papers/lux-oracle-infrastructure/sections/08-mathematical-foundations.tex

\section{Mathematical Foundations}


\subsection{Byzantine Fault Tolerance}

The oracle system maintains Byzantine fault tolerance with threshold $f$ failures:

\begin{equation}
n \geq 3f + 1
\end{equation}

Where $n$ is the total number of oracle nodes. For Lux's implementation with 21 oracle nodes:

\begin{equation}
f_{max} = \lfloor \frac{21 - 1}{3} \rfloor = 6
\end{equation}

\subsection{Outlier Detection}

Statistical outlier detection using modified Z-score:

\begin{equation}
M_i = 0.6745 \cdot \frac{|P_i - \tilde{P}|}{\text{MAD}}
\end{equation}

Where:
\begin{itemize}
    \item $M_i$: Modified Z-score for price $P_i$
    \item $\tilde{P}$: Median price
    \item MAD: Median Absolute Deviation = median$|P_i - \tilde{P}|$
\end{itemize}

Prices with $M_i > 3.5$ are rejected as outliers.

\subsection{Oracle Reputation Scoring}

Dynamic reputation scoring based on historical performance:

\begin{equation}
R_t = \omega \cdot R_{t-1} + (1 - \omega) \cdot S_t
\end{equation}

Where:
\begin{itemize}
    \item $R_t$: Reputation at time $t$
    \item $\omega$: Decay factor (0.95)
    \item $S_t$: Current performance score
\end{itemize}

Performance score calculation:

\begin{equation}
S_t = \frac{1}{1 + d^2} \cdot \frac{1}{1 + l} \cdot a
\end{equation}

Where:
\begin{itemize}
    \item $d$: Price deviation from median (normalized)
    \item $l$: Latency in seconds
    \item $a$: Availability (0 or 1)
\end{itemize}