You might be interested in this website, which in particular shows that the kind of 'bible code' rules that people like to use can be applied to any sufficiently long text such as to Moby Dick to find related words for essentially anything you want.
Also, simple mathematical analysis invariably dispels this mystery. Just for example, if your desired word is k letters, and your text has n letters from an alphabet of size c, the number of ways to pick an arithmetic progression of k indices (such as those bible codes in the linked website) is at least (n/2)·(n/2k) (i.e. pick the first index from the first half and then pick the distance between the first two indices), so you have n^2/4k possible choices and ought to expect to see your desired word roughly (n^2/4k)/c^k times, as a first-order estimation (ignoring the letter frequency distribution). For (k,n,c) = (7,10^6,26), we have (n^2/4k)/c^k ≈ 4.
For vague or unspecified rules, such as in your case, the correct way to analyze the phenomenon is to understand Kolmogorov complexity, which is an essentially objective way to measure how much information is encoded into a proposed explanation of some pattern. Informally, if the specification of the rule itself takes m characters, then it should be completely unsurprising if the rule generates a pattern of only m characters.
K-complexity is a mathematically precise form of Occam's razor, where we can define comparison between two hypotheses for some data that are each given by a program, simply by favouring the shorter program. This notion is truly objective in the sense that, if the correct explanation can be captured by a program, then once you have sufficiently large amount of data, you will eventually stabilize on favouring the correct explanation. To get the most out of this even for not that much data, use a general-purpose programming language such as Python.
Many people underestimate how long a pattern must be in order for a proposed explanation to be favoured by this K-complexity-based analysis. For example, see this post showing that the pattern of primes is probably favoured only given a rather long initial segment, in particular long enough for the desired program to check primality to win other programs.
We cannot directly apply this to your question, because there is simply too little data generated by the proposed rule. In particular, the claim is that some text encodes some data, and the rule to retrieve it must be simple enough that it actually shows that the output was really encoded in the text and not in the rule! With this in mind, the cited achievement of these numerologists is utterly disappointing. They only managed to recover 6 decimal digits despite injecting a huge amount of gematria!
We can still apply K-complexity in the following way. Since the claim is that some text T is special, and there is a rule to retrieve some data from T, then there must be a program P and parameters given by S such that:
- P(S,T) is the desired data (e.g. first m digits of π).
- S is very short. (It had better be shorter than m!)
- P(S',T') is not the desired data for any other text T' and short S'.
Clearly, you can achieve such a thing if your P is complicated enough (e.g. using gematria in an over-fitting sense). So the true challenge is to find such (P,S) with P having minimum length. To invalidate the numerology, it suffices to find some (P',S') with length(P') ≤ length(P) that produces the desired data from a different text, as it would demonstrate that the numerology captured by P fails to support the claim that T is special. jpa's answer can be viewed as an instance of this kind of invalidation!