25 Tamuz, 5758
__The Seal of G-d is Truth: Supplement__

by Doron Witztum

## A.

One of the readers, Yosef Beremez [note 1], has made the following comments
related to the measurement of Phenomenon II (described in Part A of the
Appendix):
- Randomization A was done using the same 100 permutations of the expression
")LHYM )MT". This reader suggested that there was a possible dependence
between the permutations. He also suggested redoing the experiment while:
- Using 4 different sets of 100 permutations of the expression
")LHYM )MT".
- Using a larger set of permutations (in such a case the dependence
is negligible).

- Y.B. also inquired - what would be the outcome of Randomizations
B, working with perturbations x, y, z, between -2 and 2 as done in earlier
experiments?

## B.

I performed the requisite experiments and the results are as follows:
- 1a. I repeated the measurement described in Part A of the appendix
(section A), with the variation of using 4 different sets of 100 permutations
of the expression ")LHYM )MT".

I randomly drew 4 results as described there, and repeated this process
100,000,000 times. The new rank of P2 was 156 out of 100,000,000.
- 1b. I repeated the measurement described in Part A of the appendix
(section A), with the variation of using all 5039 permutations of the expression
")LHYM )MT" (without the identity) instead of the 100.

I randomly drew 4 results as described there, and repeated this process
100,000,000 times. The new rank of P2 was 454 out of 100,000,000.
It turns out that Yosef' was correct, and I thank him for the correction.

- 2. I repeated the measurement described in Part A of the appendix (section
B), using perturbations x, y, z , between -2 and 2.
The new rank of P2 was 89 out of 10,519,600.

## C.

__ Additional Measurement__
Regarding the convergence between ")LHYM )MT" and "XTMY", the function
*c* showed an exceptionally strong result: the convergences of the ELSs ranked
ahead of all the PLSs. Therefore, we upgraded the program to be able to
work with the domain of perturbation to -20 £x, y, z £ 20. Out of the 68,921
possible perturbations for ")LHYM )MT", 54,825 do appear. The ELSs ranked
*first* out of all 54,825. It is clear from this measurement, that
in this case we have two types of convergences: the convergence of "XTMY",
which is very strong, and the convergences of the other three words, which
are also successful, but not of the same order of magnitude.

____________

Note 1. Literary name.