So you are trying to tell me that it is not POSSIBLE for me to guess all answers right in a row for infinity of tries?
If you are guessing randomly, then the probability that you are correct for infinitely many guesses is exactly zero.
That does not mean that it is impossible, but it almost surely will not happen.
It is theoretically POSSIBLE to have such a configuration of the random number generator that it will answer correctly every time.
For a truly random sequence, sure. But the probability of this is exactly zero. And the existence of such a sequence is not in any way useful.
For a deterministic pseudorandom-number generator, which you seem to have alluded to in your previous comments by mentioning "seeds" and here by mentioning "random generator settings," no, this is not necessarily possible.
Now this magic guess of ours may not work tomorrow for new input (to your point about convincing you about all input). But that's not important! what is important is, there will be a POSSIBILITY tomorrow of finding another configuration that will not only correctly solve all previous inputs, but also the current ones.
Well, sure. Just keep a list of previously found solutions to instances of the problem, and every time you solve a new instance (however you do it), add the solution to the list. Now you have a list of correct solutions for all previous inputs.
Is that what you mean by a "configuration"? If not, then how is what you mean by "configuration" different from a simple list of previously found solutions?
How it works and what the mathematics are behind it and why this particular set of random generator settings produce this mysterious and baffling property of always finding correct hamiltonian paths
But you don't have such a thing!
Even if you have a number generator that can take arbitrarily large seeds, and for every finite n there is some seed that can be used to generate the correct solutions to the first n instances of a problem that happen to be presented to it, that doesn't mean that there exists a seed that can be used to generate the correct solutions to all instances of the problem.
The fact that a statement is true for all finite values of n does not imply that the statement is true for n = ∞.