Generative Structural Theory v1.103 — A Depth-Three Probing Counterexample with No Additional Core Deletion

v1.101 showed, at the input J13a on thirteen nodes, that the probing rule of depth two does not decide "impossible" at every undecided stop with D empty (no pair of the core deleted in addition to the re-hanging of the leaf) and no feasible set, so that a common depth d as in (Q33) of v1.99, if it exists, has d ≥ 3; v1.102 found no stop missed at depth three among the one-node extensions of J13a. This text gives, at the input J15a on fifteen nodes, J13a with two of its pairs divided by new nodes, an undecided stop with D empty and no feasible set at which the rule of depth three does not decide "impossible" (Proposition ϳ1); so, if d exists, d ≥ 4 (Corollary ϳ2). The proof is a symbolic argument on this finite graph, backed by checks of reachability and of strong connectivity listed in finite check 46, its 920 comparisons, all equal, being 877 between the two implementations on the tests that both make, 36 within implementation 1 (the pairs probed found in two ways) and seven against the cut data written in the script, and the other facts being found by the one method named for each (§5); it is not a proof independent of these checks, and ruling 02 reports a check of the certificate by a program of the review's own. Seven cuts and seventeen obstructions, each with the pair of the core it bypasses and a route, exclude every feasible set. For the miss at depth three, six of the pairs probed, with the pairs not probed, make with r a strongly connected relation; the run from each of the six stops "undecided", and at its stop the rule of depth two is shown in the same way not to decide "impossible", through 29 runs at the next level and 113 runs at the level of depth one, all of which stop "undecided"; so r ∪ M ∪ (R ∖ I_3) is strongly connected. Findings of finite cases (no ground for a general statement): in the family fixed by search plan 04 after a preliminary exploration, the 250,776 labelled one-node extensions to fifteen nodes of the inputs of the 56 stops of v1.102 detected at depth three with R0 ∖ I_3 not empty, 33 undecided stops without a feasible set are missed at depth three, all made by the division of a pair, their inputs in 12 classes under the isomorphism named, confirmed by a second implementation of the drafting side (not an independent verification), and all decided "impossible" at depth four. J15a is the input of one of them, one of the candidates of least |R| among the 33, and is not known to be least in the number of nodes. The result does not show that no common depth exists, nor that depth four suffices in general, and it does not settle (Q1) of v1.76.