Since every sequence of pseudo-random numbers must be generated by deterministic means, it has to follow some underlying mathematical expression (function-like I guess). Suposse you intend to get this underlying expression in order to predict the output of the PRNG. Even if you could predict the next pseudo-random number that the expression will generate every single time for a billion iterations (say $ n$ ), you could never be sure that the process will not backfire at any given moment, as a consequence of the underlying expression being defined by some piecewise function of the kind:

$ $ \forall x ; g(x)=\delta$ $

$ $ g^{\prime}(x)=\left\{\begin{array}{ll}\delta & \text { if } x<n \ \delta^{\prime} & \text { if } x \geq n\end{array}\right.$ $

Where $ \delta$ and $ \delta’$ are distinct mathematical expressions as a function of $ x$ and n is an arbitrarily large threshold. I have to attempt such a feat (predicting the next random number that a PRNG will output) with machine learning tools, and this observation, although perhaps of triviality, may be of importance, at least to clear out that I will not be able to find any definite solutions to the task, only partial and working solutions.

My issue is that I lack solid or even basic knowledge of the fundamentals of mathematical proving, and I am not even sure if the above counts as a rigurous proof, or if there is a way to formally express the thought. My inquiry would be to know if I am mistaken in my assessment and, otherwise, to obtain a formal proof to include this in my work in a respectable manner. Any thoughts and remarks are welcomed.