Stochastic conjecture

Stochastic conjecture is the believe that mathematics remains non-contradictory at the addition of the axiom below:

Let $\mathbb N_+$ be set of natural numbers.

Let $X \in \mathbb R$ be irrational number expressed with finite combination of rational numbers and elementary functions.

Let $B \in \mathbb N_+$

Let $N_B$ be set of $n\in \mathbb N_+$ such that $0<n\le B$

Let $x_n \in \N_B \forall n\in \mathbb N_+$ such that

Let $X\in \mathbb Z$

Let $x>X+\sum_{n=1}^M \frac{x_n}{B^n}$ for all $M\in \mathbb N_+$

Let $x<X+\sum_{n=1}^M \frac{x_n}{B^n}+\frac{1}{B^M}$ for all $M\in \mathbb N_+$

Let $F : (M\in \mathbb N_+, \vec{x} \in {N_B}^M) \mapsto \mathbb R$

Then, for all $M\in \mathbb N_+$, the following relation holds:

$\displaystyle \lim_{ \begin{array}{c} R\in \mathbb N_+\\ R\rightarrow \infty \end{array} } \lim_{ \begin{array}{cc}K\in \mathbb N_+\\ K\rightarrow \infty \end{array}} \frac{1}{K} \sum_{k=0}^{K-1} F(M,\{x_{R+kM},x_{R+kM+1},x_{R+kM+2},..,x_{R+kM+(M-1)}\}) = \frac{1}{B^M} \sum_{\vec y \in {N_B}^M} F(M,\vec y)$

Meaning

The Stochastic Conjecture means, that in the positional representation of any physical irrational number, all the sequences of decimal digits are equally frequent.

Function $F(M,\vec y)$ means some statistical procedure that tries to reveal any peculiarity in the sequence of $M$ decimal digits stored in the array $\vec y$.

The stochasticity means that any irrational number mentioned at the top, can be used as a source of pseudorandom numbers. The only way to reconstruct the sequence (or to show any regularities/peculiarities in it) is to guess a number representable as a linear function of $X$ with rational coefficients. The set of such numbers has measure zero.

However, such an application menans that there is efficient way to evaluate $X$ with arbitrary amount of decimal digits required.

References

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