Theoretically if I made a neural network which made combinations of every character possible within a 500,000 character limit. Will there be one instance where we have the perfect combination giving us the perfect book which in my context is a book which contains all future predictions?
If we restrict attention to all the relevant facts of the rest of your life. Probably, they fit in about 500 pages. The answer is yes, it is theoretically possible to obtain the perfect book in this way. An way to see it is to review the Infinite Monkey Theorem, a rigurous theorem in probability:
It states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare.
but there are many practical difficulties:
However, the probability that monkeys filling the entire observable universe would type a single complete work, such as Shakespeare's Hamlet, is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero).
In addition, this theorem can be generalized to state that any sequence of events which has a non-zero probability of happening will almost certainly eventually occur (i.e. writting the perfect book), given enough time. In the case you propose, we eliminate the random factor, since the perfect book will necessarily be one of the 27^(500000) texts generated. But that number is vastly greater than the number of particles 10^80 in our observable universe. Therefore, although such a procedure, given sufficient space and time would generate, the perfect book is surely one of the least practical means of even predicting tomorrow.
Absolutely not. The finite character limit limits the book to finite predictions. Even restricting the case to 'predicting the relevant facts of your life' or in some other way, that's going to be a very ambiguous and limited set of predictions.
You should read the short story The Library of Babel by Borges, which grapples with issues of scale and consequences of a collection of books of exactly the kind you imagine.
It's interesting to consider the ontology of the Many Worlds interpretation of quantum mechanics. If it's right, every combination of probabalistic outcomes does happen in some subset of branches in the state-space. So if you could make it into a problem for a quantum computer algorithm, with some criteria to make the correct answer stand out, you could imagine directing the decoherence of all the books, into one specifically, by using destructive interference to remove all the other answers.
In practice, only relatively short strings by comparison can be tested so far, for minimal useful sifting processes, just prime factors for cryptography. I'd say all you could do even with a most-powerful-imaginable quantum computer in the future, is limit the set of books to the set of real possible futures for a defined subsystem, probably only in relative isolation, or the inputs would get too complex.
The consequences of finite time and energy, and the finite states or bit-representations of subsystems of the universe (eg measurements not wavefunctions), seem to place fundamental constraints on how much information can be represented in the macroscopic universe. You need constraints, like knowing initial conditions to limit the state-space, or even far smaller calculations than the one you propose become 'unphysical', not relisable even in principle in the universe as we understand it (but only in the multiverse of Many Worlds). I suggest the appropriate accuracy of the initial conditions of your life, to get any useful answer, would be such an unrealisable variable.
If Many Worlds is right, preserving determinism but only across the multiverse, then the book you speak of is the states of all your particles, and all those that could interact with you as limited the speed of light. But that wouldn't give you a precise prediction of your future, because all the futures would happen to every possible subjective you, with no way to distinguish between them (without a way to get destructive interference, which finite inputs constrain).
That's only about 1000 pages, depending on what you mean by "all future predictions" that's not really much. So if you make that too short, the world formula might not be able to be written down in it's full comprehendable glory leaving you with something like "42". Which for all intents and purposes could be correct, but also makes you ask what exactly the question was. While if it's too long you might not really be able to read it in a live time or before those predictions came to be.
The other problem is that you'd probably also get every other prediction of future events. Meaning not only do you have the problem that you can't read them in time, you'd also only after the fact, know which of the predictions was actually correct. Meaning you might as well have the book that contains all the predictions but you wouldn't know which one it is.
Will there be one instance where we have the perfect combination giving us the perfect book which in my context is a book which contains all future predictions?
Perhaps. However, assuming a 28-character alphabet (26 letters, space, full stop), there would be 28500 000 - 1 other books. That's over 1.0367×10723 579 books; the number's too big for the Stack Exchange post limit, and my rounding error is more books than could be read if the whole population of Earth worked on it for a thousand trillion millennia.
Even if you read all of the books, you'd have no way of knowing which was accurate.
Generating the books is the easy part. Working out which book is right is the hard part; to do that, you have to collect lots of information about the world and compare them to your candidates. Many of the books will say:
the sky will be filled with densely packed teleporting frogs on the second of august of the year two thousand and twenty two.
so you can rule these books out. Once you've discarded the obviously-wrong books, you'll have to look harder at the world, and find deeper contradictions and impossibilities, ruling more and more of the books out.
And once you've ruled out all but a single, self-consistent group of the books… you're not done. Because you're only checking books up to 500 000 characters long; it might be that none of them hold only true predictions! Your job is never done. We call this endless task science.
Theoretically if I made a neural network which made combinations of every character possible within a 500,000 character limit.
You don't need a neural network for that, it's just a plain set of business rules and a lot of memory and processing resources.
Will there be one instance where we have the perfect combination giving us the perfect book
"Perfect" is subjective. So, any book will do for some specific subject. You don't need many books, you just need one book and many people.
which in my context is a book which contains all future predictions?
Any prediction can be broken, there's no way to have a set of predictions that can't be broken, so, the book is impossible.
Notice that "future predictions" is a pleonasm. All predictions are about the future.