Generative Structural Theory v1.96 — Beyond the Binary Class: Tree-Constrained Residual Feasibility

The probing rule of v1.93, written without cuts in v1.94, decides "impossible" at an undecided stop (A, M) of the forcing procedure of v1.92 for one-sided re-hangings of a transit leaf exactly when r ∪ M ∪ (R ∖ I) is not strongly connected, with R = A ∖ M and I the pairs whose run of the procedure ends "impossible". (Q27) of v1.93 asks whether some input has a stop without a feasible set that the rule does not decide "impossible"; such an input would answer (Q27) "yes" and deny the completeness of the rule. v1.95 proved the rule complete in the binary class and gave an abstract system, Q7, in which it misses impossibility but whose candidate pairs do not lie within the pairs of a tree. This text shows that the tree constraint alone does not remove such stops in abstract systems: in the system K7 on seven nodes, with q = H ∪ {v}, the obstructions defined by the bypass routes of the pairs of H, and R every pair of a tree that the rules making U admit, the procedure stops undecided at once, the rule leaves the stop undecided, and no feasible set exists (Proposition ϧ2: the absence of a feasible set by a symbolic argument from six cuts, the obstructions, the minimal candidate sets and the runs partly by a finite exhaustive check). K7 is not an input, and its relation is not the relation r ∪ M of any state of an input: at an input every cut separates a from b, or s_d from t_d when a pair d is deleted, so that r ∪ M becomes strongly connected with at most two pairs added, while q of K7 needs at least four pairs added (Lemma ϧ1, Proposition ϧ2 (d)); systems made from K7 by adding nodes or pairs are not taken up. With no pair deleted, a feasible set exists exactly when there is a route from a to b with no repeated node whose pairs in R hold no relative obstruction (Lemma ϧ1 (d)), a feasible set being free to hold other pairs. The results of this text are thus two: the miss in a tree-constrained abstract system (K7, Proposition ϧ2) and the reaching conditions proper to the inputs (Lemma ϧ1). (Q27) is not settled. Findings of finite cases and of draws (no ground for an answer to (Q27) or to the new question (Q30)): in the saved outputs of finite check 22 of v1.93 (each pair of a class representative minimal relation and a labelled tree on seven nodes or fewer, every choice with (N1) and (N3)), the 1178 undecided stops without a feasible set, all on seven nodes, all have a deleted pair; in the samples of v1.95 on eight nodes (20000 labelled pairs, 13143 undecided stops) and nine nodes (6000 labelled pairs, 4888 undecided stops), run again with the old outputs reproduced byte for byte, the undecided stops with no pair deleted (743 and 202) all have a feasible set, the stops without a feasible set (19 and 11) all have a deleted pair and are decided "impossible" by the rule, and every cut at every undecided stop separates a from b or s_d from t_d; in exploratory searches of systems of the form of the inputs with the tests (N1)-(N5) not made, which are not inputs, 40902 undecided stops with no pair taken out all have a feasible set. The new question (Q30) asks whether every undecided stop of an input with no pair deleted has a feasible set; only if it is answered "yes" does the open part of (Q27) lie at the inputs with a deleted pair. (Q27), (Q28) and (Q29) of v1.93 and v1.95, (Q30), (Q22) of v1.88 and (Q1) of v1.76 are open.