1. Define Scale as Logarithmic Topological Dimension

Definition 6 “Scale (D_T) refers to the relative size of any Radiation Source.”

-The Unified Theory of Energy

D_T(n) = 10^n, \quad \text{where } n \in [-35, 35]

2. Effective Dimensional Projection Function

D_{\text{eff}}(n) = f(D_T(n), D) 

3. Example: D_T and Physical Object Mapping

\text{At } n = -35: \quad \text{Electron} \sim \text{Red Giant} \quad (\text{massive, extended}) \\
\text{At } n = 0: \quad \text{Human scale} \quad (\sim 1 \text{ meter}) \\
\text{At } n = 35: \quad \text{Universe} \to \text{Pointlike} \quad (D = 0)

4. Recursive Dimensional Identity

\text{Let } U(n) \text{ be a universe at scale level } n \\

\Rightarrow \exists \, P(n-35) \in U(n): \quad P(n-35) \sim U(n+35)

5. Time Scaling Function (for dynamic behavior)

T(n) = \alpha \cdot 10^{-n}, \quad \text{where } \alpha \text{ is a base unit of time}

6. Scale-Relative Perception Function (conscious observer)

\mathcal{P}(n) = \frac{1}{1 + |n - n_0|}, \quad \text{where } n_0 = 0 \text{ (human reference scale)}

7. Euler's Validity Condition

e^{i\omega t} \text{ is valid only if } D_T(n) \approx 10^0 \Rightarrow n \in [-1, 1]

8. Recursive Field Model (speculative)

\Phi(x, n) = \int \psi(x, n') \cdot K(n, n') \, dn', \quad \text{where } K(n, n') \text{ is a scale coupling kernel,}