Generative Structural Theory v1.91 — Beyond Individually Forbidden Pairs: Joint Bypass Obstructions for One-Sided Leaf Moves

(Q24) of v1.90 asks, for the one-sided re-hangings of a transit leaf under the deletion bound two, when a set of tree pairs can be chosen that restores strong connectivity while avoiding every combination of two or more tree pairs that together close a second route for a core pair. 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 bypass obstructions and of the cuts that need a tree pair, and draws from it a further 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 and let (N1) and (N3) of v1.89 hold; U is T minus (F ∪ {v}), F being the tree pairs that close a second route by themselves (v1.90). An obstruction of a pair e = (s_e, t_e) of H is an inclusion-minimal C ⊆ U with t_e reached from s_e in (H minus {e}) ∪ {v} ∪ C. Lemma ϝ1: a set B ⊆ U gives a second route for e exactly when it holds an obstruction of e; every route through an obstruction C uses all of C, so that |C| is the number of pairs of U on a simple such route, and |C| ≥ 2; the least size of an obstruction of e is the least number of pairs of U on a route from s_e to t_e, an inclusion-minimal obstruction being possibly larger. Theorem ϝ2: a step deleting D exists exactly when some B ⊆ U makes H ∪ {v} ∪ B strongly connected on V' and holds no obstruction; from such a B, every inclusion-minimal subset keeping strong connectivity gives a step. Lemma ϝ3: strong connectivity asks, for every nonempty proper set W of nodes left by no pair of H ∪ {v}, a pair of U leaving W; the pairs that are the only such pair for some W form a set Q, decided by reachability and held by every strongly connecting B. Corollary ϝ4: hence the necessary condition (N5), decided by reachability: Q holds no obstruction; two such cuts whose only candidates form an obstruction prove that no step exists. These statements hold for every finite node set. Theorem ϝ2 restates the joint condition; the search over the sets B remains, no polynomial decision is claimed, and (Q24) 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 ϝ2 equal to the direct test at all 7492 choices. At each of the 16 choices where (N1)-(N4) hold and no step exists, (N5) fails: two pairs, each the only candidate of a cut, form an obstruction, held by every strongly connecting set within U. On all minimal relations of four nodes (decided directly) and of five nodes (the step taken by the condition of Theorem ϝ2, with no direct test), wherever (N1)-(N4) hold, (N5) decides the step, and so does the test of the obstructions of two pairs alone. Neither does so in general: two examples on six nodes outside the records, decided directly, show a choice with no step whose only strongly connecting set avoids every obstruction of two pairs but holds two of three, and a choice with no step where (N5) holds, every strongly connecting set holding an obstruction but none common to all. These are findings of finite cases, not general statements; the 21192 residual instances are not enumerated or run again here.