Introduction In order to define a Theomatic factor, one must define a certain context in which to determine this factor. Such a context is defined by a number of things:
A pool of all possible phrases is constructed for a given context and the numerical value of each phrase in this pool is found by adding up the numbers associated with the letters in the phrase. All of the phrase
values are then examined to determine all of the factors that divide into them according to the required accuracy. When a factor is found that divides many more phrases than would be expected in a random environment according to
the chosen statistical metric, this factor is called a Theomatic Factor and the corresponding subject a Theomatic Subject. Basic Insight In any search for a Theomatic factor in a given context, there will generally always be quite a few factors that divide into at least one of the topic's phrase sums; this is not unusual. Some factors will obviously divide into more phrase sums than others simply by chance. If a large number of potential factors, say N, is considered when searching for a Theomatic factor in a random context (where no real Theomatic factors exist), one will generally find at least one factor that divides into so many phrases that the chances of this occurring randomly are 1 in N . This type of result is quite normal. For example, if we consider 1000 factors in our search for a Theomatic factor, looking at all possible divisors from 2 to 1001 for each phrase sum we examine, we would normally expect to find some factor among all these that divides so many of the topic's phrases that the event would have only a 1 in 1000 chance of occurring randomly. Finding such a factor would not imply that it is a Theomatic factor because it performs just like one would expect it to perform in a random context. This is due to the fact that we have considered so many possible factors when choosing F.
When one observes that a certain factor F chosen in this manner divides many more phrases than the expected 1 to 1000 odds, giving a result much
more unusual than what would be expected in a random chance scenario, one may properly call this factor a Theomatic
factor. What Theomaticians would do is look for such factors in the Bible, interpret their significance by the mathematical relationships of these factors to each other, and leverage such mathematical relationships to obtain insights into biblical topics and themes that might not otherwise be apparent. Ultimately, therefore, Theomatics potentially impacts how one would interpret Scripture. To justify such use of this phenomenon, as the author clearly does in his publications, the odds for random occurrence for any such factor must indeed be small enough to imply divine intervention, or significantly smaller than 1:1000.
Finding Factors Though there may be several Theomatic factors embedded in any given biblical context, each of these Theomatic factors must (by definition) divide into significantly more phrases than one would expect if Theomatics were not designed by God, that is, if the context were merely random. Therefore, by definition, in order to select even one valid Theomatic factor, one must understand what the expected properties of such factors would be in a random environment, or at least have a way of comparing observed results with theoretically random results. This reasoning is supported by the author's statement in Lk15 p. 22: "How did the (factor) come about? The simple answer is  it became self evident. The null hypothesis would tell us that this could 'never' happen if Theomatics was untrue." The "null hypothesis" referred to by the author is the assumption that Theomatics does not exist, that the numerical structure he has observed is merely a product of random chance. Accepting the null hypothesis is accepting the idea that the numerical structures the author has found can reasonably be attributed to random chance; that is, the results are not so unusual that accepting this assumption is unreasonable. Rejecting the null hypothesis is accepting the idea that God must have designed the mathematical structure because there is no other rational way to explain its existence. The fact that no person or group of people could have constructed a text with such unlikely numerical patterns woven throughout it is presumed to be selfevident, therefore rejecting the null hypothesis is equivalent to accepting the concept that the Bible must have been designed by a supernatural Being: God.
The author's statement above establishes the idea that a random factor will never (for all practical purposes) be found in any given context that outperforms the proposed Theomatic factor. Finding another factor F_{2
} that does outperform the claimed Theomatic factor F implies that F_{2} is also a Theomatic factor. If any such factor does not significantly exceed the 1 to N
odds we have mentioned, it must be classified as a random one; it is not a Theomatic factor by definition. Difficulty A problem surfaces in this type of research if a metric is chosen that does not conveniently lend itself to calculating the probability of random occurrence. This is exactly what the author has most often done; he has generally chosen metrics for which determination of the statistical probability is very difficult, if not impossible. He often considers metrics like "phrase quality," "explicit, clear references" and the average word length (WLA) for hits. Using such metrics, the author claims that his chosen factors divide phrases which "make more sense" in connection with his chosen subject than those divided by other factors. In the context of such obscure metrics, the author claims that the patterns he has discovered occur way beyond what the laws of random chance will allow, yet concedes difficulty in determining the probabilities mathematically. Essentially, the author claims to have discovered certain factors that do significantly exceed the 1 to N odds, but he does not know this for certain... it just seems to him that the results he observes are extremely unlikely. He claims to have found hundreds of these Theomatic factors, clear evidence that the Bible is supernaturally designed, yet he does not ever publish the actual probabilities involved. The author feels that there are two possible ways to overcome this difficulty: show that no other factors come even remotely close to the results of the Theomatic factors in the standard context, or randomize the
letternumber assignments to show that these other contexts give random results which are significantly inferior to results obtained in the standard context. Establishing Validity In T&SM, p. 614, the author notes that the most thorough way to determine if a proposed Theomatic factor is statistically significant in a given context is to search for all possible factors in each context and note that the results of each nonTheomatic factor are far less significant than the Theomatic factor(s). Concerning this approach, however, the author claims (T&SM, p. 619, italics his): "It is a mathematical fact, however, that this method is only effective on a limited number of references . It burns itself out after about 6 to 8 references. What this means is that it will not really be to the advantage of the person trying to debunk Theomatics, to use this method over an extensive number of references." What the author is saying here is that it is not actually possible in a typical Theomatic scenario to test all possible factors on all possible phrases in the phrase pool and determine that his claimed Theomatic factor is indeed the most significant factor in the standard context. He claims that the necessary mathematical calculations are simply too extensive to perform in a reasonable amount of time, even by a high speed computer. The question naturally arises then, how does the author go about locating a Theomatic factor? If he must know what the statistical significance of a factor would be in a random environment in order to identify his suspected factor as a Theomatic factor, or at least know that this factor significantly outperforms any other factor in the same context, and it is not possible for him to determine this, then how does he ever select a Theomatic factor? In light of the presumed infeasibility of the most thorough approach, the author proposes to resolve this difficulty by randomizing the letternumber mapping giving rise to factor F in its context a few times, and then testing for similar results from F (or from some arbitrarily similar factor) in the new random contexts. This is recommended as more practical than actually determining the true statistical significance of the behavior he has observed in the claimed Theomatic factor... and just as valid an approach. In Lk15, the author follows this approach for a particular factor in a given context, publishing results of several such random tests, in each test randomizing the letternumber assignment and looking for results from that particular F which is thought to be significant in the standard context. Since he is not able to find this F to be significant in five or six such random contexts, he concludes that results with F in the standard context demonstrate evidence of Theomatic design (i.e. that Theomatics is nonrandom). The author states explicitly that this type of conclusion is mathematically valid (T&SM 620),
"The important fact to realize here, is that if there is nothing to Theomatics, any one random assignment would have just as good a chance of producing the same feature count, as what the Theomatic's number/letter equivalencies
have been able to produce. If we are not dealing here with anything unique or special, each random assignment that is tried, should beat out the results of Theomatics 50% of the time." The Fatal Error The author's proposed less computationally intensive approach, in which he presumes that half the randomly constructed contexts should outperform the standard context if his discovery is merely random, violates the assumption mentioned above: the Theomatic factor F is that unique factor which is the most statistically significant factor out of some large pool of possible factors. If the context were indeed random (if Theomatics were not true), and given that results from factor F are at least somewhat unusual by definition (on the order of 1 in N), this same factor certainly might not perform as well in a few arbitrarily selected random contexts, and it should come as no real surprise when it does not. One would need to examine at least N such contexts without obtaining comparable results before even beginning to conclude that a particular factor is evidencing nonrandom behavior. Finding such a potential factor to yield insignificant results in a few random contexts does not identify it as a Theomatic factor. Each unique context will have a unique most statistically significant factor simply due to randomness. Each such factor will only yield similar results in one in every N random tests on average, where N is the number of possible factors considered when choosing F. It is evident that this N is significantly larger than 100, yet the author only explores half a dozen random scenarios in his validation of Theomatic factors. One cannot validate the nonrandomness of Theomatics the way the author proposes.
It is very disappointing that the author has attempted to validate his findings as he has, and that none of his more mathematically oriented associates have caught this flaw in his logic. The error does indeed invalidate all of the
author's published works. It is such a pity. We do understand that this may be difficult to see... and for the author to accept. Regardless, the brutal fact remains  his approach is invalid. An Illustration For example, suppose we are looking for tall men, and we have gone to considerable trouble to obtain a random sample of a thousand men in which to search for a tall man: we picked them blindly out of a worldwide registry and got them all together for our test. We measure a few of the men in our sample and then we happen to notice a very tall man in our sample, much taller than those we have just measured, and we pick him as our best candidate for a tall man: he just "stands out" and is evidently the tallest man in our sample. We grow weary of measuring all the rest of the thousand men in our sample, but we think our candidate is the largest man in our sample simply because he is so tall. In fact, he is so tall we actually think that he is a much larger man than we should have been able to find in a random sample of a thousand men (even though we don't really have any clue how tall such a man would be on average). In fact, we are willing to claim that he is so very tall that our sample could not have been random at all... the hand of God must have been on our experiment because we think we actually have the tallest man in the world!! ... though we don't really have any clue how tall the tallest man actually is either. To prove our hunch that he is actually an unbelievable "catch," we start some random testing with individual men we come across who are not in our original sample, comparing our specimen's height with the height of each "random" man we observe. How many random men should we examine in order for our conclusions about our candidate to seem reasonable? A half dozen? The fact that we make five or six such comparisons with our subject and find him to be much taller than another man each time we make a comparison does not mean that our selection of such a tall man is unusual. We chose him  from a fixed (and supposedly) random sample of 1000 men  because he was tall. On average, in conducting such an experiment, we would expect to find someone taller than our subject only once in every thousand or so comparisons... that is, if we were astute enough to actually select the tallest man in our sample without measuring all of the men (which, the author claims, is simply too difficult to do). If we did not, and there are other men in our sample that are actually taller than the man we chose, we would on average then find a man of comparable height more often in our search. If we get really serious about our work and test all of the remaining men in our sample, and we confirm that "our man" is indeed the tallest man in our sample, do we have now evidence that he is such a tall man that we can conclude someone planted him in our sample by design? Can we say that we have found an "unusually" tall man in our search? Only if we have some feel for how tall such a man is expected to be on average if we did this experiment a large number of times. Perhaps we persist in our testing and compare our subject with 10,000 randomly selected men and our man is still the tallest man among them all. Perhaps then we might begin to suspect that we have found an unusually large man in our sample of a thousand men. If we kept on, and compared him with 100,000 men... and he was still the tallest man among them all... we would consider this to be very unlikely, and begin to wonder if our initial sample was indeed a random one. Then, if we simply got out our Guinness Book of World Records and found out who the tallest man actually is, and it was indeed the man we had selected... well, perhaps then we have a miracle! In other words, we would begin to think that someone had planted him in our sample, that our sample was designed. This illustrates the essence of the Theomatic proposition: the discovery of an event that is so unusual it is impossible to accept the notion that the event
is merely random. If we wish to know whether we have actually discovered a man much taller than we could have statistically been expected to find in our selection, we must start by asking, "What would the average height be of the
tallest man in a random sample of 1000 men? How does our subject compare with that height?" This is the concept operating at the heart of our analysis of Theomatic significance, and defines what statisticians call a
Maximum Order Statistic (MOS). Detailed Explanation The author's proposition requires that the most significant factor (by some predefined measure, the statistical metric) has been chosen from among a large pool of factors. By definition then, the author's Theomatic factor represents an instance of a maximum order statistic, or MOS. The MOS represents the most significant data point in any given sample. It is not merely describing a particular data point in a particular phrase pool sample, it is generally describing the most significant factor in any similar setting. Each factor has a certain significance associated with it in each Theomatic experiment. Each time we observe the most unlikely factor in a particular context, we are looking at an instance of the MOS. Back in our Theomatics context, as the author looks for a random letternumber assignment (gemmatria) which happens to yield the same results as the standard gemmatria for the given factor F , he will, on average, need to test N random gemmatria before finding another "successful" letternumber assignment (giving similar results as in the standard gemmatria) if his chosen factor is merely a random one, where N is the number of potential factors in the pool from which F was selected. This is because the factor F is the most significant data point in his collection of N factors in that particular test: it is an instance of the MOS for that particular experiment. If the gemmatria is changed, some other factor would be another instance of this same MOS. If this type of Theomatic experiment was performed many, many times with random gemmatria, keeping all of the other contextual parameters constant in each trial, and the significance of the most significant factor in each trial was determined, and the average significance of all such factors was calculated after all trials were complete, this average would approximate the expected significance of the MOS for that experimental context. It is this significance that should be used to compare against our candidate for a Theomatic factor, and each such factor will occur as the most significant factor on average once in every N trials if Theomatics does not exist, where N is the number of possible factors considered when F is selected. In order to demonstrate Theomatic design, to show that his discovery cannot rationally be considered the product of chance, the author must test so many more than N random assignments without finding comparable results to this chosen factor that we would finally become convinced that his initial find was from a nonrandom sample, that Theomatics is designed by God and not a random phenomenon.. In other words, it is a fact that there does exist a most significant factor F in each Theomatic context (i.e. in each combination of subject, chosen passage, accuracy, phrase rules, unique letternumber
mapping and factor pool size) simply due to the nature of randomness. This, in itself, does not indicate supernatural design in the relevant context but is expected in a random environment. The factor that is most significant in
one context due to random chance is certainly not expected to be as significant in some other since its statistical significance was dependent on the gemmatria which produced it. One must expect to have to look through at least
N such gemmatria before finding another one in which F is significant even if Theomatics does not exist. Hence, it is illogical to choose a statistically significant F
from the standard gemmatria, find that this same F is not statistically significant in a couple other random gemmatria, and then conclude that F has the significance of design
in the standard context. By definition, F is only expected to be statistically significant in about 1 in N such gemmatria, not in every other context as the author claims. Kurt Fettelschoss, the German statistican collaborating with the author in responding to our work, has published a denial of the validity of the use of the MOS as a benchmark for statistical comparison in Theomatics, stating (of us): "He is applying the 'law of large numbers.' The reason for applying the 'MOS' method is the assumption, that the theomatic factor cannot be taken out of the sample and, therefore, has to be identified by the testing of a sufficiently large number of samples." This remark indicates a complete misunderstanding of our approach, for he states that we employ the MOS in order to obtain the Theomatic factor F when the factor should be obtained directly from the sample instead. However, we do not employ the MOS as Kurt suggests: we do obtain the Theomatic factor directly from the sample. We use the MOS simply as a benchmark for randomness, what randomess looks like when compared to the Theomatic factor. This cannot be inappropriate, for the technique is based on the same mathematical structures as the probability calculations that Kurt himself employs in validating the author's claims, and is simply another way of presenting the same information. Kurt's conclusion, that the MOS does not apply to Theomatics, is based upon this misunderstanding and is therefore in error. The MOS certainly may
be used to verify whether the author's claim of nonrandomness is valid, as explained above. Any equivalent standard may be used as well (though the author seems to prefer using none). As Kurt correctly states, developing
such a measure does relate to the "law of large numbers." However, it is a "large number" of random gemmatria that should be utilized to understand randomness in this context (rather than to detect the factor itself). This is
exactly what we pursue in our research, where we observe results consistent with the mathematics. Conclusion In order to actually classify any factor F as a Theomatic factor one must either:
The first option is definitely preferred, but will only be useful for relatively straightforward metrics. Since the author generally elects to use convoluted metrics, we are thus generally forced into the second option... unless there are several other apparently random factors comparable to the author's claimed Theomatic factor in his chosen context. When the author's factor is indeed the most significant, one must in such case conduct enough random trials to determine the average number of trials needed to obtain successful results, and find that this average is much larger than N. This second option therefore implies a huge number of trial mappings, on the order of hundreds of thousands, to test a single proposed Theomatic factor, even if Theomatics does not exist. If  as the author simply asserts without any hint of proof or evidence  it is infeasible to perform the basic calculations necessary to determine the performance of all possible factors in the standard letternumber assignment when the metric permits, we may conclude that it is certainly infeasible to investigate hundreds of thousands of random letternumber assignments to validate such a factor. This is evidently much more difficult than the author realizes. The author gives no indication that he has ever tested any particular Theomatic factor in more than a dozen random contexts in order to determine its validity, even though his factor pool is clearly very much larger. In light of the above facts, we may conclude the following:
Resolution For the purpose of promoting Theomatics as evidence of supernatural design in the Scripture, conveniently, and perhaps obviously, the author's claim that it is mathematically infeasible to locate all possible factors in a Theomatic instance is false: sufficient computational capability has existed for decades to make this effort relatively painless and precise. For a single Theomatic context, certainly, it is possible to perform such calculations by hand: astronomers have been performing comparable tasks for centuries. The author certainly has this capability and clearly understands how to perform such calculations. We are simply at a loss to explain why he did not. We have developed a simple computer program
which successfully completes a robust Theomatic test on 1000 unique phrases in about ten seconds on a lowend laptop computer. It finds all possible factors, determines the statistical probability of the simple hit results from each, and sorts them in descending order of significance. This was not difficult. Additionally, the program can randomize the letter number mapping millions of times, perform the above search on each mapping, and catalogue the results of each... but running such a random test does indeed take some time.
Research Requirements In order to validate Theomatics, we must decide on a standard metric that will be used across all proposed subjects and factors. If this metric permits determining the exact probability of occurrence, one need only determine this probability for each factor and find that it is much smaller than 1/N. If not, if the metric does not lend itself to mathematical analysis, then it is necessary to randomize the letter number assignment enough times for each context to demonstrate that the Theomatic factors produce results that are not achievable by the MOS (maximum order statistic) under any reasonable number of random assignments (where the MOS's standard is the average significance of the most significant factor when many contexts are considered). In either case, with a simple metric or a complex one, such tests should be performed for all subjects where Theomatic occurrences are suspected. Any test revealing a factor that
ultimately outperforms the author's claimed Theomatic factor implies discarding the author's factor for the new one. If any subject does not contain any factor at all that significantly outperforms the MOS, the topic itself
should be discarded as an invalid Theomatic topic the way that Theomatics is currently defined. The cumulative results of any remaining Theomatic occurrences must then be combined into a single representative statistic, and the
probability of this statistic occurring randomly determined mathematically. This is the only valid means by which Theomatic design may be demonstrated. Objective Guidelines To construct such a completely rigorous test, as the author himself observes, certain logical guide rules must be followed to exclude subjectivity. Robust rules are evident, and appear to be essential in establishing an objective and unanswerable scientific proof of the nonrandom nature of Theomatics. These rules appear to be slightly more restrictive than those generally employed by the author, especially in the area of phrase construction. We propose the following:
To seek to establish the validity of the Theomatic claim in a most robust fashion, numerous tests should be performed on numerous biblical subjects according to the above rules. Various statistics may obviously be used concerning what features to test, such as the number of hits, or the number of phrases producing hits, or word length averages (WLA), etc. Using multiple independent statistics, as the author often does (as in requiring a certain hit performance, certain clustering result, with a given WLA), is discouraged since this unnecessarily complicates determination of the joint probability (we are not aware that he has ever done so). Instead, the phrase construction rules should be used where possible to achieve the same affect (e.g. set a global phrase length that maximizes Theomatic significance instead of measuring each of hits, clustering, and WLA since hit significance and WLA are not independent). In each test, establish the context consistently from subject to subject by only changing the subject topic, then determine the factor F
that yields the most significant result for each topic in that context. Combine all topiclevel results into a single representative statistic and compare this result with that of the MOS
in a random context. The best statistic/context combination from among the various sample formation and test statistic options should be used to express the probability that Theomatics is random. Factor Bound To make this task somewhat more tractable, at least initially, a reasonable bound N might be set on the largest F allowed for all subjects in the test. This will not damage results so long as no known Theomatic factor is excluded by such a bound. The only affect of such a constraint is the possible omission of very large and currently undetected Theomatic factors located between N and the true factor bound (being half the largest phrase sum encountered in the entire test). If the claimed Theomatic factors are valid and less than N then no harm is done by this artificial bound. If it turns out that known Theomatic factors are less statistically significant than apparently random ones found in testing, and that the most statistically significant factors appear to be such as would occur randomly in such a population of factors bounded by N, then we might eventually relax this bound to remove all doubt about the randomness of Theomatics, with a view to finding VERY large and yet undiscovered Theomatic factors which show Theomatics is indeed a designed phenomenon. This would imply that the author's claimed Theomatic factors are all invalid and that the true patterns of Theomatics remain undiscovered. While this could theoretically be true, such a step is certainly unnecessary for our present purpose, which is simply to determine if Theomatics exists as promoted by the author. Motivation for bounding F lies in the fact that, on average, if the choice of the Theomatic factor is bounded by N, one should be able to find a random factor that yields a "probability of random occurrence" of 1:N for a given context simply due to randomness. This property should hold so long at the phrase sums encountered in testing are generally an order of magnitude larger than N (permitting freedom of divisibility). Finding an appropriate value for N is somewhat subjective, but testing can be repeated for various choices to determine its affects. As phrase sums often appear (from
the author's many examples and from our research) to be between 1000 and 10,000, and since factors as large as 888 are employed (being the value of the name of Jesus), the largest upper bound for N
that appears reasonable might be around 1000. We believe it is unlikely that such a bound excludes any valid Theomatic factor for any subject.
Statistical Analysis Given a choice for N, F can be seen as an instance of a maximum order statistic (MOS): the most statistically significant event in a random sample of N events, where each possible factor from 2 to N+1 produces an element of this sample. That is, every possible factor F in a Theomatic test can be viewed as defining a uniform random variable. When this factor is applied to numerous phrases for a specific subject there will be a certain number of successes, or "hits," that occur. The probability of a success in each trial is a uniform random variable whose parameters depend upon the size of the factor. Odds of obtaining a hit will be C/F, where C, the number of valid remainders, is one more than twice the cluster radius R: C=2R +1, the cluster radius R being the farthest that a "hit" phrase value can be from a multiple of F. For each possible phrase a "hit" or "miss" will be obtained for each of the N random variables (the N factors). Considering S phrases will create a sample of S elements from the binomial distribution of each possible factor. The actual number of hits among all S elements for each factor is an instance of a binomial random variable, an event in each factor's respective binomial distribution with a certain probability of occurrence. For a simple hit count metric, the probabilities of these events can easily be determined and compared. The factor with the lowest probability of occurrence for each subject may be chosen as a potential Theomatic pattern F for that subject from among N factors, and therefore viewed as an MOS from a random sample of size N. If a more complex metric is selected, then validation may involve randomizing the letternumber assignment M times, where M is an order of magnitude larger than N (say NxN). Each assignment is considered a trial and assignments giving results comparable to some standard are counted as successes. A simple comparison of how various factors perform in these random contexts gives sufficient indication of randomness. A Theomatic factor must have results that are achievable in a proportion of random assignments that is an order of magnitude smaller than 1/N . Regardless of the metric used to define significance, probabilities indicating significant nonrandom design need to be significantly smaller than that expected from the mean of the MOS in this context. To properly compare the performance of Theomatics with the MOS, the probability of F can be viewed as resulting from a similarly significant event in a standard normal distribution by determining the standard normal event that would have produced this same probability. This value can then be compared with the distribution of the MOS from random samples of size N from a standard normal population. The probability distribution of this MOS is thus our means of testing for randomness in Theomatics, giving us a benchmark in its mean since this mean is exactly the result we would expect to find in any test of Theomatics if Theomatics is indeed random. Each Theomatic subject will imply an instance of the MOS in its primary Theomatic factor F, and all such instances taken together as a sample will have a mean and an associated probability based on the known behavior of this standard normal MOS. This information can then be used to determine the total probability that Theomatics occurred randomly. The distribution of the MOS is known whenever the probability density of the population is known. We have the ability to determine the distribution of the MOS from a standard normal population for any size N. The distribution of this order statistic is always bellshaped and asymmetrical, being positively skewed (having a longer "tail" to the right and the "hump" of the bell leaning to the left). The mean of the MOS from random samples of size N=1000 from standard normal populations is known to be 3.25, with a pvalue of .000577, or odds of 1 in 1733. What this means is that the best factor in any Theomatic context where the factor pool ranges from 2 to 1001 is expected to have odds of 1 in 1733 of occurring. This MOS random variable also has a standard deviation of about 0.35, which we have determined experimentally. We will use this choice (1000) for N in our research until we see a better choice for N. The probability of the random occurrence of the resulting overall Theomatic statistic, which is the standard normal value having the same probability as
the mean of the relevant pvalues obtained from testing Theomatic subjects, can be determined mathematically based on its placement in the MOS's distribution. Clustering The clustering phenomenon is also noted by the author to be significantly nonrandom, and is used by the author extensively in his defense of Theomatic significance. Clustering refers to the fact that a phrase sum need not be an exact multiple of the factor being tested: the author allows the multiple to be off by one or two, and calls this a "cluster." For example, using a cluster radius of two the following numbers are multiples of both 10 and 5: 48, 49, 50, 51, 52. Clusters play a key role in the detailed probability and statistical analysis in the Luke 15 study since the author believes that a significantly higher proportion of the hits are off by less than two than what one would expect, so it is appropriate to include such a statistic in our benchmark for randomness. For the purpose of a standard benchmark, we will assume "average" clustering in the MOS, determined
experimentally from ten thousand samples of 20 uniformly distributed hits. The average Chi Square pvalue for this random sample is .5, so we will use this in combination with the pvalue above to obtain a final pvalue of: .000577
x .5 = .000289, or odds of 1 in 3466. This translates to a standard normal event of 3.44, which has also been verified by simulation. Measuring Theomatic Significance For each result in a Theomatic instance, the final pvalue (P) obtained from the product of the probabilities of the hits (P_{H}) and the clustering (P_{C}) will be used to derive the statistic O relating general Theomatic significance to that of the MOS with average clustering. We define O as follows: the arithmetic inverse of the probability of a standard normal event being greater than (M3.44)/0.35. This is actually the average number of random scenarios one would need in order to find a result of like significance to the Theomatic factor if Theomatics were random. Finally, to combine all Theomatic factors into a single statistic, we note that the mean M of the standard normal event corresponding to the mean of P obtained from all Theomatic subjects will be compared with the theoretical mean of 3.44 using the Standard Limit Theorem and a standard deviation of 0.35. While it is not precisely clear to us that the distribution of M has the same standard deviation as that of the MOS (it is actually unlikely that it does have the same standard deviation: simulation suggests a value of 0.032; however, the use of 0.35 does yield intuitively appealing results for O; the 0.032 value does not), it is clear that if 0.35 is not the correct value for the expected standard deviation of M this fact is irrelevant for the purpose of comparing Theomatic results since this error will affect all results linearly. Once the correct value for the standard deviation is determined, one can correct O simply by multiplying it by the ratio of 0.35 divided by the correct standard deviation. The O statistic calculates to 2 using this approach in a random context, which is as expected (i.e. every other test should give a "better than average" result). We suggest that an accumulated probability giving odds of the occurrence of Theomatics lower than 1:10^{9}
be regarded as failure to demonstrate nonrandomness in Theomatics. This is a standard with which no one may legitimately contend, and should be quite easily attained if any reasonable number of Theomatic factors actually exists.
