To transition from Newtonian mechanics to the Unified Theory of Energy (UTE) and its equation of General Recursivity, we recontextualize gravitational energy as a multidimensional and scalable equilibrium between Gravitation (G) and Radiation (R). Here's the step-by-step derivation:

Step 1: Express Newtonian Potential Energy

Begin with Newtonian gravitational potential energy:

U = -\,G\,\frac{m_1\,m_2}{r}

This is a static, 3D description. The UTE generalizes this by integrating over all Degrees of Surface Interaction (D) and interaction times (DT​).

Step 2: Define Energy Density in UTE

In UTE, energy E is the integral of the product of Gravitation G(r,D,DT​) and Radiation R(r,D,DT​), constrained by equilibrium (G=R) at the Surface:

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

Here, δ(G−R) enforces energy conservation at equilibrium.

Step 3: Specialize to 3D Newtonian Limit

For Newtonian gravity, equilibrium occurs in D=3. Define:

G(r,3,D_{T}) = G\,\frac{m_{1}m_{2}}{r},\quad R(r,3,D_{T}) = G(r,3,D_{T})

The delta function collapses the integral to D=3:

\delta(G - R) \rightarrow \delta(D - 3)

• At the third Degree of Surface Interaction (explicitly D=3), the gravitational energy function G(r,3,D_{T}) is defined explicitly as the classical Newtonian gravitational potential form (-G m_{1}m_{2}/r)

  -\,G\,\frac{m_{1} m_{2}}{r},

  except we keep it positive and explicit for general usage here; the negative sign for potential energy would typically come in separately as needed.

• Radiation R(r,3,D_{T}) at the third Degree of Surface Interaction explicitly equals the gravitational energy function G(r,3,D_{T}) at the Surface, distance r from the theoretical center of the Radiation Source (e.g. r_Sun=696,000km), reflecting the equilibrium condition of the Unified Theory of Energy framework explicitly.

Step 4: Evaluate the Integral

Substitute G=R and integrate:

E = \int_{3}^{3} \int_{0}^{\infty} \left[\,G\,\frac{m_{1}m_{2}}{r}\,\right]^{2} \cdot \delta(D - 3)\,dD_{T}\,dD

The spatial integral reduces to a single term at D=3:

E = G_{\text{Newton}}\,\frac{m_{1} m_{2}}{r}\int_{0}^{\infty} dD_{T}

For static fields, Scale (D_T)​ integrates to a constant:

E \propto -\,G_{\text{Newton}}\,\frac{m_{1} m_{2}}{r}

The proportionality symbol "∝\propto" clearly indicates that the integral over D_{T}​ (Scale (UTE) or “Topological Dimension” (Mandelbrot, et. al.)) yields a constant in static fields, thus explicitly recovering Newton’s classical gravitational potential energy (with negative sign indicating bound energy states).

Step 5: Generalize to All Scales

The Unified Theory of Energy extends the previous considerations by explicitly summing (integrating) contributions over all Degrees of Surface Interaction (D) and Scales (D_{T}). Within this framework, singularities are naturally suppressed due to the equilibrium condition enforced by the delta function δ(G−R)\delta(G - R), which ensures balanced energy exchange precisely at the interaction Surface (r). The upper boundary, D_{\text{max}}​, is currently six in the UTE, but more probable is 2π, and likely 4π, or beyond, reflecting complete and recursive dimensional interactions inherent to the structural hierarchy of our universe.

Conclusion

The UTE equation reduces to Newtonian energy when D=3 and G=R at the Surface (e.g. r_Earth=6371km), while inherently allowing for multiscale gravitational-radiative equilibria.

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