This article introduces the concept of General Recursivity—a logical structure arising from the Unified Theory of Energy, establishing a comprehensive mathematical description of energy as inherently recursive and scale-dependent. By defining energy explicitly as interplay between Radiation, Gravitation, and Particulate Motion across infinite scales, the theory integrates classical physics with recursive fractal geometry and quantum-scale interactions, offering refined frameworks for defining, estimating, and calculating potential energy.
Current scientific approaches, rooted in simplified binary distinctions and mechanistic models, fail to capture the inherent recursive complexity of energy interactions at varying scales. The Unified Theory of Energy suggests a generalized framework incorporating three mutually dependent states of energy—Radiation, Gravitation, and Particulate Motion. This paper formalizes the recursive interplay between these states mathematically, providing a concise expression that captures inherent self-regulation, recursion, and multidimensionality.
Energy manifests recursively in three distinct but interdependent states:
Radiation (R): Energy projected outward for absorption.
Gravitation (G): Energy stored as potential within mass structures.
Particulate Motion: Kinetic (inertial) energy that modulates mass structures and radiation interactions.
These three states are interwoven recursively—each depends fundamentally upon the others at all scales.
Theorem (Energy State Theorem):
Energy exists only as Radiation, Gravitation, and Particulate Motion simultaneously; no state exists independently.
This theory introduces Scale (topological dimension D_T) and Degrees of Surface Interaction (D) explicitly, producing a generalized recursive energy equation:
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)\, dr\, dDThis generalized equation integrates fractal geometries and multi-dimensional recursion, acknowledging that energy interactions inherently depend upon the intricate structure and complexity at the interacting surfaces.
Gravitation explicitly depends recursively upon stored radiation:
G(r, D, D_T) = \frac{r^D \cdot M(r)}{D_T} \cdot \exp\left(-\int_{0}^{r}\frac{G(r')}{R(r')}dr'\right)M(r) represents cumulative mass structure at radius r.
The exponential ensures finite gravitational potential energy by recursively balancing emitted and stored radiation at all scales.
Radiation inherently integrates fractal recursive structure through the gradient of D_T:
R(r,D,D_T) = \nabla_D D_T \cdot M(r) + \frac{\hbar c}{r^2}\left(1 - e^{-D_T/D}\right)Radiation emission is thus explicitly scale-dependent and intrinsically quantum at small r, bridging classical and quantum scales naturally.
Energy exchange explicitly occurs where gravitational storage and radiative emission balance—on the Surface ( r=r_{\text{Surface}} ):
\delta(G - R)\big|_{r=r_{\text{Surface}}} = 0This interaction condition defines the surface as the fundamental interface of recursive energy dynamics.
Your theory explicitly incorporates fractal geometry—where Scale is not just size but the recursive depth of nested Coordinate Systems:
r_{\text{effective}} = r \cdot \prod_{n=1}^{D_T} \left(1 + \lambda_n\right)Here, recursive fractal expansion emerges naturally through wavelengths \lambda_n, ensuring finite containment and self-consistency across scales—from subatomic to cosmic.
Energy exchanges recursively evolve complexity in discrete "Degrees":
Degree (D)Energy Interaction Result
D=0 Neutral recursive orbit states
D=1 Atomic bonds; electromagnetic interactions
D=2 Formation of atmospheric gases
D=3 Liquids; complex proteins and nucleic acid synthesis
D=4 Unicellular life—separation from surface (cellular walls emerge as Hypersphere)
D=5 Multicellular organisms, specialized biological systems, perception
D=6 Highly specialized mobile organisms; emergence of self-awareness
This categorization links energy scales explicitly and recursively from simple particle exchanges to living organisms and conscious awareness.
Using explicit recursive definitions, we predict Earth's surface gravitational potential (setting D_T=1, D=3, Earth radius r≈6.37×10^6 m):
G_{\text{Earth}} \approx \frac{(6.37\times 10^6\,\text{m})^3 \cdot M_{\oplus}}{1}\cdot e^{-\int_0^{R_{\oplus}}\frac{G(r')}{R(r')}dr'}This explicitly recursive integral demonstrates Earth’s gravitation as dependent upon its own internal recursive energy dynamics, relating macro to microscopic levels.