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Challenge

The author proposes the following challenge in connection with his published results from Luke 15. He writes: "Please send me:

1. the one random assignment from the Majority Text,
2. with 46 hits from Luke 15,
3. all composed of short phrases not averaging over 2.37 WLA,
4. all of which include a word pointing to the two brothers,
5. with a multiple factor any number the size of 90, and
6. the same identical cluster distribution p-factor as Theomatics, for all the 46 hits."

His claim with respect to this challenge is: "My verdict is that you will not be able to come even remotely close to these results.  The only way you could possibly do it would be to program a computer to go through thousands and millions of number/letter arrangements, which would only prove ultimately that the Theomatics p-factors are are indeed very high. Theomatics certainly did not find its results in the same manner (it is limited to the one arrangement of historical record), so that would be a completely unfair way to try and match or beat Theomatics."
 

Response

We have met this challenge successfully, and quite easily, as follows:

1. We took the text published by the author.
2. We required every phrase included in the test to:
      - contain one of the references in Luke 15 noted by the author in his study, 
      - be no more than 4 words in length, and
      - consistently conform to our phrase construction rules.
3. We randomized the gemmatria (letter-number assignment) 25,000 times, such that letters with single-digit values in the standard gemmatria retained single-digit values, and likewise those with double-digit and triple-digit values retained values of comparable magnitude.
4. We located every factor less than 1000 in each of the random gemmatria which:
      - was at least 95% as large as the author's factor (all factors > 85),
      - had as many hits as in the author's context (53, not 46),
      - had as good a WLA on total hits as in the author's context (2.74,  not 2.37).
      - had as unikely a Clustering Distribution as in the author's context

As noted in analysis of the author's errors, the statistics the author formally requires that we meet in accepting his challenge are invalid since he did not actually obtain these results himself.
 

Results

There are 175 instances of gemmatria in the 25,000 random gemmatria examined, approximately 1 for every 143 attempts, where a factor surpasses the actual number of hits that occur in the author's context (H₄), has as good a Word Length Average (WLA_H4 ) and as unlikely a clustering distribution.

Notice below the distribution of these 175 instances by factor. The top row in the following chart is the factor, the second row is the number of successful random gemmatria for that factor among the 25,000 gemmatria tested, followed by the percent related to that factor among all 175 instances.

Observe that 61% of these instances (108 of 175) involved either factor 90 or factor 100, and that remaining instances heavily favor the smaller factors. If the environment were purely random one would expect a distribution as in the last two rows above, much more evenly spread, mildly favoring the smaller factors (since their probability of success is proportionately higher), and tapering off toward the larger ones.

This unusual distribution does not follow the expected pattern, and can only be due to the affect of manipulating variables in phrase construction and the types of letters generally found in variables; it has absolutely nothing to do with Theomatic design since every single gemmatria considered was random. This implies that the language itself, coupled with the use of variable manipulation techniques, implies a context that is highly non-random, biased toward small multiples of 10. This violation of the randomness assumption tends to invalidate any indication of Theomatic significance, even if it happens to be observed.

The author, in an afterthought to the above requirements, proposed the following addendum: "In addition to the 46 hits and the requirements you must meet, of those 46 hits 39 of them will have to all fall within the three word phrase limit and produce an ultimate p-factor of .000000003828, or 1 in 262 million (along with the clustering).  In case you have not read the last part of my report, that is precisely what Theomatics produced."

Of the above 175 instances, 31 of them, or 1 in 806, have at least as many 3-word hits (35), have as good a WLA (2.2286), and exhibit as good a clustering result (.1864) in these three word phrases as the author requires (corrected, of course, for his errors). These instances are given in the table below.

The top row gives benchmark results from the author's proposed Theomatic factor 90. The trial that the gemmatria was constructed is given first, followed by the random gemmatria producing the result (syntax explained below), the factor (F), the number of hits (H₄) in 4 words or less, the WLA for these hits (WLA ₄), the clustering Chi Square p-value for 4-word phrases (CS₄), followed by the same three parameters for phrases of 3 words or less. Also, the probability of the hits (PH₄), the joint probability of hits and clustering (P₄), N in the ratio 1:N (N₄), and the O statistic (O₄) are shown for 4 word phrases. The reference pool of 683 phrases is given here. The actual hits for each of these 31 factors are given here. 

Each gemmatria is given in three sections, delimited by a decimal. The first section shows letters with single-digit values, the middle section those with double-digit values, and letters with triple-digit values are given last, as shown for the standard gemmatria in the following chart.

In each random gemmatria, placing the re-arranged letters in this chart will give the random letter-number mapping. The letters vau (standard value 6) and koppa (standard value 90) are omitted by the author. Those assigned to 200 in the standard gemmatria (S and J) are both assigned to the same value in the random gemmatria.
 

Analysis

Clearly, the author's verdict was incorrect. Approximately 1 trial in 800 gives results comparable to the author's results in his initial context, not 1 trial in millions as the author expected. This is much less frequent than one would expect in a purely random environment. This indicates that there is a more significant factor in the standard gemmatria than the author discovered.

It is noteworthy that of the 31 instances outperforming 90, 21 of them (68%) are multiples of 10, 11 instances (35%) are factor 90 and 9 (29%) are factor 100. Clearly, there is something about the phrase construction rules and the structure of the language itself that favors such factors.
 

Comments

A number of comments are in order with respect to this challenge and the above results.

• Factor size, in itself, does not affect the statistical significance of a factor since the odds of success are naturally incorporated into the relevant statistic: P_H . The author's insistence that factors be of appropriate size and obtain a certain number of hits appears rooted in the fact that smaller numbers will naturally divide successfully more frequently, and that this fact must be accounted for when selecting a factor, which is certainly true. This ties his 2^nd and 5^th requirements together in a statistical sense.
  
  Clearly, requiring factors to be a certain size apart from the above concern, or that they obtain  the actual number of hits requested regardless of being larger in magnitude, as the author has in advancing two distinct requirements in #2 and #5 -- which are fully represented in the P_H statistic -- is unreasonably subjective and arbitrary. It actually violates the spirit in which he located the claimed Theomatics factor to begin with, which was merely based upon the statistically unlikely number of hits obtained, and had nothing to do with the factor's actual size. The author gives no indication in his publications that certain factors should be sought or discarded merely due to their size. In fact, he publishes results with Theomatic factors as small as 17 (TII, p 50-75, the key Theomatic pattern for Fishes and Fishing).
   
• The author has formally required that both the hit performance and the clustering observed in the 90 factor be matched exactly by a random context. We interpret this to mean, "matched or exceeded in significance." Requiring the exact same probability from two random events is evidently absurd; we cannot believe he intended this, and give him the benefit of the doubt.
   
• One must observe that requiring many particular properties of a single event to be matched (or exceeded) in a comparison of two apparently random events is unreasonably restrictive. For such a comparison to be appropriate it must finally be based upon a single representative statistic indicating the overall likelihood of the entire event.
  
  In other words, in our opinion, the author's challenge technically contains reasoning comparable to the following: "I flipped a coin 100 times once and got 62 heads. The first 4 flips were heads, so at least the first four of yours must be, and of my first 12 there were 8 heads, so you have to have at least that many in your first 12 flips. At 26 flips I had 16 heads, so you have to have that many heads by flip 26.... You get five or six chances at most to try this (after all, I only tried once). If you can't, then you have to agree that what I did was miraculous." We hope the analogy speaks clearly enough: hindsight is 20:20.
  
  Not permitting a final statistic to be the metric used in Theomatic comparisons, which allows some factors to be ranked higher than the benchmark which are much better in one category though somewhat worse in another, is heavily biasing the challenge in the author's favor. There are, in fact, 4 factors in the standard gemmatria which outperform 90 in clustering in the 4-word phrases, implying that the Theomatics factor would not have been chosen if any of these other random factors had been noted first based upon the author's own biased methodology. These 4 factors are shown below, with the number of  hits and percent of total for each radius type, followed by the p-factor:
  

  F	0	1	2	0%	1%	2%	CS
  90	13	21	19	25	40	36	0.6792
  85	14	16	16	30	35	35	0.2090
  86	7	12	17	19	33	47	0.6456
  87	8	19	12	21	49	31	0.4545
  89	8	16	21	18	36	47	0.6592

• If the O statistic were simply used in this challenge, which we have shown to be a valid metric in this context, being consistent with the manner in which Theomatic factors should be chosen, we would not have ever randomized the gemmatria since 20 factors outperform factor 90 in the standard gemmatria in this more appropriate test, as shown in the first general testing result. This metric O, or an equivalent one, must ultimately be used to validate general Theomatic significance, so it should be the standard upon which the challenge is made.
  
  Even so, the author's challenge (as interpreted) has been met quite successfully and very easily, in our opinion. This is due to the very statistically bland nature of the final results he has presented. If the author's results had indeed been somewhat statistically significant, the task would have been much more difficult.
   

Conclusion

It is evident that the factor 90 is not statistically significant in this context: it has an O value of 1.00. The benchmarks set by the author, considered both independently and simultaneously, are actually below what is achievable by randomness, which is 2 in a normally distributed environment -- a benchmark achieved in half of our random gemmatria, or by 1 in every 1,388 attempts. This is about twice what we would expect in a purely random context, indicating that the text is not actually behaving in a random manner, as we have observed, reinforcing the legitimacy of these experimental results.

Matching all of the components of the author's benchmark simultaneously took a few more trials on average, slightly more than the number of trials that would generally be required to match the general significance of the Maximum Order Statistic, even though such a challenge is heavily biased in the author's favor such that he himself could not accept it on behalf of his Theomatic factor if such a challenge were similarly presented to him.

Further, the distribution of the factors themselves is entirely nonrandom, clearly favoring the author's factor outside a Theomatic context, indicating that the language structure and the affects of variable manipulation in phrase construction violate our assumption that the context itself behaves in a random manner. Multiples of 10 are clearly favored in this context regardless of the gemmatria chosen. Even with this advantage, the Theomatic factor noted by the author was outperformed by randomness from any perspective.

One must conclude that the author has proposed an instance of Theomatics, apparently one of his very best, that fails in every respect to demonstrate any type of inherent design. One cannot appropriately reject the null hypothesis that Theomatics is random based upon these results.

Comments are certainly welcome.

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