Generative Structural Theory v1.98 — A Probing Counterexample with No Additional Core Deletion

(Q31) of v1.97 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 always decides "impossible". This text answers it "no". The input J11a on eleven nodes (the input J10 of v1.97 with one pair of h and one edge of S each divided by a new node; the leaf l = 3, side 1, D = ∅) meets every condition of an input; the procedure stops undecided at (U, ∅); no set B ⊆ U is feasible, by a symbolic argument from four nested cuts and four relative obstructions, three of them of three pairs; and the probing rule decides the stop "undecided" (Proposition ϫ3; the conditions of the input, the stop and the outcome of the rule are checked by finite computation with every case tried). So the part of existence of (Q27) of v1.93 has the answer "yes": there is an input with an undecided stop without a feasible set that the probing rule does not decide "impossible". Whether such an input exists with fewer nodes is open, and J11a is not claimed to be a least one. Proved for every input: at an undecided stop, the run of the procedure from (A, M ∪ {c}) stops in its first round unless c lies in a relative obstruction of two pairs, so that at a stop without a feasible set the set I of the rule lies within the pairs of such obstructions (Lemma ϫ1); and, with D = ∅, a condition the rule needs to decide "impossible" at a stop without a feasible set and a condition under which it does, the second of which accounts for the detection at J10 (Corollary ϫ2). The input J11b on eleven nodes has an undecided stop with D = ∅, k = 2 and no feasible set, by two cuts and four relative obstructions of two pairs (Proposition ϫ4), so that the feasible set that v1.97 guarantees at k = 1 is not guaranteed at k = 2. Findings of finite cases (no ground for a general statement): among the one-node extensions of J10 named by a search plan (1482 labelled pairs (h, S) on eleven nodes, a family made from J10 and not every pair on eleven nodes; 8212 choices with D = ∅), there are 2915 undecided stops, 355 of them without a feasible set (26 with k = 2), and 2 of these, J11a and one other, are not decided "impossible"; the plan was fixed after J11a had been found by an exploratory scan of the same family. The probing rule applied again at the undecided stops of its runs (depth two) decides all 355 "impossible", a finding of this family only; the new question (Q32) asks whether it always does, the soundness of that rule to be proved first. (Q31) is answered "no", and (Q27) "yes" in its part of existence; the rest of (Q27), on a least input, and (Q28) of v1.93, (Q29) of v1.95, (Q32), (Q22) of v1.88 and (Q1) of v1.76 are open. J11a is not a counterexample to the two-deletion chains of (Q1); it fixes a limit of the local rule studied for them.