Generative Structural Theory v1.100 — Beyond the Sufficient Condition: Detection Mechanisms for Depth-Two Probing

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), v1.99 defined the probing rule of every depth, proved it sound, and gave a sufficient condition for the rule of depth two to decide "impossible" at an undecided stop (Corollary ϭ3); in its search, 25 stops detected at depth two and not at depth one did not meet that condition. This text describes symbolically, for every input and every D, the first two rounds of the forcing run from a stop with one candidate forced (Lemma ϯ1): the first round takes out exactly the partners of the candidate in relative obstructions of two pairs; the second ends at rule (a) exactly when some cut has all its candidates among these partners, and otherwise forces exactly the pairs left as the only candidate of some cut and ends at rule (c) exactly when some relative obstruction lies within the candidate and the pairs so forced; so the run ends "impossible" within two rounds exactly when a condition RT holds. With RT in place of the structure of v1.98 Corollary ϫ2 (b) at the inner stops, the condition of Corollary ϭ3 becomes a weaker sufficient condition for the rule of depth two to decide "impossible" (Corollary ϯ2), which that of Corollary ϭ3 implies. Findings of finite cases (no ground for a general statement): the condition RT was formulated from the inputs J11a and J11c before the search plan was fixed, and tested on the 25 stops, which fall into 21 classes up to isomorphism; the condition of Corollary ϯ2 holds at 24 of them, the cuts on which it holds and the ground of RT for each candidate being recorded and checked again on their own, and does not hold at one, A12-04, which the rule of depth two decides: there both cuts closed at depth two have a candidate at whose stop every cut closed at depth one has a candidate whose run ends only in the third round of the forcing procedure. That is a third round of a forcing run, not a third depth of probing. (Q32), (Q33), the new auxiliary question (Q34), whether one constant, not depending on the input, bounds the number of rounds j* at every stop decided "impossible" at depth two (the stops not detected at depth two lying outside it), 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; the results are statements on the local rule studied for the two-deletion chains of (Q1) and do not settle them.