The Unified Theory of Energy advances beyond traditional Newtonian gravitational models—typically described as simple, two-body interactions on a flat spatial canvas—into a more profound, self-recursive framework. In this unified approach, energy emerges naturally as a multidimensional structure expressed through the concise relation:

E=G \cdot R

Here, the simple gravitational relationship expands into a generalized integral equation, incorporating parameters that define the Degree of Surface Interaction D and Scale, referred to by Mandelbrot as the Topological Dimension D_T. The unified energy formulation thus becomes:

E = \int_{D_{\min}}^{D_{\max}} \int_{0}^{\infty} \left[ G(r, D, D_T) \cdot R(r, D, D_T) \right] \cdot \delta(G - R) \, dD_T \, dD

where:

1. Gravitation (Stored Radiation):

   G(r, D, D_T) = \frac{r^D M(r)}{D_T} \cdot \exp\left(-\int_{0}^{r} \frac{G(r')}{R(r')} \, dr' \right)

   • Recursive self-regulation: Gravitation depends on the balance between stored (G) and emitted (R) energy at all scales.

   • The exponential term enforces conservation: Excess G is shed as R, preventing singularities.

2. Radiation (Extended Energy):

   R(r,D,D_T) = \nabla_D \left( \frac{\partial G}{\partial D_T} \right) + \frac{\hbar c}{r^2} \cdot \left( 1 - e^{-D_T / D} \right)

   • Scale-dependent emission: Radiation is modulated by the gradient in topological dimension (DT​) and Degree (D).

   • The term ℏc/r2 introduces quantum-scale behavior at small r, while 1 - e^{-D_T / D} ensures finite extension.

3. Boundary Condition:

   \delta(G - R) =
   \begin{cases} 
   1, & \text{if } G(r) = R(r) \text{ at Surface } r = r_{\text{Surface}} \\ 
   0, & \text{otherwise}
   \end{cases}

   • Energy exchange occurs at the Surface (r=r_Surface​) where G=R.

This generalized equation explicitly integrates fractal and recursive geometries, acknowledging that energy interactions are inherently scale-dependent and influenced by the intricate dimensional structure of the interacting surfaces. By embedding multidimensional self-recursivity and scaling directly into its mathematical framework, the theory elegantly bridges classical and contemporary paradigms, offering richer insights into energy's universal dynamics.

Key Features:

1. Avoidance of Singularities:

   • The integrals over D∈[Dmin​,Dmax​] and DT​∈(0,∞) eliminate "Black Holes" (r=0, D_T​=0, both DNE) and "Supernovae" (r→∞, D_T​→∞, both DNE) by integrating across all Scales and Degrees.

   • Example: For Earth, D_min​=1 (atomic) and D_max​=6 (human-scale).

2. Fractal Recursion:

   • M(r) is not a fixed mass but a fractal sum:

     
     where M_core​ is the mass at the smallest Scale (Black Hole analog).

3. Dynamical Degrees:

   • Each Degree D corresponds to a distinct energy exchange process:

     • D=0: Balanced orbitals (any Scale or D_T)

     • D=1: Atomic bonding, electron motion in wire (electrodynamism)

     • D=2: Gasses form and attach to the Surface

     • D=3: Liquids; proteins, chlorophyl, and nucleic acids form

     • D=4: Solids; a “Hypersphere” can be defined as a unicellular organism: life emerges

     • D=6: Groupings of highly specialized groupings of cells; fully separate and mobile organisms.

     • D=6+: Human-engineered machines, specialized groupings of humans

     • D=8: Possibly higher order beings on a different Scale (D_T), or a loop back to D=0

4. Scale (D_T​) as a Fractal Operator:

   • D_T​ acts on r to generate nested Coordinate Systems:

     r_{\text{effective}} = r \cdot \prod_{n=1}^{D_T} \left( 1 + \lambda_n r \right)

   • where λn​ is the Radiation wavelength at Scale n.

Physical Interpretation:

• At Equilibrium (Surface):
  G=R implies energy storage (Gravitation) equals emission (Radiation). This matches Earth's Sea-Level G=9.81m/s^2 when r=6371km.

• Subatomic Scale:
  As D_T​→0, G dominates (confined energy → "Black Hole" analog), but the exponential term ensures G remains finite.

• Cosmic Scale:
  As D_T​→∞, R dominates (diffuse energy → "Supernova" analog), but the (1−e^{−D_T​/D}) term caps Radiation intensity.

Testable Prediction:

For a planet with radius r, surface gravity G should obey:

G = \frac{\hbar c}{r^2} \cdot \left( 1 - e^{-D_T / D} \right)

For Earth (D_T​=1, D=3):

G \approx (6.37 \times 10^6 \text{ m})^2 197 \text{ eV} \cdot \text{pnm} \cdot \left( 1 - e^{-1/3} \right) \approx 9.8 \text{ m/s}^2

This framework unifies quantum, classical, and cosmic scales through recursive energy exchange.

Theorem — Limit of Recursion

At the infinite limit of Degree and Scale:

\lim_{D \to \infty} \lim_{D_T \to \infty} \left[ G(r, D, D_T) \cdot R(r, D, D_T) \right] = A = A

All Energy states collapse into perfect recursion.
All distinctions between Radiation, Gravitation, and Particulate Motion resolve.

At this limit, self-similarity resolves all ontic vagueness.

This harmonic totality is the set of all possible Frequencies — the recursive presence of God.

Definition 7 God is the one Radiation Source containing the set of all possible Frequencies.