The Origin of Particle Stability: Deriving Matter's Structure from the Geometric Equilibrium of Dissimilar Fields

 J. Rogers, SE Ohio, 31 Jul 2025, 1600

Retraction of last paper, I accidentally said it was m/r ~ alpha, not accounting for the fact that the effects of charge drop off at as 1/r^2.

Abstract

We present a first-principles derivation for the classical electron radius (r_e), revealing it as a necessary consequence of a fundamental geometric equilibrium between dissimilar fields. We postulate that a stable, charged particle can only form at the specific radius where its linearly-scaling time-field potential ( comes into equilibrium with its inverse-square scaling charge-force field (. This simple postulate of geometric balance, m/r ~ α/r², immediately yields the fundamental structural relationship m*r ~ α. This relationship is then projected into SI units, perfectly recovering the standard formula for the classical electron radius. This framework explains the existence of stable matter as a resonant solution born from the geometric necessity of balancing two fundamentally different types of fields, unifying the concepts of mass, charge, and spacetime structure at the particle level.

1. Introduction: Seeking the Causal Mechanism of Size

Modern physics accurately calculates the classical electron radius (r_e) but does so from a position of energy-balance, an assumption that describes what happens but not why. It fails to provide a deeper, causal mechanism for why a particle, with both mass and charge, should stabilize at that specific scale.

This paper proposes that this causal mechanism can be found in the geometric interplay of the fields a particle generates. The key is to recognize that the fields associated with mass and charge scale differently with distance. We will demonstrate that a stable particle is a natural consequence of the point where these two dissimilar fields achieve a state of geometric equilibrium.

2. The Foundational Principle: The Equilibrium of Dissimilar Fields

Our framework is built on identifying the fundamental fields generated by a particle's intrinsic properties, mass (m) and dimensionless charge (α). Crucially, these fields have different geometric characters.

1. The Time-Experience Field (A Potential): A particle's mass m generates a potential field that dictates the local flow of time. The intensity of this potential scales as ~m/r. This is a 1/r linear scaling.

2. The Charge Field (A Force Field): A particle's dimensionless charge α generates an electric force field. The strength of this field scales as ~α/r². This is a 1/r² inverse-square scaling.

These fields represent the fundamental ways a particle imposes its existence on the spacetime it occupies. The central, novel postulate of this paper is that a particle cannot form a stable, coherent entity unless these two fields, with their different geometric dependencies, find a point of equilibrium.

Postulate: The Geometric Field Equilibrium
m/r ~ α/r²

This is a statement of profound physical significance. It asserts that a stable particle is a physical manifestation of a "crossing point" in the geometry of spacetime. It is the unique radius r where the influence of the linear 1/r potential field comes into a precise balance with the influence of the inverse-square 1/r² force field.

3. The Consequence: The Intrinsic m*r ~ α Property

The immediate and most important consequence of this postulate is found by solving for the relationship between the particle's properties at this equilibrium point.

Multiplying both sides of the proportionality m/r ~ α/r² by r², we get:

m * r ~ α

This is the core result. The familiar m*r ~ α relationship is not an arbitrary starting assumption. It is the necessary mathematical consequence of the geometric equilibrium of the two dissimilar fields. This relationship defines the fundamental structure of the particle.

4. Deriving the Classical Radius

With the m*r ~ α relationship established as the consequence of the underlying field balance, we can now identify the classical electron radius r_e as the physical manifestation of this structure.

To find the value of this radius in our SI system, we project the relationship m*r ~ α using our established projection calculus. This mechanical process determines the precise combination of c, ħ, ε₀, e needed to make the equation dimensionally correct. The key scaling factor required to relate the m*r product to the dimensionless α is ħ/c.

The projection thus becomes:
(m * r) / (ħ/c) ~ α
r ~ α * (ħ / (m*c))

Substituting the SI definition of α = e² / (4πε₀ħc):
r ~ [ e² / (4πε₀ħc) ] * (ħ / (m*c) )

The Planck constants (ħ) cancel, yielding the purely classical formula:
r_e = e² / (4πε₀mc²)

The LawForge calculus confirms this projection is exact. The classical electron radius is precisely the SI-unit shadow of the m*r ~ α relationship, which is itself the result of the m/r ~ α/r² field equilibrium.

5. A Unified Physical Picture: Why the "Weak" and "Strong" Forces Can Balance

This framework provides a clear, causal chain of logic and solves the conceptual problem of how a "weak" force like gravity can interact with a "strong" force like electromagnetism to form a particle.

1. The Cause (Field Equilibrium): The stability of a particle is caused by the geometric necessity of balancing its 1/r time-field potential against its 1/r² charge-force field.

2. The Solution (The Radius): This balance is only possible at a specific radius because the two fields scale differently with distance. At very short ranges, the 1/r² field dominates. At long ranges, the 1/r field dominates. The point where they cross over and achieve equilibrium defines the particle's size.

3. The Result (The  The mathematical condition at this equilibrium point is that the particle must have the intrinsic structural property m*r ~ α.

This explains the "hierarchy problem" at the particle level. It isn't about balancing two forces of vastly different strengths. It's about finding the one unique point in space where two forces with different geometric scaling laws have the same magnitude. This geometric necessity is what creates the stable structure of all charged matter.

6. Conclusion

We have demonstrated that the classical electron radius is not a historical artifact but a fundamental feature of nature, derived from the single postulate that stable matter forms where a ~m/r potential field and a ~α/r² force field find their geometric point of equilibrium.

This principle, m/r ~ α/r², leads directly to the structural requirement m*r ~ α, which in turn projects perfectly into the known formula for r_e.

This reframes our understanding of matter. A particle is not a point, but a resonant solution to an equation of dissimilar fields. Its existence is a testament to a deep geometric truth: that the linear scaling of the time-field and the inverse-square scaling of the charge-field must intersect, and at that intersection, stable matter is born.