by Doron Witztum

**Introduction**- An Investigation of Phenomenon II
- An Investigation of Phenomenon I
- The Results Overall for Both Experiments
- Appendix A: Calculations Relating to Phenomenon II.
- Appendix B: Calculations Relating to Phenomenon I

__Introduction
__

In his article, "A Skeptical Look at the Torah Codes," published in
Jewish Action, Prof. Barry Simon cited a famous dictum of the Sages, "The
seal of G-d is truth." This dictum has its origins in a Midrash (Song of
Songs Rabbah 1:9) concerning the verse in Jeremiah (10:10),
"WH' )LHYM )MT" (HaShem is a G-d of truth.) "What is truth?" the Midrash
asks. The commentary Yefeh Toar explains that the question is: What does
the verse mean by ")LHYM )MT" (a G-d of truth')? and that Rav Bibi in
the name of Rabbi Reuven in the Midrash answers that "the seal of G-d is
truth."

On this same verse, another Midrash tells us, King David discovered that God refers to Himself as ")LHYM )MT" (Eliyahu Rabbah 8). We refer to God the same way in the Mussaf prayers on Rosh HaShannah: "Purify our hearts to serve You in truth.. "KY )TH )LHYM )MT (because You are a G-d of truth.)" This is also the reason why it is forbidden to interrupt between the words ")LHYKM" (your God) and ")MT" (truth) when reciting the Shma.

__An Idea for an Experiment
__

Let us investigate whether the theme, "The seal of G-d is truth" ( XWTMW
$L HQB"H )MT ) is encoded in letter-skips in Genesis, by looking at possible
convergences between the expression ")LHYM )MT" (a G-d of truth)
and the subject "XWTM" (seal).

We will allow Prof. Simon to be our guide not only in selecting our theme: in the article mentioned above he writes, regarding my experiment on the topic of "Chanukah" (a more thorough presentation of which will appear B(Z"H at this site): "Not only are there lots of potential word combinations, but because of the nature of Hebrew, there are variants.." As he proceeds to explain, Hebrew words can be written ML), in a "full" form, or XSR, a shorter form. They can also be written with or without the definite article H. In fact, our current topic can be expressed in all of the following ways:

- The word "seal" is found in the Tanach written both XWTM and XTM.

- To each of these the definite article can be affixed: HXWTM or HXTM
("the seal").

- Similarly, a suffix can be added, rendering XWTMW and XTMW ("His
seal").

- And since we are dealing with Genesis, G-d, as the Author, has every
right to use the forms XWTMY and XTMY ("My seal"), as well.

Altogether we end up with the following eight forms:

1. XWTM

2. XTM

3. HXWTM

4. HXTM

5. XWTMW

6. XTMW

7. XWTMY

8. XTMY

Let us use all of them in our experiment.

In our articles, "Equidistant Letter Sequences in the Book of Genesis", "Equidistant Letter Sequences in the Book of Genesis: II. The Relation to the Text," and "A Hidden Code in Equidistant Letters in the Book of Genesis: the Statistical Significance of the Phenomenon" (all three articles coauthored with Eliyahu Rips and Yoav Rosenberg; the last in Hebrew), we noted two phenomena which can be observed in Genesis. When one examines expressions occurring as ELSs, which are minimal over large segments of the text, one finds that:

1. There is a tendency for conceptually related expressions occurring in this form to converge with one another.

2. There is also a tendency for these expressions to converge with conceptually related expressions occurring in the text as a string of consecutive letters.

In our previous articles we have reported on various experiments, some of which explored the first phenomenon, and others of which dealt with the second phenomenon. The expression )LHYM )MT can only be found in Genesis in the form of ELSs, and not as consecutive letters. All eight variations of the word "seal" can be found as ELSs, and several of them can be found in consecutive letters, as well.

Thus we can conduct two types of experiments with respect to our theme: one examining Phenomenon I and another examining Phenomenon II. I will present our findings in the order in which the experiments were conducted: First Phenomenon II, and then Phenomenon I.

__An Investigation of Phenomenon II
__

**1. Measuring the Convergences
**

Of the eight forms listed above, only five of them appear as consecutive
letters in Genesis: XWTM, XTM, HXTM, XTMW and XTMY.

We can investigate how each of these five converges with the expression
)LHYM )MT, appearing as ELSs in the text.

The expression )LHYM )MT converges successfully with every one of
the five forms listed above. The values of the function c (using the perturbations
-2 £x, y, z £ 2) were:

for XWTM - )LHYM )MT: 4/98

for XTM - )LHYM )MT: 3/98

for HXTM - )LHYM )MT: 2/98

for XTMW - )LHYM )MT: 5/98

for XTMY - )LHYM )MT: 1/98

The form XTM is contained within the forms HXTM, XTMW and XTMY. Since we are dealing with words in consecutive letters, every convergence of HXTM, XTMW or XTMY with an expression occurring as ELSs, will also be a convergence of XTM with that same expression. Thus convergences involving XTM cannot be considered truly independent of the convergences involving the other forms. For this reason I decided to ignore this form when making my computations (had I not done so the results would have improved).

**2. The Randomizations**

For the remaining four convergences I calculated
the values P1 and P2, and evaluated the overall significance using two
different randomizations: Randomization A and Randomization B.

Randomization A is used to measure whether the expression )LHYM )MT tends to converge with the four target expressions more than would be expected to occur by chance.

In this randomization we compare the values of c for the original convergences (that is, between )LHYM )MT and the target expressions) with the values of c for convergences between "substitutes" for the expression )LHYM )MT, and the target expressions.

Randomization B is used to measure whether the expression )LHYM )MT tends to converge with the target expressions more than it does with other expressions in the text.

In this randomization we compare the values of c for the original convergences (that is, between )LHYM )MT and the target expressions) with values of c for convergences between )LHYM )MT and "substitutes" for the target expressions.

In Appendix A, I will describe how each of these randomizations was performed.

**3. The Overall Significance
**

In Appendix A it will be explained how I arrived at the following values
for P2:

Using Randomization A, the significance was: p = 4 x 10-8.

Using Randomization B, the significance was: p = 9.5 x
10-8.

It will also be explained in the Appendix why I preferred not to base my calculations on Randomization A in this case.

I selected P2 to perform the randomization because in this case it was
the superior result (actually, it was clear from the start that P1 could
not be very small in this case, since there were only 4 values for c);
nevertheless, since we originally had two statistics to choose from, let
us double the result. Applying Randomization B to P2 and multiplying by
2 yields a significance of:

p = 1.9 x 10-7.

This figure served as the basis for the level of significance which I published in the Jewish Action article.

__An Investigation of Phenomenon I
__

(At the time the Jewish Action article was written, the following calculations had not yet been made.)

**1. Measuring the Convergences
**

Here are the values received for function c (for perturbations where
-2 £ x, y, z £ 2):

for XWTM - )LHYM )MT: 17/98

for HXWTM - )LHYM )MT: 84/98

for XWTMW - )LHYM )MT: 15/98

for XWTMY - )LHYM )MT: 13/98

for XTM - )LHYM )MT: 83/98

for HXTM - )LHYM )MT: 9/98

for XTMW - )LHYM )MT: 7/98

for XTMY - )LHYM )MT: 18/98

Although here again the form XTM is included in the forms HXTM, XTMW and XTMY, and the form XWTM is included in the forms HXWTM, XWTMW and XWTMY, nevertheless, unlike the previous case where we were dealing with target words of consecutive letters, here there will not necessarily be a dependency. In my estimation, any such dependence will have a negligible effect on the final results. In any event, in the calculations which follow, as we evaluate the overall significance, I will offer both possibilities: using all 8 forms, and using only 6 forms - i.e. omitting XTM and XWTM. Let the reader choose which ever method he prefers.

**2. The Randomizations: **

I computed
the values of P1 and P2 for all eight convergences, and evaluated the overall
significance using the two randomizations, A and B, parallel to my procedure
for Phenomenon II. In Appendix B I will describe how each of these randomizations
was performed.

**3. The Overall Significance:
**

In Appendix B I will explain how I arrived at the following values
for P1:

Using Randomization A, the significance was: p = 0.00052 (omitting XTM
and XWTM the significance was: p = 0.00043).

Using Randomization B, the significance was: p = 0.0098 (omitting XTM
and XWTM the significance was: p = 0.0062).

It will also be explained in the Appendix why I preferred not to base my calculations on Randomization A in this instance.

As before, we had two statistics to choose from, P1 and P2. Since in this case the result for P1 was better, we take it and double the result. Applying Randomization B to P1 and multiplying by 2 yields a significance of:

p = 0.0197

(Omitting XTM and XWTM yields a significance of: p = 0.0124).

__The Results Overall for Both Experiments
__

Using Randomization B, the level of significance for Phenomenon I was:
p = 1.97 x 10-2, and

for Phenomenon II: p = 1.9 x 10-7. (If XTM and XWTM are omitted, the
results for Phenomenon I are: p = 1.24 x 10-2).

Note that although both experiments were performed using function c, as described in previous articles, nevertheless, as I wrote in my article, "Does Tolstoy Really Love Brendan McKay?", section D, both experiments could have been evaluated using the methodology which has been labeled "BEST" (which differs in that instead of taking a summation of all convergences, it takes only the most successful convergence).

__Appendix A: Calculations Relating to Phenomenon II.
__

**I. Randomization A:
**

A. In order to evaluate the overall result for the four convergences
we use a randomization. In the article, "A Hidden Code in Equidistant Letters
in the Book of Genesis: the Statistical Significance of the Phenomenon,"
we describe how this can be carried out; cf. this site, the article, "Does
Tolstoy Really Love Brendan McKay?" section B. Here is how I applied the
randomization in the case at hand:

1. I calculated the convergences between XWTM and each of 100 permutations
of the expression )LHYM )MT (permutations in which the letters comprising
the expression were rearranged).

2. I calculated the convergences between HXTM and each of 100 permutations
of )LHYM )MT as above.

3. I calculated the convergences between XTMW and each of 100 permutations
of )LHYM )MT as above.

4. I calculated the convergences between XTMY and each of 100 permutations
of )LHYM )MT as above.

(None of the above permutations included the original expression).

Four results (values of c) were drawn-- one from 1, one from 2, one from 3, and one from 4 -- and the values of P'1 and P'2 for the four results were calculated.

I repeated this procedure for all 100,000,000 possible combinations.
P2 ranked 4th among all the values of P'2.

Obviously, the value for P1 could not be very small, because there
were only 4 results to work with.

B. The measurement just described is liable to be deficient owing to two factors: 1. Because of the repeated use of the expression )LHYM )MT, it is subject to the "Charisma" effect (see the article mentioned above, "Does Tolstoy, etc." section D.).

2. In calculating function c, a comparison was made with only 125 perturbations, which limits the precision of the results. One could expect that if the number of perturbations were increased we would achieve greater "resolution".

In order to increase this resolution I calculated values of c for perturbations x, y, z, between -8 and 8. (8 was the greatest value which my program allowed at the time this experiment was conducted). These are the results:

For XWTM - )LHYM )MT: 207/3921.

(For XTM - )LHYM )MT: 33/3921).

For HXTM - )LHYM )MT: 109/3921.

For XTMW - )LHYM )MT: 149/3921.

For XTMY - )LHYM )MT: 1/3921.

In order to avoid any possibility of error resulting from the "Charisma" effect, we will also evaluate the results using Randomization B.

**II. Randomization B:
**

To evaluate the probability for the four convergences of )LHYM )MT
with: XWTM, HXTM, XTMW and XTMY, we can perform a randomization using permutations
of the letters of XWTM, etc. However, it turns out that for the word XWTM
only 7 permutations appear in the text; for HXTM only 5 permutation appear;
and for XTMW and XTMY, 7. This miniscule number of permutations allows
for only a limited randomization, as there are only 5x7x7x7 = 1715 possible
combinations. We can proceed as follows: We make the following observations:
1. The word XWTM appears as a string of consecutive letters exactly once
in Genesis.

2. The word HXTM also appears as a string of consecutive letters exactly
once in Genesis.

3. The word XTMW appears 3 times this way.

4. The word XTMY appears twice this way.

Let us do the following:

1. Check for all appearances of the permutations of the word XWTM
as strings of consecutive letters in Genesis. We then draw one of these
appearances and measure it's convergences with )LHYM )MT.

2. Check for all appearances of the permutations of the word HXTM
as strings of consecutive letters in Genesis. We then draw one of these
appearances and measure it's convergences with )LHYM )MT.

3. Check for all appearances of the permutations of the word XTMW
as strings of consecutive letters in Genesis. We then draw three of these
appearances and measure their convergences with )LHYM )MT.

3. Check for all appearances of the permutations of the word XTMY
as strings of consecutive letters in Genesis. We then draw two of these
appearances and measure their convergences with )LHYM )MT.

(The set of permutations will not include the four original words).

In this way we will obtain 4 values of c for convergences 1-4. We will then compute the overall measure of proximity P'2, and compare it with the value of P2.

I repeated this procedure for all 10,519,600 possible combinations. P2 ranked 1st out of all the values of P'2.

__Appendix B: Calculations Relating to Phenomenon I
__

**I. Randomization A:
**

A. To evaluate the overall results, here too we will use a randomization.
In the article, "A Hidden Code in Equidistant Letters in the Book of Genesis:
the Statistical Significance of the Phenomenon," we describe how this can
be carried out; cf. this site, the article, "Does Tolstoy Really Love Brendan
McKay?" section B. . Here is how I applied the randomization in the case
at hand:

1. I calculated the convergences between XWTM and each of 100 permutations of the expression )LHYM )MT (permutations in which the letters comprising the expression were rearranged). Function c was calculated as follows: XWTM is only taken as ELSs, while the permutation of )LHYM )MT is taken as ELSs and as PLSs (perturbed letter sequences). In other words, the ELSs of the permutation of )LHYM )MT compete with the PLSs of it over the more successful proximities to the ELSs of XWTM.

In the same manner:

2. I calculated the convergences between HXWTM and
each of the same 100 permutations of )LHYM )MT.

3. I calculated the convergences between XWTMW and each of the same
100 permutations of )LHYM )MT.

4. I calculated the convergences between XWTMY and each of the same
100 permutations of )LHYM )MT.

5. I calculated the convergences between XTM and each of the same 100
permutations of )LHYM )MT.

6. I calculated the convergences between HXTM and each of the same
100 permutations of )LHYM )MT.

7. I calculated the convergences between XTMW and each of the same
100 permutations of )LHYM )MT.

8. I calculated the convergences between XTMY and each of the same
100 permutations of )LHYM )MT.

I then randomly drew 8 values of c: one from 1, one from 2, etc. For these 8 results I calculated P'1 and P'2.

I repeated this procedure 1,000,000 times. P1 did better than P2; it ranked 522nd out of all the values for P'1. (Omitting the forms XTM and XWTM, the ranking of P1 was 443rd out of 1,000,000 values of P'1).

The measurement just described is also liable to suffer from the "Charisma" effect, owing to the repeated use of the expression )LHYM )MT. Therefore, here too we will examine the results using Randomization B, which is based on permutations of XWTM, HXWTM, etc.

**II. Randomization B:
**

1. I calculated the convergences between each of the permutations of
the word XWTM, and the expression )LHYM )MT. Function c was calculated
as follows: the permutation of XWTM is only taken as ELSs, while )LHYM
)MT is taken as ELSs and as PLSs. In other words, the ELSs
of )LHYM )MT compete with the PLSs of it, over the more successful proximities
to the ELSs of the permutation of XWTM.

In the same manner:

2. I calculated the convergences between each of the permutations of
HXWTM and the expression )LHYM )MT.

3. I calculated the convergences between each of the permutations of
XWTMW and the expression )LHYM )MT.

4. I calculated the convergences between each of the permutations of
XWTMY and the expression )LHYM )MT.

5. I calculated the convergences between each of the permutations of
XTM and the expression )LHYM )MT.

6. I calculated the convergences between each of the permutations of
HXTM and the expression )LHYM )MT.

7. I calculated the convergences between each of the permutations of
XTMW and the expression )LHYM )MT.

8. I calculated the convergences between each of the permutations of
XTMY and the expression )LHYM )MT.

(The permutations do not include the original 8 words).

I randomly drew 8 values of c -- one from 1, one from 2, etc -- and calculated values of P'1 and P'2 for the 8 results.

I repeated this procedure 1,000,000 times. Again P1 did better than
P2, ranking 9847th out of all the values of P'1. (Omitting XWTM and XTM,
P1 ranked 6206 out of 1,000,000 values of P'1).