The Issue Of Near-Minimality
Author: Keith York
This article is property of the author and may not be reprinted or distributed without permission.
March 16, 2000
The article The Truth About The Bible Codes contains the following statement concerning criteria for judging whether a matrix is a valid code array or not. "What criteria distinguish an array consisting of a random pattern of ELS's occurring by chance from an array consisting of deliberately encoded ELS's? There is a general consensus among Bible code experts that significant or valid arrays share the following characteristics. They consist of two or more related words that demonstrate the foreknowledge of God, where these words are both compact and in close proximity to each other in a two-dimensional array. In addition, near-minimality in skip distance of an ELS is considered significant." The last sentence of this statement is the subject now under consideration.
Before considering near-minimality, we must first distinguish between a two-dimensional Bible codes array (which is the subject of the above paragraph) and a one-dimensional Bible codes cluster. As The Truth About The Bible Codes states, "There have been a few cases in which a word can be found as an ELS more frequently than expected by chance in a small localized part of the text when that word is particularly relevant to the passage in question." This is what I term a one-dimensional cluster. By this I mean that the statistical effect being measured is not dependent upon any parameters that arise from a two-dimensional array such as geometrical compactness or proximity within the array. Rather, the statistical effect comes from the one-dimensional nature of the surface text. A particular ELS will have a given skip distance regardless of the line length of an array it may happen to be in. Also, two ELS's will have a set distance from each other in the surface text independent of how close or far apart they are in a two-dimensional array. By contrast, parameters such as geometrical compactness and proximity within an array depend upon the line length of an array. Jeffrey Satinover's Cracking The Bible Code (1997, William Morrow and Company, Inc.) gives a number of examples of this phenomenon I call one-dimensional clustering on pp. 33-40, 120-129, and 143-147. The Torah Codes And Israel Today (1996, Lev Eliyahu) by Robert Haralick and Matityahu Glazerson also contain examples of this. That book's statistical analysis consists of finding the shortest passage in which every one of a list's related words can be found as ELS's and then looking at parameters such as shared letters and shared skip distances. Ed Sherman and Dave Swaney of www.biblecodecritic.com take the following approach. Given a list of ELS's, designate the most improbable as the focal code and analyze the probability that each other ELS of a given skip distance or less would come at least as close as it does to the focal code in the surface text by chance. As mentioned above, skip distance and surface text proximity are one-dimensional parameters in that they are independent of an array's line length. Geometrical compactness and proximity of ELS's within an array are two-dimensional parameters in that they are dependent on an array's line length.
What the above paragraph implies is that there are a number of different classes of Bible codes, some of which are more suited to one type of analysis and some of which are more suited to another type of analysis. Code clusters are simply the term I apply to those Bible codes most suitable to analysis of one-dimensional parameters, while code arrays are simply the term I apply to those Bible codes most suitable to analysis of two-dimensional parameters. Given this fact, the following question is asked of two-dimensional code arrays: how important is near-minimality of skip distance? Is it essential that every ELS in an array be near-minimal? This is what the research team of Doron Witztum, Eliyahu Rips, and Yoav Rosenberg (hereafter referred to as WRR), who published "Equidistant Letter Sequences in the Book of Genesis" in Statistical Science in 1994 claim. Furthermore, most of the examples shown in Jeffrey Satinover's Cracking The Bible Code (1997, William Morrow and Company, Inc.) involve ELS's which are near-minimal in skip distance. However, this view has not been universally held among codes proponents. A contrasting position is that it is only essential for the central term (and maybe a few others) to be near-minimal in skip distance, while other ELS's are required only to be geometrically compact and in close proximity. That has been the view often presented at this and other sites, and a view that I myself once held.
Even if it is essential that every ELS in an array be near-minimal in skip distance, just what exactly does "near-minimal" mean? How far does an ELS have to get from being minimal to cease being "near-minimal"? I have recently come up with what I believe to be an answer that is at once satisfactory (in that it accounts for what is observed in numerous arrays), defensible (in that it is in agreement with what WRR have been proposing and that the limits do not appear to be arbitrary), and easily stated (the rules are easily stated, easily understood, easily applied, and unambiguous).
If you remember from WRR's 1994 paper, they searched for the 10 shortest skip distance occurrences in Genesis of each ELS in their search list. They did not consider surface text appearances (those with + 1 skip distance), but only those whose absolute skip distance was greater than or equal to 2. The Tanach (the Christian Old Testament) is 15.3 times the length of Genesis, so it stands to reason that the 153 shortest skip distance occurrences of any word in the Tanach (not counting surface text appearances) would be equivalent to the 10 shortest in Genesis.
WRR ignored surface text appearances in their paper for technical reasons, but there have been reported arrays by Satinover and others where surface text words were considered an integral part of the array. Thus when surface text appearances of a word are part of an array, they should be considered near-minimal.
WRR also elaborated on another aspect of near-minimality in their 1994 paper. For any occurrence of an ELS there is a value T, which they designate the domain of minimality. T is the amount of text (or fraction of Genesis in their paper) for which no shorter skip distance occurrence of this particular ELS occurs. The farther the occurrence is from being near-minimal, the smaller the domain of minimality is likely to be. If an ELS spans a portion of text which contains a shorter skip distance occurrence of that same word, then the longer skip distance occurrence does not have a domain of minimality. (In other words, say 'dog' is your ELS with d at letter position 100, o at 125, and g at 150. Say that a shorter skip distance occurrence of 'dog' occurs within this span of text with d at position 110, o at position 116, and g at position 122. Since the shorter skip distance occurrence of the word is completely contained within the portion of text spanned by the longer skip distance occurrence of the word, it can be said that the longer skip distance ELS of 'dog' does not have a domain of minimality.) This suggests a third criterion for an ELS to be considered "near-minimal". If the portion of text spanned by an ELS occurrence of a word does not also completely contain a shorter skip distance occurrence of that same word, then it is to be considered as "near-minimal".
The three-fold test can be stated like this. If an ELS meets any of the three following criteria, it is to be considered as "near-minimal" in skip distance: (1) it is a surface text appearance, i.e. having a +1 skip distance; OR (2) it is one of the 153 shortest skip distance occurrences (not counting surface text appearances) of that word in the Tanach; OR (3) the portion of text spanned by that ELS does not completely contain a shorter skip distance occurrence of that same word.
For the time being, the preceding three-fold test will be considered as a prerequisite for all future publication of arrays submitted to this site. In other words, before a codes researcher submits an array for publication at this site, he should check that every ELS in the array meets one of the three criteria of "near-minimality". If it does not, one can always submit it to another site for consideration, and potential codes researchers are encouraged to do so. The exception to this rule will be if the submitting person is able to provide statistical analysis that convinces me that the ELS's in his array that are not "near-minimal" are indeed statistically significant. I realize that this is a major departure from past practice at this site and that I speak for myself and not necessarily each and every staff member at this site nor even other codes proponents whom we respect and whose sites we provide links to. Though every ELS in an array being near-minimal will not ensure that the array is indeed statistically significant, this threshold requirement will serve to "weed out" those arrays which upon detailed analysis would prove not to be statistically significant.
Can I state dogmatically that ONLY "near-minimal" ELS's are significant in code arrays? No. It is clear from published and statistically analyzed results that this is not the case for code clusters. It may also be the case that "near-minimals" are only one subclass of arrays, just as arrays and clusters are two broad classes of codes. However, that only "near-minimals" are valid is the claim of some of the most prominent codes researchers concerning code arrays, and it is my belief that this is an aspect of the codes which those of us who are codes "popularizers" have ignored for too long. I have increasingly come to the conclusion in recent months that "near-minimality" is indeed an important aspect of code arrays. I hope to further demonstrate the rationales behind this conclusion in some soon-to-come articles and case studies that I will publish here. I thus would ask the reader to stay receptive to this concept and stay tuned for future articles. Is "near-minimality" an essential prerequisite for an ELS to be a valid part of a code array? I do not know. However, what I do know is this. It may be that there is another subclass of code arrays that do not use "near-minimals", but this can only be demonstrated by the use of statistical analyses and/or control text studies. Thus I challenge other codes proponents at their sites to state their case using statistical analysis and control studies. Codes proponents such as Witztum, Rips, and Rosenberg and their associates such as Harold Gans rely heavily upon statistics in their analysis of the codes, but besides the people at www.biblecodecritic.com (who examine code clusters rather than arrays), most of the rest of us have ignored statistics. That should change. Let the debate begin.
As a final note, it has to be noted that other prominent codes researchers disagree with me. While I will use this site to present my perspective on the codes, I also believe it is only fair to the reader to provide links to selected sites with other perspectives when I believe they have posted noteworthy items. Thus the News section will include not only articles presenting my perspective but also important developments presenting others' perspectives. In this way, I hope to balance two roles: (1) advocacy of what I believe to be an accurate view of the nature of the codes, and (2) notification to the reader of important developments in codes research. While it may be a fine line to walk, I will try my best to be fair.
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