One can say more on the possible stopping rule for the second test and where the P2 ratio we see came from. This is related to the effect of adding (or deleting) of one appellation. (Remember, each Rabbi has several appellations.)
The ratio of the P2 scores is still considerably lower than the usual effect on the P2 scores obtained by a single addition of a single appellation. But the square of the ratio obtained by adding one appellation seems quite close (still a bit on the lower side) to the ratio we experience.
Indeed in the second list there are 44 appellations whose deletion decreases the P2 score, 23 whose deletion increase the P2 score and 35 appellations which are dummies. (The ``dummies'' do not participate in any pairs of ELS and therefore they have no effect on the score.)
For 12 out of the 44 appellations whose deletion decreases the P2 score, the amount of decrease is smaller than the square of 1.1217 (roughly 27 % ). This is the case for 9 out of the 23 appellations whose deletion increases the score (39 %).
This is compatible with an optimization process of repeated additions of appellations which stops when reaching a score which passes the P2 score of the first test. Indeed, in such an optimization process the ratio between the resulting P2 score and the P2 score of the first test is likely to be in the neighborhood of the square root of the effect of adding the last appellation. to the P2-score.
To see this pass to the logarithm of the P2-score and note that when we gradually add quantities and stop when we pass a threshold T then we can expect the difference between the outcome and T be in the neighborhood of the half the last quantity that was added.
(There are reasons to believe that smaller improvements will be delayed to later in the optimization, because of the dependencies of the pair distances it is more profitable to optimize in places where there are already excellent pair-distances.)
We cannot tell if this optimization was blunt cheating or if there was an innocent process (but totally wrong, of course,) were the criteria for including or rejecting appellations were formed using the distances observed in the Book of Genesis.
The realization of the large degree of freedoms concerning the appellations by Dror Bar-Natan (et al.) was a major turning point in understanding the work of Witztum, Rips and Rosenberg. Bar-Natan [1] discovered that the P2-score of the dates of the Rabbis in the second list with respect to the appellations of these Rabbis which were not chosen by WRR is significantly smaller than the the P2 score of the same dates with the same appellations randomly permuted.
In the example we gave in Section 2 Bar-Natan's finding is analogous to finding a significant negative correlation between height and salary for people who could have been considered as having academic education by the assistant but nevertheless were rejected.