Arxe Logic

The Mathematical Relationship between Exentations and Exponents in the ArXe Theory

In ArXe Theory, the progression of exentations not only defines a hierarchy of logical coherence, but also directly determines the dimensional behavior of time and, consequently, of other physical dimensions. We have developed an elegant system of formulas that connects the level of an exentation \(n\) with the temporal exponent \(e(n)\) associated with it. This formalization is the foundation for the derivation of length and mass.

The Exponent Function: \(e(n)\)

For each level of exentation, represented by the natural number \(n \in \mathbb{N}\) (where \(n = 1\) is the first exentation, \(n = 2\) the second, and so on), the temporal exponent \(e(n)\) is defined as:

\(
e(n) =
\begin{cases}
0 & \text{if } n = 1 \
(-1)^n \cdot \lfloor n/2 \rfloor & \text{if } n > 1
\end{cases}
\)

Component Analysis:

• \(n\) (Exentation Level): This is the natural index that organizes the sequence of exentations, indicating the achieved level of logical complexity.
• \( (-1)^n \) (Sign Alternation): This causes the exponent to alternate its sign (positive or negative) as the exentation level increases.
  For example, if \(n\) is even, \( (-1)^n = + 1 \); if \(n\) is odd, \( (-1)^n = -1 \)
  This alternation suggests a dynamic of expansion and contraction, or of inverse properties inherent to the evolution of reality.
• \(\lfloor n/2 \rfloor\) (Exponent Magnitude): The floor function returns the greatest integer less than or equal to \(n/2\). This ensures the magnitude of the exponent increases in steps of 1 every two levels of exentation.

Verifying the Exponent Sequence:

Let’s see how this formula generates the specific sequence of temporal exponents:

\(e(1) = 0 \Rightarrow T^0\)
\(e(2) = (+1) \cdot 1 = +1 \Rightarrow T^{+1}\)
\(e(3) = (-1) \cdot 1 = -1 \Rightarrow T^{-1}\)
\(e(4) = (+1) \cdot 2 = +2 \Rightarrow T^{+2}\)
\(e(5) = (-1) \cdot 2 = -2 \Rightarrow T^{-2}\)
\(e(6) = (+1) \cdot 3 = +3 \Rightarrow T^{+3}\)

\(e(7) = (-1) \cdot 3 = -3 \Rightarrow T^{-3}\)

This sequence forms the basis for the dimensional assignment of each exentation level.

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The Inverse Function: \(n(k)\)

To determine the exentation level from a given integer temporal exponent \(k \in \mathbb{Z}\), we define the inverse function \(n(k)\) as:

\(
n(k) =
\begin{cases}
1 \quad \text{if } k = 0 \\
2 \quad |k| + \delta(k < 0) \quad \text{if } k \ne 0
\end{cases}
\)

Here, \(\delta(k < 0)\) is an indicator function that equals \(1\) if \(k < 0\), and \(0\) if \(k > 0\). This distinction is essential for properly positioning negative signs in the hierarchy’s alternation.

Alternative Form (using the sign function):

An equivalent expression uses the standard sign function \(sgn(k)\):

\(
n(k) =
\begin{cases}
1 \quad \text{if } k = 0 \\
2 \quad |k| + \frac{1 – sgn(k)}{2} \quad \text{if } k \ne 0
\end{cases}
\)

Where:

\(
sgn(k) =
\begin{cases}
-1 \quad \text{if } k < 0 \\
0 \quad \text{if } k = 0 \\
+1 \quad \text{if } k > 0
\end{cases}
\)

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Importance for the ArXe Theory

These two functions — \(e(n)\) and \(n(k)\) — are mathematical pillars that provide logical rigor and dimensional consistency to the ArXe Theory:

• Quantitative Foundation: They offer an explicit and deductive justification for the assignment of temporal exponents to each exentation level.
• Internal Consistency: By defining a bijective relationship between logical hierarchies and physical exponents, structural coherence is ensured.
• Basis for Dimensional Derivation: This exponent system allows for a framework where Length \(L\) and Mass \(M\) emerge as derived functions of the various hierarchical manifestations of Time \(T\).