# Determining the Monte Carlo constant

In the same way that the high-precision measurement of the fundamental physical constants represents a rigorous test of the Standard Model, a precise determination of the Monte Carlo constant represents a fundamental test of algorithmic information theory. In fact, the Monte Carlo constant defines the exact boundary of the Monte Carlo hypothesis.

If \(X_{2N} \in \{0,1\}^{2N}\) represents the prime-encoding of \([1,2N] \subset \mathbb{N}\), we may find the minimal description length of \(X_N\) relative to a class of universal function approximators \(F_{\theta}\):

\begin{equation} \hat{f} = \arg\min_{f \in F_{\theta}} [H(X_N \lvert f) + H(f)] \end{equation}

where

\begin{equation} \hat{f}(x_n) = x_{n+1} \implies \delta_{\hat{f}(x_n),x_{n+1}} = 1 \end{equation}

and we note that relative to \(\hat{f}\), if \(x_k = X_{2N}[N+k]\) the Monte Carlo sequence \(\phi_{N}\) is given by the true-positive rate:

\begin{equation} \phi_N = \frac{1}{N_1} \sum_{n=1}^{N-1} \delta_{\hat{f}(x_n),x_{n+1}} \cdot \delta_{\hat{f}(x_n),1} \end{equation}

where

\begin{equation} N_1 = \sum_{n=1}^{N-1} \delta_{\hat{f}(x_n),1} \end{equation}

Now, given that \(\phi_N\) is decreasing and bounded it converges to the limit-computable constant:

\begin{equation} \phi^* = \lim_{N \to \infty} \phi_N \end{equation}

known as the Monte Carlo constant. As prime-encodings are algorithmically random, we know that \(\phi^* \in (0,\frac{1}{2})\).