I am not convinced your answer applies to the question.

The question you asked was:

"Do you know of any other computational number that can derive such an array of digits that demonstrate no pattern forming?"

I suggested sqrt(2). Can you find some pattern in its digits? It's certainly computational. Newton gave an algorithm for finding square roots over 300 years ago. The square roots of other nonsquare numbers form an infinite class of candidates for you to consider. As do e, the base of the natural log; or phi, the golden ratio. What is your objection to those examples?

*"For randomness to be truly random, pattern formation must be possible"* in other words there should be no constraint either way.

If I give you the binary sequence 10101010101010101... that alternates, that's a pattern. Surely you would not call such a sequence random. The meaning of a random sequence is that no pattern ever shows up. If there's a pattern it's not random. You are operating from a definition of "random" that's the opposite of what everyone else means by that word.

However if a number sequence runs into millions of digits then pattern formation is a statistical necessity.

Find the pattern in sqrt(2) and then we can talk about this. If there's a pattern then the number's not random. Patterns are the conceptual opposite of randomness. You're just making up your own definition, like defining "up" to mean "down" and then claiming that gravity causes things to fall up.

This is why I asked the question that I did.

Pi appears to have an ability when derived to in excess of a million digits no ability to form patterns with in that sequence.

This defies normal statistical requirements for the term usage of "randomness". [I believe]

No it's actually the opposite. "Almost all" real numbers show no pattern whatsoever, where almost all has its usual technical meaning. The set of numbers having a pattern has measure zero in the real numbers. Numbers with patterns are extremely unusual. It's just the case that they're generally the number's we're familiar with, like 1/3 = .33333... Rational numbers are very rare in the scheme of things, as are numbers showing patterns.

Pi is typical in the sense that it's irrational and its digits show no pattern.

Pi is not an accident, as it appears to be specifically devoted to prevent pattern formation.

The vast majority of real numbers show no pattern. This has been mathematically proven. Pi isn't very special at all. The only reason people think pi is special is because it's the first "mysterious" irrational number people are exposed to in high school. But in the scheme of things pi is just a typical irrational.

So therefore one could conclude that the formation of Pi which is a mathematical "emulation" of intrinsic natural phenomena is indicative of some fundamental of universal structure. And is a pattern in itself, but only when taken holistically.

Only if you define "up" to be "down" and toss in a little crankology like "universal structure" and "holistically." Those are phrases with no mathematical meaning in this context.

**Challenge:**

To formulate an algorithm that produces and "infinite" sequence of digits that MUST NOT allow the formation of any patterns what so ever.

My bet is that the only way to do this is to use Pi as no other method would be sufficient.

I've already suggested sqrt(2); the square roots of all the other non-square natural numbers; the constant e; and the constant phi. But we know mathematically that the patternless numbers are by far the more common real numbers; and that the numbers that show any pattern at all are exceedingly rare.