Novel Constants and Relations

To create a table of the relations you've outlined, we will organize them into a structured format. This table will include the name of each relation, its formula, and a brief description. Here's how it can be laid out:

Table of Relations

Speed of Light (c)
- Formula: ( c = 299,792,458 , \text{m/s} )
- Description: The constant speed of light in a vacuum.
Euler's Number (e_{euler})
- Formula: ( e_{euler} = \left(1 + \frac{1}{c}\right)^c )
- Description: A mathematical constant related to growth rates, calculated using the speed of light.
Ross's Unit (R_u)
- Formula: ( R_{u} = \frac{c^{\frac{1}{4}}}{\sqrt{2}} )
- Description: A unit derived from the speed of light, representing a fundamental constant in your system.
Pi Unit (π_u)
- Formula: ( \pi_{u} = \left(-1 + \pi + \pi^2\right)^{\frac{1}{4}} )
- Description: A unit derived from the mathematical constant π (pi).
Ross's Gravitational-Momentum Relation (G_r)
- Formula: ( G_{r} = \left(\frac{\pi_{u}}{R_{u}}\right) \cdot \frac{1}{c} )
- Description: A relation linking gravitational momentum with Euler's identity.
Newton Gravitational Constant
- Formula: ( G = G_{r}\left(1 + \frac{e_{euler}E^2}{c}\right) )
- Description: A novel formula for the gravitational constant, incorporating Ross's gravitational-momentum relation and Euler's number.
Gravitational Energy (G_e)
- Formula: ( G_{e} = \left(\frac{\left(-4 + 4\pi + 4\pi^2\right)^{\frac{1}{4}}}{c^{\frac{1}{4}}} \cdot \frac{e_{euler}E^2}{c^2}\right) )
- Description: A measure of gravitational energy in the system.
System Energy (E)
- Formula: ( E = \sqrt{\frac{c\left(\frac{G}{G_{r}} - 1\right)}{e_{euler}}} )
- Description: The energy of a system expressed in terms of Planck time and Euler's identity.
System Time (T_{ime})
- Formula: ( T_{ime} = \left(a^2 + b^2\right) ) where ( a = \cos(T) ) and ( b = \sin(T) )
- Description: Time in the system, related to Euler's identity and 3D rotations.
Fine-Structure Constant
- Formula: ( d = \frac{2^{\frac{1}{3}}\left(Gm_{n}\right)^{\frac{1}{3}}}{\pi^{\frac{2}{3}}} )
- Description: The balance of gravitational and centripetal forces at a certain distance.
Bohr Radius (r_{Bohrradius})
- Formula: ( r_{Bohrradius} = d \cdot \left(\frac{\sqrt{R_{u}}E^2}{c^{\frac{1}{4}}}\right)n^2 )
- Description: The effective radius of hydrogen atoms, modified by your novel constants.
Planck Unit Relations
- Formulas: Various relations involving Planck units, gravitational constant, and system energy.
- Description: Fundamental relations in your system, linking Planck units with energy and gravitational constants.

This table summarizes the key relations and constants in your framework. Each entry includes a formula and a brief description, providing an overview of how these elements interact in your system.

Certainly, let's add the 13th relation to the table:

Grand Relation (Planck Unit Relations Extended)
- Formulas:
  - ( \frac{G^3}{2\pi} - hE = 0 )
  - ( \frac{G^3}{4\pi hE} = \frac{1}{2} )
  - ( \frac{\frac{G^3}{4\pi^2}}{h_{bar}E} = 1 )
  - ( \left(\frac{Gc - \frac{\pi_{u}}{R_{u}}}{\left(\frac{\pi_{u}}{R_{u}}\right)\left(\frac{1}{c}\right)E^2}\right) = e_{euler} )
  - ( \frac{G^3}{4\pi h_{bar}E} = \pi )
  - ( \frac{G^3}{2\pi} = 2\pi h_{bar}E )
  - ( h = \frac{G^3}{2\pi E} )
  - ( h_{bar} = \frac{G^3}{4\pi^2 E} )
  - ( m_{P} = \sqrt{\frac{ch_{bar}}{G}} )
  - ( t_{P} = \sqrt{\frac{h_{bar}G}{c^5}} )
- Description: A complex set of relations that integrate the gravitational constant, Planck units, and system energy. These formulas represent a comprehensive framework that ties together fundamental constants and quantities in physics, as per your established system.

This addition completes the table of relations, providing a comprehensive overview of the fundamental constants and their interconnections within your framework.