Generative Structural Theory v1.99 — Depth-Two Probing with No Additional Core Deletion

(Q32) of v1.98 asks whether, for the one-sided re-hangings of a transit leaf studied since v1.91, at an undecided stop of the forcing procedure of v1.92 with D = ∅ (no additional core deletion: no pair of the core h0 deleted in addition to the re-hanging, which itself deletes one pair of h) and no feasible set, the probing rule of v1.94 applied again at the undecided stops of its runs (depth two) always decides "impossible". This text defines the rule of every depth d, the candidates, the cuts and the relative obstructions being taken afresh at each stop, and proves for every input and every D: the rule of every depth is sound (at "impossible" no feasible set exists), the set of the candidates it detects grows with the depth, and the rule of depth m = |R| decides "impossible" at every undecided stop without a feasible set (Lemma ϭ1); at the stop of the run that forces a candidate c, a relative obstruction of three pairs holding c becomes one of two pairs when its other pairs stay (Lemma ϭ2); and a sufficient condition for the rule of depth two to decide "impossible" (Corollary ϭ3). At the input J11a of v1.98, missed at depth one, the rule of depth two decides "impossible" (Proposition ϭ4), through the cut W4 = {4, 6, 7, 8}: the run forcing (4, 5) ends "impossible", and at the stop of the run forcing (6, 1) two relative obstructions of three pairs of v1.98 have become obstructions of two pairs, which with two more close the cuts W3 and W2; the argument is symbolic and rests on facts of that input checked by machine with every case tried. The bound m = |R| depends on the input; (Q32) is not answered, and the new question (Q33) asks whether some depth, the same for all inputs with D = ∅, always suffices. Findings of finite cases (no ground for a general statement): in a search fixed before it was run, over the one-node extensions of J11a and of J11c, the other stop missed at depth one in v1.98 (2037 labelled pairs (h, S) on twelve nodes in each family, families made from one input each and not every pair on twelve nodes), 634 and 675 undecided stops with D = ∅ have no feasible set; 358 and 352 of them are missed at depth one, and none at depth two; the condition of Corollary ϭ3 holds at all but 25 of those detected at depth two only. (Q32), (Q33), the rest of (Q27) of v1.93, (Q28) of v1.93, (Q29) of v1.95, (Q22) of v1.88 and (Q1) of v1.76 are open; Lemma ϭ1 is a statement on the local rule studied for the two-deletion chains of (Q1) and does not settle them.