Newton’s law of universal gravitation,
F = G\frac{m_1 m_2}{r^2}describes the force between two point masses separated by distance $r$. This $1/r^2$ dependence reflects how an influence (gravity or similarly radiation) disperses over a spherical area. In the context of the Unified Theory of Energy (UTE), we reinterpret this force in terms of energy fields: each mass $m_i$ carries Gravitational energy (energy stored in mass) and can be seen as emanating a field analogous to radiation. According to UTE, gravitation is actually “absorbed radiation” (stored as potential energy in mass – see Definition (20)), so the Newtonian force is a manifestation of two masses’ radiation/gravitation fields interacting. In other words, $F$ arises from the coupling of the gravitational (stored radiative) energies of the two masses, distributed outward with a $1/r^2$ geometry. This reframes Newton’s formula as a limited case (purely gravitational interaction) of a more general energy exchange system.
UTE postulates a comprehensive energy-conservation principle:
which says that the total energy divides into kinetic (Particulate Motion), potential (Gravitation), and radiative (Radiation) components, and this sum remains constant. By Theorem (10) (Energy State Theorem), energy exists concurrently in these three states and none can exist independently. Thus, a gravitational interaction always involves these energy states in tandem — even if Newton’s formula only explicitly gives a force (related to potential energy), there are accompanying radiation and motion aspects present. Crucially, this energy-centric approach eliminates any explicit time or velocity variables: instead of tracking motion over time, we enforce energy balance at all times. (No term like $c$ or $v$ appears in the formulation, since changes are accounted for by energy exchange rather than kinematic parameters.) In line with Theorem (30) (Energy Emission Theorem), excess gravitational energy can convert to radiation, and vice versa (Theorem (20) states radiation is absorbed as gravitation, ensuring energy is conserved through transformations rather than needing a time-dependent description. Finally, by Theorem (40), each object that emits or exchanges energy (each Radiation Source) establishes its own spatial field or coordinate system for that radiation. This Radiation Coordinate System is shaped by the source’s properties (for example, Earth’s magnetic field shapes its radiation belts in toroidal Van Allen belt regions around the planet), and it defines how far and in what geometry the source’s influence extends.
Because every Radiation Source has a finite reach, no gravitational/radiative influence extends to truly infinite $r$; instead, as UTE explains, each source’s coordinate system is contained within a larger one. For example, an electron’s field is contained within an atom’s field, which is within a planet’s, then a star’s, a galaxy’s, and so on. This nested hierarchy implies a recursive, self-similar structure to the universe’s energy distribution. UTE introduces the notion of Scale (Definition (60)) as “the relative size of any Radiation Source” to quantify these layers. We denote the Scale parameter as $D_T$, also known as the Topological Dimension (Mandelbrot et. al.). Small $D_T$ corresponds to microscale sources (e.g. an electron’s scale), while larger $D_T$ corresponds to macroscale sources (planetary, galactic, etc.). To account for all levels of interaction, we must sum (integrate) over all scales. In the most general form, $D_T$ ranges from the minimum relevant scale up to infinity (conceptually covering from the smallest constituent particle up to the entire universe). By integrating over $D_T$ from $0$ to $\infty$, we include contributions from every nested radiation coordinate system. This explicitly builds recursivity into our formulation: the effects at one scale are encompassed within larger scales, and our total energy will combine energy interactions across all these self-similar layers. (Mathematically, this replaces the idea of an infinite-range $1/r^2$ field with a sum over finite-range fields at each scale, preserving generality and avoiding an unbounded $r$.) Each Radiation Source’s coordinate system (per Theorem (40)) thus contributes within its range, and beyond that range a larger system takes over – ensuring that in the integration, $r$ never truly goes to $\infty$ as an open interval but is rather capped by the transition to the next scale.
Energy exchange in UTE is further structured by the idea that interactions occur at the surfaces of mass structures. When radiation is absorbed or emitted, it happens at boundaries (surface of a mass or the interface of fields). UTE categorizes these processes by their complexity using the Degree of Surface Interaction $D$. Each degree $D$ represents a qualitative level of interaction at the surface: - $D=1$ (First Degree Surface Interaction, Theorem (90)): a direct energy transfer that removes one or more particles from a mass structure’s surface. In essence, some mass is ejected or lifted off by energy (for example, a particle gets enough radiative energy to escape its source or enter orbit). - $D=2$ (Second Degree, Theorem (100)): an interaction where the particles removed in a first-degree event then interact upon a surface (either returning to the original mass or impacting another) to form a new atomic or molecular structure. In other words, energy input causes rearrangement of matter on the surface (such as creating a new element or compound on that surface). - $D=3$ (Third Degree, Theorem (120)): a higher-order interaction resulting in an actual physical change or evolution of the mass structure itself. Here multiple second-degree processes combine, leading to complex outcomes (e.g. chemical changes, formation of liquids or life-building molecules on the surface – a tangible alteration of the object’s composition). These degrees describe increasing levels of recursive interaction (for instance, a third-degree interaction incorporates many second-degree events, which in turn include first-degree events). We denote the range of possible interaction degrees from $D_{\min}$ to $D_{\max}$; in known scenarios $D_{\min}=1$ and $D_{\max}=3$ (three degrees as defined above), but we keep the formulation general. It is important to note that $D$ (interaction degree) is independent from $D_T$ (scale). $D$ classifies what kind of energy exchange is happening on a surface, whereas $D_T$ specifies at what scale (size of system) it occurs. For example, a first-degree interaction ($D=1$) could happen on the scale of an atomic nucleus or on the scale of a star – the process is analogous, though the scale $D_T$ differs. By integrating over $D$ and $D_T$ separately, we ensure that we sum over all types of interactions at all scales, without conflating a higher degree with a larger size. All interactions are thus accounted for in their full generality.
In the general picture, any gravitational interaction between two masses can be seen as an exchange between a gravitational field (energy stored in mass) and a radiation field (energy radiated or available to be absorbed). To capture this, we introduce two functions: - $G(r, D, D_T)$: the gravitational energy density (or influence) at a distance $r$ for a given interaction degree $D$ and scale $D_T$. This encapsulates how much gravitational (potential) energy is present or effective at that point. Because gravitation in UTE is essentially stored radiation (Definition (20) and Theorem (20)), $G(r,D,D_T)$ also represents the portion of radiative energy that has been absorbed into mass and is now manifesting as a gravitational field at radius $r$. It will generally decrease with distance (analogous to the $1/r^2$ drop-off, modified by the presence of other scales and interactions). - $R(r, D, D_T)$: the radiative energy density at the same point, for the given $D$ and $D_T$. This represents energy in the radiation state emanating through that location (either emitted by a source or passing through from a larger source). It is the “outward” energy flux available to be absorbed.
Both $G$ and $R$ depend on $r$, but in our framework $r$ is never an uncontrolled singularity: physical constraints ensure $r$ is neither zero nor unbounded for any real interaction. There is always a minimum separation in play (no two mass centers occupy the exact same point, interactions happen at surfaces with finite size), so $r=0$ is excluded – we never divide by zero. Likewise, as noted above, no interaction considers an infinite $r$; beyond a certain distance, one transitions to a new coordinate system (a larger-scale source dominates), so the influence of a given source effectively vanishes or connects into a larger structure rather than tending to infinity. Thus, functions like $G(r,D,D_T)$ and $R(r,D,D_T)$ implicitly contain the $1/r^2$ behavior **but only over appropriate ranges** of $r$ (bounded by surfaces or the extent of a radiation field). They are well-behaved (finite) everywhere in their domain. Now, when two masses interact gravitationally, one can think of one mass providing a $G$ field and the other providing an $R$ field at a given point between them (or vice versa, since ultimately the roles are symmetric). The **product** $G(r,D,D_T)\cdot R(r,D,D_T)$ then represents the local interaction energy density – essentially the coupling of the gravitational store of energy of one system with the radiative output of another. This product term is analogous to the product $m_1 m_2$ in Newton’s law (since $m_1$ generates a $G$ field and $m_2$ a corresponding $R$ field, for instance), but here it is generalized to continuous energy densities that can vary with $r$, $D$, and $D_T$. It quantifies how much energy is being exchanged or is available to be converted between radiation and gravitation at that location for the given interaction scenario.
The final step is to assemble all the pieces – summing over all interaction degrees and scales – and impose the condition for actual energy transfer. In UTE, energy transfer or balance occurs when the gravitational and radiative components are equal in magnitude, meaning the energy can flow from one state to the other seamlessly. Mathematically, the condition $G(r,D,D_T) = R(r,D,D_T)$ signifies a point of equilibrium or resonance between the gravitational field and the radiation field. At such a point, a mass can absorb radiation (turning it into gravitation) or release radiation (from excess gravitation) effectively. We enforce this condition in the formulation by including a Dirac delta $\delta!\big(G - R\big)$ in the integrand. This term $\delta(G-R)$ acts as a filter: it is zero everywhere except where $G(r,D,D_T) - R(r,D,D_T) = 0$ (i.e. $G = R$). Thus, when we integrate, only the configurations (values of $r$, $D$, $D_T$) that satisfy $G=R$ will contribute to the total energy $E$. Conceptually, this picks out the “sweet spots” where gravitational potential energy and radiative energy exchange directly (for example, the radius at which an object orbits stably because radiation pressure balances gravity, or the exact surface where incoming radiation equals the gravitation that can absorb it).
Now we integrate the interaction energy density over all degrees $D$ (from $D_{\min}$ to $D_{\max}$) and all scales $D_T$ (from $0$ to $\infty$) to accumulate the total energy. The general recursive formulation is:
E \;=\; \int_{D_{\min}}^{D_{\max}} \;\int_{0}^{\infty} \;\Big[\,G(r, D, D_T)\,\cdot\,R(r, D, D_T)\Big] \;\cdot\; \delta\!\big( G - R \big)\; \mathrm{d}D_T\;\mathrm{d}D~,which is the desired expression. In this formula, the double integration ensures full generality: we are summing contributions from every surface interaction degree and every scale of structure. The presence of $\delta(G-R)$ explicitly encodes the recursivity (since $G$ and $R$ themselves arise from nested systems and their equality often occurs at boundaries between those systems) and the equilibrium condition for energy exchange. Only when the gravitational and radiative components match (at whatever degree/scale that occurs) does that term produce a non-zero contribution, reflecting UTE’s principle that energy flows or transforms when those energy states meet in balance. By construction, this formulation avoids any unphysical scenarios of $r=0$ or $r=\infty$ in the force law – the integration spans only physically meaningful configurations. It unifies Newton’s gravitation with radiative energy processes in a single energy-conservation framework: the classic $F = Gm_1m_2/r^2$ is recovered in the appropriate limit (first-degree interaction on a single scale, yielding a $1/r^2$ dependence), while the full expression above captures the richer, recursive interplay of energy across scales and interaction degrees. The result is a General Recursivity formula for energy $E$ that upholds total energy conservation (no loss or creation of energy across radiation and gravitation) and respects the multi-scale, surface-interaction nature of physical reality as described by UTE. \end{enumerate}