Generative Structural Theory v1.102 — Depth-Three Probing on a Specified Family of One-Node Extensions of J13a

For the one-sided re-hangings of a transit leaf studied since v1.91, with D = ∅ (no pair of the core deleted in addition to the re-hanging), v1.101 showed at the input J13a on thirteen nodes an undecided stop without a feasible set that the probing rule of depth two does not decide "impossible", so that a depth common to every input, if it exists ((Q33) of v1.99), is at least three. This text tests the rule of depth three on a specified family: the one-node extensions to fourteen nodes of J13a, 3525 labelled pairs (h', S') with every choice of the leaf and the side, D empty. A preliminary exploration of the first 40 pairs was run first and its results were seen; the plan and the code of the main check (search plan 03, finite check 39) were then fixed and digested by machine before the run. At every undecided stop the existence of a feasible set was decided exactly, by trying every set, apart from any rule of probing. Of the 1395 undecided stops without a feasible set in the family, 571 are decided "impossible" at depth one, 367 first at depth two, and 457 first at depth three; depth three was computed at these 457 stops only, and at the other 938 it decides "impossible" by the monotonicity of the rule in the depth (v1.99 Lemma ϭ1 (d)). So no undecided stop without a feasible set in the family is missed at depth three. Findings of finite cases (no ground for a general statement): the 457 stops missed at depth two are checked again by a second implementation by the same side, not an independent verification; at 401 of them every pair probed is taken out by the inner verdicts of depth two, and at 56, each obtained by one further division of a pair of J13a, one or two probed pairs are not, while a cut whose candidates are all taken out still exists. The finding concerns the undecided stops without a feasible set in this family only; it does not say that depth three suffices on other inputs, and it does not answer (Q33): if a common depth exists, it is at least three, as before. (Q33), (Q34), (Q35), 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.