Generative Structural Theory v1.90 — Protected Cuts for One-Sided Leaf Moves under Budget Two

(Q23) of v1.89 asks for a structural description of the joint condition of the one-sided re-hangings of a closed transit leaf under the deletion bound two: when some set B of tree pairs restores the strong connectivity of (h0 minus D) ∪ B ∪ {v} on V' with no pair of h0 minus D made bypassable (v1.89 Theorem ϙ2). This text does not settle (Q1) of v1.76, the two-deletion chain question, or (Q22) of v1.88; it puts the joint condition in the form of protected cuts, cuts that keep the core pairs essential, and draws from it a necessary condition decided by reachability alone. For a fixed side and deletion set D (empty, or {d} with d a pair of the core h0), let H = h0 minus D. A protected cut of a pair e of H is a set of nodes that e alone of H leaves and the virtual pair v does not leave. Lemma ϛ1: the protected cuts of e = (s', t') are the sets closed in (H minus {e}) ∪ {v} holding s' and not t'; when nonempty, the family of protected cuts of e has a least and a greatest member; and a family with one protected cut for each pair of H exists exactly when the conditions (N1) and (N3) of v1.89 hold. Theorem ϛ2: a step deleting D exists exactly when v is not in H and some protected family X makes H ∪ {v} ∪ A_X strongly connected on V', A_X being the tree pairs other than v that leave no cut of X; an inclusion-minimal B within A_X then gives the step, the pairs of H being kept essential by their cuts and the essentiality of v not being asked. Corollary ϛ3: assuming (N1) and (N3), the union of the sets A_X over all protected families is T minus (F ∪ {v}), F being the tree pairs that close by themselves a second route for some pair of H; hence the necessary condition (N4), decided by reachability: H ∪ {v} ∪ (T minus (F ∪ {v})) is strongly connected, that is, some set of tree pairs other than v that restores strong connectivity avoids F. Proposition ϛ4: what (N4) leaves are second routes through at least two tree pairs together. These statements hold for every finite node set. Theorem ϛ2 restates the joint condition; the search over protected families remains, no polynomial decision and no gain in speed is claimed, and (Q23) is answered in part only. Finite checks on the records of v1.88 at six nodes (364 instances, 534 leaves and 7492 choices of leaf, side and deletion set; not every relation on six nodes) find the form of Theorem ϛ2 equal to the direct test in both directions (548 choices with both, 6944 with neither). Of the 1524 choices where (N1)-(N3) hold and the joint condition fails, (N4) fails at 1508. At the 32 leaves of each side where the specific pair is in h0 and no step deletes it, (N4) fails when it is deleted: every strongly connecting set of tree pairs holds a pair of F, not necessarily the same pair for every set. At the leaves of the 52 instances with no one-sided step, every deletion set fails by one of (N1)-(N4). At the 16 choices of leaf, side and deletion set left, every strongly connecting set within T minus (F ∪ {v}) has a second route through two of its pairs. These are findings of the six-node records, not general statements; the 21192 residual instances are not enumerated or run again here.