 residual instances, P_FGHJNX^first reaches a leaf of cases (i)–(iii) in 19688 instances, compared with 19680 for P_FGHJN^first. All eight additional successes occur through one initial application of X, and no previous success is lost. The remaining 1504 instances comprise 26 of class I and 1478 of class II. These counts are finite-enumeration results; a counted success is arrival at a leaf admitting the stated reduction, not an instance for which (Q1) has been proved. The second half of (Q17) is answered only for one-step bidirected reattachment at the tree neighbour. Other forms of transformation, preparatory chains, and the general question (Q1) remain open. No polynomial-time decision procedure for (C'') is claimed.