# Our Hypothesis

This states our predictions about what our research intends to find through scientific research methods.

01.

## Particles move via the discrete Bernoulli Process.

- The variance is dependent on velocity, but no reference frame has been defined.
- We asked if it is possible to measure this reference time.

\begin{align}\overline{x(N)} = (2\beta -1)\delta x N \notag\end{align}

\begin{align}\overline{x(N)} = (2\beta -1)c t \notag\end{align}

\begin{align}(2 \beta -1) = v/c \ \ \ \ \text{ Mathematically nice } \beta \in [0,1] \ v \in [-c,c] \notag\end{align}

\begin{align}(\Delta x (N)_{\beta})^2 = 4 \beta (1 - \beta) (\delta x)^2 N \notag\end{align}

\begin{align}(\Delta x (N)_{v})^2 = \left (1 - \left ( \frac{v}{c} \right )^2 \right ) (\delta x)^2 N \notag\end{align}

## The variance is dependent on velocity, but no reference frame has been defined

02.

## Variance of proper time describes a Jitter term.

- The conclusion is a new insight to special relativity in that the variance of the space-time location is minimized in the time dimension in a particular frame of reference.

\begin{align}(\Delta x (N)_{v})^2 = \left (1 - \left ( \frac{v}{c} \right )^2 \right ) (\delta x)^2 N \notag\end{align}

\begin{align}(\Delta p (N)_{v})^2 = \frac{\left (1 - \left ( \frac{v}{c} \right )^2 \right ) (m c)^2 }{N} \notag\end{align}

\begin{align} (\Delta x)^{2} = (\Delta x(N)_{v})^{2} + \left(\frac{\Delta p(N)_{v}t}{m}\right)^{2} \notag\end{align}

\begin{align} (\Delta s)^{2} = \left(\frac{v}{c} \right)^{2} \frac{\hbar t}{m} \notag\end{align}

## This conclusion is a new insight to special relativity in that variance of the space-time location is minimized in the time dimension in a particular frame of reference.

03.

## Our Solar System.

\begin{align}\overrightarrow{R_{E}} = R_{E} \cos (-2\pi f_{\text{year}} t + \varphi ) \hat{x} + R_{E} \cos (-2\pi f_{\text{year}} t + \varphi ) \hat{y} \notag\end{align}

\begin{align}\overrightarrow{R_{e}} = R_{e} \cos (-2\pi f_{\text{SR}} t + \vartheta ) \hat{x} + R_{e} \cos (-2\pi f_{\text{SR}} t + \vartheta ) \hat{y} \notag\end{align}

\begin{align}\overrightarrow{R_{y}} = v_{z \perp} t \hat{y} \notag\end{align}

\begin{align}\overrightarrow{R_{\text{Laboratory}}} = \overrightarrow{R_{E}} + \overrightarrow{R_{e}} + \overrightarrow{R_{y}} \notag\end{align}

\begin{align}|v_{e}| = 2 \pi f_{\text{SR}} \cdot R_{E} \notag\end{align}

\begin{align} (\Delta s)^{2} = \frac{2\hbar t}{m c^{2}} v_{z \perp} v_{e} \cos(2 \pi f_{SR} t - \vartheta) \notag\end{align}

Which is consistent with our data.

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