The pair distances

A technical remark: The program computing the distances using the algorithm of WRR had repeated minor modifications (or debugging) and we do not have now the program used for the numbers appearing in the Statistical Science article and not in the earlier preprints.

At present there are two variants of the program. One variant (REAL_GEOMETRIC = OFF) is just a debugged (or slightly modified) version of the original one and the other variant (REAL_GEOMETRIC = ON) represents a slight methodological change which occurred after the paper appeared in Statistical science. The distributions of P2 ratios in both these variants are fairly close together (at least in the range of interest to us) and the 0.0092 figure is the probability of P2 ratio below 1.1217 in both of them. The individual P2 scores for the original partition did modify. The P2 scores in the modified version of the original concept is $1.741530 \cdot 10^{-09}$ for the first list $1.516102 \cdot 10^{-09}$for the second and the ratio is 1.1487. The probability to be below this ratio is 0.0113. (For the new variant, which is irrelevant, the P2 score of the first list is $1.428 \cdot 10^{-09}$ for the second list it is $2.005 \cdot 10^{-09}$ the ratio is 1.40394 and the chance to be below it is 0.0278.)

We think 0.0092 is the right number to take and in any case the WRR preprint contains the original pair-scores so it is possible to insert them to the computer and use the original data.