Generative Structural Theory v1.89 — Beyond Existence Conditions: Cut Criteria for One-Sided Leaf Moves under Budget Two

(Q21) of v1.88 asks for a structural description of the condition (O_s) under which a one-sided re-hanging of a closed transit leaf at its tree neighbour, keeping one pair of the leaf, exists within the deletion bound two (v1.88 Lemma ϗ1). This text does not settle (Q1) of v1.76, the two-deletion chain question, or (Q22) of v1.88; it puts (O_s) in the form of directed cuts and of reachability, draws from it conditions on the pair deleted that do not depend on the tree pairs added, and explains by a bypass why the step fails where it does. Lemma ϙ1 restates the condition (E) of v1.88, for a fixed deletion set D (empty, or {d} with d a pair of the core h0) and a fixed set B of tree pairs, as four conditions on the directed cuts of the other nodes V'. Theorem ϙ2: a step deleting D exists exactly when the virtual pair v is not in h0 minus D and some set B of tree pairs makes (h0 minus D) ∪ B ∪ {v} strongly connected on V' with no pair of h0 minus D bypassable; that each pair of B be essential can be dropped, an inclusion-minimal such B meeting it. Corollary ϙ3 gives three necessary conditions, (N1)-(N3), each decided by reachability without trying any B. Proposition ϙ4: at a closed transit leaf, the specific pair, (p, a) at side 0 or (b, p) at side 1, when in h0, is bypassed by v exactly when b reaches a in h0 without it; then no step has D empty, and every step deletes the specific pair or a pair on every such path; this alternative is a necessary condition only: deleting the specific pair itself is not required in general, and is not shown to work. These statements hold for every finite node set. Lemma ϙ1 and Theorem ϙ2 restate (E); (E) is not replaced by "g ∪ {v} minimal"; no condition that decides the existence of B without trying sets B is given, so that (Q21) is answered in part only, and no gain in the cost of the decision is claimed. Finite checks at six nodes, on the records of v1.88 (364 instances, 534 leaves and 7492 choices of leaf, side and deletion set), find the reachability form equal to the direct test in both directions and the counts of v1.88 again. At each side, the specific pair is in h0 at 184 leaves and bypassed at 144 of them; a step deleting it exists at 152, and at 32 deleting it fails although (N1)-(N3) hold, by the joint condition: every set of tree pairs that restores strong connectivity makes a pair of the core bypassable. These are figures of each side, not numbers of distinct leaves over the two sides. Each of the 52 instances with no one-sided step has one leaf of this kind; at each side, every deletion set fails by (N1)-(N3) at 36 of these leaves, and some deletion set fails by the joint condition alone at 16. These are findings of the six-node records, not general statements; the 21192 residual instances are not enumerated or run again here.