Generative Structural Theory v1.95 — Residual Feasibility after Cut Probing

The probing rule of v1.93, written without cuts in v1.94, decides some undecided stops (A, M) of the forcing procedure of v1.92 for one-sided re-hangings of a transit leaf: with R = A ∖ M and I the pairs c whose run of the procedure from (A, M ∪ {c}) ends "impossible", it decides "impossible" exactly when r ∪ M ∪ (R ∖ I) is not strongly connected. It is sound; whether it is complete is (Q27) of v1.93. This text proves it complete on one class of stops and shows two abstract systems where the argument does not reach. A relative obstruction is the set of the pairs of R on one route without repeated nodes, in (r ∪ M ∪ O) ∖ {e} for a pair e of H, and has at least two pairs at an undecided stop (Lemma ϥ2). At an undecided stop in the binary class, where every minimal candidate set and every relative obstruction has exactly two pairs, the probing rule decides "impossible" exactly when no feasible set exists (Proposition ϥ3, by a graph of implications between the pairs and Lemma ϥ1); so no impossible stop of that class escapes probing, and at a stop known to lie in it the existence of a feasible set, and so of a step, is decided in polynomial time; whether membership in the class is decided in polynomial time is the new question (Q29). In the abstract system P(4, 3), with candidate sets of three pairs and obstructions of two, and in the abstract system Q7 on seven nodes, whose relation is H ∪ {v} and whose obstructions are routes as in Lemma ϥ2, the probing rule leaves undecided a stop without a feasible set (Propositions ϥ4 and ϥ5). Neither system is an input of the setting, and neither answers (Q27): (Q27) and (Q28) are open outside the binary class, and no boundary of complexity is fixed. Findings of finite cases (finite check 24; not exhaustive and no ground for an answer to (Q27)): at every undecided stop on four to six nodes, at the 1178 inputs of v1.93 on seven nodes (1120 of them in the binary class), in a sample of 20000 labelled pairs on eight nodes (13143 undecided stops, 3757 outside the binary class) and in a sample of 6000 labelled pairs on nine nodes (4888 undecided stops, 1411 outside the binary class), Lemma ϥ2 holds at every relative obstruction found, probing agrees with the existence of a feasible set in the binary class as Proposition ϥ3 proves, and no stop without a feasible set is left undecided or decided "success" by probing. (Q22) of v1.88 and (Q1) of v1.76 are open.