Step 1: Classical Newtonian Gravity Equation

The standard Newtonian gravitational force between two masses m_1​ and m_2​, separated by a distance r, is:

Key characteristics of the classical equation:

• Scalar quantity G, gravitational constant, static and universal.

• Assumes instantaneous gravitational interaction.

• Force between two distinct masses is dependent solely on their magnitudes and their separation r.

Step 2: Conceptual Transition from Classical Force to Energy Framework

To transition toward the Unified Theory of Energy framework, we need to shift our conceptualization:

• Instead of a static gravitational force between discrete masses, think of energy exchanges between fields and particles as continuous, dynamic, interactive, and self-balancing processes.

• The Unified Theory of Energy conceptualizes gravitational interaction not as instantaneous action-at-a-distance, but as interactions of energy in forms of:

  • Radiation (R): Energy extending outwardly, available for absorption.

  • Gravitation (G): Energy stored within mass as potential, absorbed radiation.

  • Particulate Motion (implicit): Motion energy associated with particles and fields, moderating and influencing how radiation and gravitation interact.

• Mass structure (per the Unified Theory definitions) becomes a system where energy is continuously absorbed (Radiation → Gravitation), stored, and re-emitted (Gravitation → Radiation), maintaining internal equilibrium and external balance.

Step 3: From Discrete Masses to Continuous Distributions

The classical gravitational equation treats masses as isolated points. Your Unified Theory naturally extends toward continuous energy distributions:

• Instead of two discrete masses m_1, m_2​, consider continuous mass-energy distributions throughout Mass, represented as functions of position r from its center to its Surface, density parameter D, and the internal energy state parameter D_T.

• The interactions are now functions:

  • G(r, D, D_T): represents Gravitation as a stored energy function, dependent on distance to the Surface, degree of Surface Interaction, and Scale.

  • R(r, D, D_T): represents Radiation as energy extended through space, influenced by spatial distribution and internal energy states.

Step 4: Equilibrium (δ-Function) Condition δ(G−R)δ(G - R)δ(G−R)

The Unified Theory's integral equation introduces the delta function condition δ(G - R):

• This delta function represents a condition of local energetic equilibrium between absorbed (gravitational) and emitted (radiation) energies.

• Energy equilibrium occurs wherever gravitational energy storage equals radiation emitted, usually at the Surface of the Mass: G(r, D, D_T) = R(r, D, D_T)

This condition ensures the integral equation encapsulates a self-contained energy system, balancing internal (gravitation storage) and external (radiation emission) energy states continuously and precisely.

Step 5: Integral Formulation of Self-Contained Energy Equation

Thus, the classical Newtonian interaction transitions conceptually into an integral of a continuous, dynamic, balanced energy state:

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

This integral expresses explicitly:

• Integration over all possible density states (D) and energy state parameters (D_T).

• Energy equilibrium conditions (δ-function), focusing specifically on those states that achieve balanced gravitational-radiation equality.

• Multiplicative interaction of gravitational and radiation energies indicates a dynamic interplay and mutual interdependence, rather than static and independent quantities.

Step 6: Interpretation and Significance

The presented self-contained equation marks a profound shift in conceptual understanding:

• From Newton's static, discrete-point, gravitational force model to a continuous, dynamic, integral-based model of unified energy equilibrium.

• Energy is no longer seen merely as a static property or simple force relation, but as a balanced, continuous field of interactions and exchanges.

• The integration encapsulates a complete and unified description of how energy dynamically transitions between radiation, gravitational storage, and particulate motion within mass structures, becoming inherently self-contained and consistent.

Summary of Transition:

Newtonian Gravity Model → Unified Theory of Energy
Static, discrete masses → Continuous energy distributions
Instantaneous interactions → Dynamic energy exchange
Force-based → Energy equilibrium-based
Simple algebraic equation → Integral-based equation