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Unified Derivation of Inverse-Square Interactions under the Geometric Extremum Principle (GEP) Framework

Abstract

Based on the Geometric Extremum Principle (GEP) and taking spatial curvature as the fundamental field variable, this paper, together with the Maximum Information Efficiency (MIE) axiom, rigorously derives the mathematical form of long-range interactions through the calculus of variations. The core action of the system is defined as the volume integral of the squared curvature current density. Through Dirichlet energy variation, the governing equation for the curvature field in source-free space is obtained. Combined with spherically symmetric boundary conditions and the definition of force as the curvature gradient, the inverse-square laws of universal gravitation and Coulomb force are derived ab initio—without empirical parameters or extra assumptions. This work elevates long-range interactions from empirical laws to a necessary geometric consequence, providing core support for the Multi-Origin Curvature (MOC) framework and unified geometric extremal physics.

Keywords

Geometric Extremum Principle (GEP); Maximum Information Efficiency axiom; curvature current; Dirichlet energy; Laplace equation; inverse-square law; unified field theory

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1. Introduction

In classical physics, the inverse-square forms of Newton’s law of universal gravitation and Coulomb’s law originate from experimental observation and induction. Existing field theories can only describe their mathematical characteristics through Gauss’s flux theorem, but cannot answer from fundamental principles why interactions strictly follow the inverse-square law. General relativity interprets gravity as a geometric effect of spacetime curvature, but it does not incorporate electromagnetism into the same geometric framework, nor does it directly derive the decay form of forces from an extremum principle.

Based on the Geometric Extremum Principle (GEP) and the Maximum Information Efficiency (MIE) axiom proposed by the author, this paper constructs a unified field framework with spatial curvature K as the sole fundamental field variable. Mass and charge are uniformly defined as local sources of curvature, and the interaction field is identified as a diffusive curvature current. Through a corrected definition of the action and a rigorous variational derivation, the inverse-square law is obtained as a purely theoretical result, achieving a low-energy geometric unification of gravitational and electromagnetic forces.

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2. Core Definitions and Axioms of the GEP Framework

2.1 Definitions of Fundamental Physical Quantities

In the GEP framework, the fundamental property of physical space is the local curvature K(\boldsymbol{r}). All physical interactions originate from the spatial distribution, gradient, and flow evolution of the curvature field. The core physical quantities are defined as follows:

1. Curvature field: a scalar field K(\boldsymbol{r}) describing the local geometric bending of space. Mass and charge are point sources of the curvature field—they excite the curvature field.
2. Curvature current density vector: describes the tendency of curvature to diffuse from high-curvature regions to low-curvature regions. It is defined as the negative gradient of the curvature field:
\boldsymbol{J} = -\nabla K
\]
This definition ensures that the curvature current always flows in the direction of decreasing curvature, consistent with the stability requirement of physical space.
3. Geometric definition of interaction force: the long-range force experienced by a test particle is proportional to the magnitude of the curvature gradient—i.e., force directly reflects the spatially inhomogeneous distribution of curvature:
F \propto |\nabla K|
\]
This definition abandons the empirical assumption of a “force field” in traditional field theory and reduces force entirely to a geometric gradient effect.

2.2 Maximum Information Efficiency (MIE) Axiom

As the central axiom of the GEP framework, the Maximum Information Efficiency axiom states: the stable distribution of the curvature field in physical space must simultaneously satisfy maximization of information transfer efficiency, minimization of field dissipation, and smooth, curl-free flow lines. Mathematically, this means the action of the stable field takes an extremum (minimum), and the evolution and distribution of the field strictly obey the variational extremum condition.

Based on this axiom, and together with the corrected core rule of this paper, the action of the system in the GEP framework is the spatial integral of the squared curvature current density. Its mathematical form is uniquely determined by the MIE constraint.

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3. Construction of the GEP Action and Variational Derivation

3.1 Strict Definition of the Action

Based on the MIE axiom and the definition of the curvature current density, the action S for a stable curvature field in source-free physical space is defined as the volume integral of the squared curvature current density:

S = \int |\boldsymbol{J}|^2 \, dV

Substituting \boldsymbol{J} = -\nabla K (the square removes the sign) gives the final form:

S = \int (\nabla K)^2 \, dV

This action is the standard Dirichlet energy in mathematics—the optimal form for describing smoothness and minimal dissipation of a field. It fully satisfies the MIE constraints of curl-free, conserved, low-dissipation behavior and forms the core mathematical basis of the GEP framework.

3.2 Variational Extremum and Field Governing Equation

The stable curvature field distribution satisfies the action minimum condition, i.e., the functional variation is zero:

\delta S = 0

Performing the standard variation of the Dirichlet energy action, together with boundary conditions for source-free space (the field gradient vanishes at infinity), directly yields the source-free governing equation:

\nabla^2 K = 0

This is the three-dimensional Laplace equation. Its physical meaning: in source-free space, the curvature field is divergence-free, curl-free, and dissipation-free, strictly satisfying conservation constraints—fully self-consistent with the MIE axiom.

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4. Spherically Symmetric Solution and Derivation of the Inverse-Square Law

4.1 Spherically Symmetric Boundary Conditions and General Solution

In physical space, isolated point masses and point charges are spherically symmetric curvature sources. The corresponding curvature field distribution is strictly spherically symmetric, depending only on the radial distance r and independent of angles. In three-dimensional spherical coordinates, the spherically symmetric general solution of Laplace’s equation is:

K(r) = A + \frac{B}{r}

where A and B are constants determined by physical boundary conditions.

4.2 Physical Boundary Conditions

According to geometric constraints of physical space: at infinity, there is no curvature source, space is flat, and the curvature field strictly vanishes. That is:

\lim_{r \to \infty} K(r) = 0

Substituting this into the general solution directly gives A = 0. Hence the physically admissible solution reduces to:

K(r) \propto \frac{1}{r}

This solution is the unique physical solution of the source-free Laplace equation for a point source, corresponding to the long-range curvature field excited by a point source.

4.3 Calculation of the Curvature Gradient and Derivation of the Inverse-Square Law

Based on the GEP definition of force F \propto |\nabla K|, we compute the radial gradient of the spherically symmetric field K(r) \propto \frac{1}{r}. For a spherically symmetric field, only the radial component exists, and the gradient reduces to a one-dimensional derivative:

\nabla K = \frac{dK}{dr} \hat{\boldsymbol{r}}

Differentiating the curvature field:

\frac{dK}{dr} = \frac{d}{dr}\left( \frac{B}{r} \right) = -\frac{B}{r^2}

Taking the magnitude of the gradient (ignoring direction and sign):

|\nabla K| = \frac{|B|}{r^2}

Thus the magnitude of the curvature gradient is strictly inversely proportional to the square of the distance. Combining this with the geometric definition of force F \propto |\nabla K| directly yields:

\boldsymbol{F} \propto \frac{1}{r^2}

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5. Physical Meaning and Theoretical Value

5.1 Essence of the Inverse-Square Law

The derivation above proves that the inverse-square form of universal gravitation and Coulomb forces is not an empirical law but a necessary geometric consequence of three-dimensional physical space, the Maximum Information Efficiency axiom, the Dirichlet energy extremum constraint, and spherical symmetry.

The exponent “2” in the inverse-square law directly reflects the geometry of three-dimensional space: the gradient of a spherically symmetric field decays as r^2, which is a direct manifestation of spatial dimensionality. The form of force is completely determined by the extremal distribution of the curvature field, requiring no empirical parameters or extra assumptions.

5.2 Theoretical Unification

In the GEP framework, the only essential difference between gravitation and Coulomb forces is the nature of the curvature source (mass as the source for gravitational curvature, charge as the source for electromagnetic curvature). Their field distributions and force laws follow the same geometric extremal rules. This provides a low-energy unification of long-range interactions and offers a seamless core derivation for the Multi-Origin Curvature (MOC) framework and unified geometric extremal physics.

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6. Conclusion

Based on the refined Geometric Extremum Principle (GEP) and constrained by the Maximum Information Efficiency axiom, this paper establishes the core action as the volume integral of the squared curvature current density. Through Dirichlet energy variation, the field equation reduces to Laplace’s equation. Combined with spherically symmetric physical boundary conditions, the 1/r curvature field distribution of a point source is rigorously derived. Finally, using the geometric definition of force as the curvature gradient, the inverse-square law of interaction is obtained purely theoretically, without empirical assumptions.

This derivation closes the logical loop from “geometric extremum of space” to “form of long-range forces,” elevating the classical inverse-square law from an experimentally induced rule to a necessary geometric consequence of three-dimensional space. At the same time, it achieves a unified geometric interpretation of gravitational and electromagnetic forces, providing a new, self-consistent mathematical and physical foundation for unified field theory.

In source-containing space, the Laplace equation can be generalized to the Poisson equation \nabla^2 K = -4\pi \rho (where \rho is the curvature source density), which further reproduces the Poisson equation forms of Newtonian gravitational potential and Coulomb potential, fully compatible with experimental conclusions of classical field theory while having deeper geometric foundational support.

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