Generative Structural Theory v1.107 — An Eighteen-Node Example with Least Detecting Depth Six

The input M 1 on eighteen nodes, with D empty, has an undecided stop of the forcing procedure at which no feasible set exists, and the least detecting depth of that stop is exactly six: the rule of depth five does not decide "impossible" there, and the rule of depth six does. So, if a common depth d as in (Q33) of v1.99 exists, d ≥ 6. v1.106 showed that on the family G of v1.105 (one-node extensions to seventeen nodes) the rule of depth five decides "impossible" at every undecided stop with D empty and no feasible set (at 201 of them by certificates, at the others by the counts of a finite check and the monotonicity in the depth), and that depth four does not suffice (J17a). This text goes one node further, to eighteen nodes, from three of the 201 stops of v1.106 (candidates 50 = J17a, 13 and 14), and asks whether some stop is not detected even at depth five. The main search made every one-node extension of the three inputs (25,047 labelled pairs, no reduction by isomorphism) with all their 152,525 choices of the leaf and the side with D empty, and found 15,305 undecided stops with no feasible set: 15,293 decided "impossible" at some depth from one to five by the search, and 12 not decided at depth five (MISS5_CANDIDATE). For the first of the 12, the input M 1 (TID 564), it is proved, by three conditions each checked by machine on a file anyone can check again, that its stop is undecided (P1), that it has no feasible set (P2, a tree of 24,871 nodes covering every candidate set), and that the rule of depth five does not decide "impossible" there (P3, evidence of 203 nodes). Two lemmas make the checks sufficient: a lemma, by induction on the depth, that evidence which passes its checks keeps a strongly connected set of pairs outside the set I_d of the rule, and a lemma that the two prunings of the tree cover every set of pairs. Then, at the stop of M 1, a certificate of depth six of v1.106 form (54 nodes) passed its checker with its controls (Proposition 9); with (P3) and the monotonicity in the depth, the least detecting depth of M 1 is exactly six (Corollary 10). The input, the two pieces of evidence of depth five, the certificate of depth six and the checkers are given in a verification appendix that checks both on the same input. Apart, at each of the 2,870 stops first decided "impossible" at depth five by the search, a certificate of depth five of v1.106 form was built and checked by machine. Not claimed: that eighteen nodes is least for a stop missed at depth five; that no common depth exists; that six is a common depth; anything about the other 11 stops of the list beyond their finding by the search.