The Invariant Interaction: One Force Law, Four Geometries

J. Rogers, SE Ohio

Abstract

We demonstrate that there is only one force interaction law in the universe, which operates through the reciprocal product of natural ratios. By defining the field intensity (I) as a 1:1 scaling of a geometric count over length, we show that the "four fundamental forces" are merely four different geometries of the count being processed by the same invariant engine. This framework eliminates "action at a distance" and all arbitrary force constants, revealing a perfectly symmetric, local, and mechanically intuitive reality.

1. Introduction: The Illusion of Four Forces

For centuries, physics has described nature through separate forces: gravity, electromagnetism, the strong nuclear force, and the weak nuclear force. Each comes with its own mathematical description, coupling constants, and peculiar behaviors. This fragmentation has led to profound puzzles: Why are the forces so different in strength? What is the origin of the arbitrary-seeming constants? And how can forces act instantly across empty space?

This paper presents a radical simplification: there are not four forces, but four geometries. These geometries are concrete, physical configurations of the universal substrate—the fundamental "fabric" of reality. They serve as blueprints that determine how discrete quantities ("counts") project their influence. The interaction itself is always the same: a local, reciprocal product of two field intensities.

What we perceive as fundamentally different forces are merely different geometric channels through which the same invariant engine operates.

2. The Engine: The Law of Natural Ratios

In the substrate, physical interaction is not a "force" in the Newtonian sense. It is the reciprocal product of two local field intensities. There is only one law:

$${\mathrm{<span\; style="background-color:\; white;">F</span>}}_{\text{<span style="background-color: white;">nat</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">I</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">I</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

This law is a Universal Identity. It has no coupling constants, no "strengths," and no preferences. It simply calculates the resulting stress from the intersection of two intensities. It is 1:1 and strictly local.

Think of it as the universe's only arithmetic operation for interaction: take two numbers (intensities) that are locally present, multiply them, and obtain the resulting action. No telepathy, no magic constants—just pure, simple arithmetic.

3. The Field: The 1:r Projection

A field is not a separate substance; it is the geometric projection of a discrete source count into the substrate. The intensity (I) of a field at any point is always defined by the ratio:

$$\mathrm{<span\; style="background-color:\; white;">I</span>}<span\; style="background-color:\; white;">=</span>\frac{\text{<span style="background-color: white;">Count</span>}<span\; style="background-color:\; white;">\times </span>\text{<span style="background-color: white;">Geometry</span>}}{\text{<span style="background-color: white;">Length</span>}}$$

Where:

• Count: The discrete quantum number (integer for charge or mass). 

• Geometry: The specific "lever arm" or "scaling factor" of that interaction type—a fixed property of the substrate's architecture. The geometry of the mass is its natural mass value.

• Length: The spatial separation from the source.

The $1\mathrm{/}{r}^2$ dependence observed in classical force laws is not fundamental; it emerges as the product of two $1\mathrm{/}r$ field projections meeting in the reciprocal interaction ($I_1\times I_2$).

4. The Four Geometries: Concrete Physical Blueprints

The "four forces" emerge because the substrate permits four distinct geometric scaling factors. These define how the "Count" projects its intensity. Crucially, these geometries are not abstract mathematical spaces but concrete, physical configurations.

4.1 The Strong Geometry (The Substrate's Grain)

• Geometry factor ($sg$): $1.0$

• Physical picture: Direct gear mesh. Imagine two interlocking gears with perfect, tooth-to-tooth contact—a 1:1 transmission with no mechanical disadvantage.

• Field: $I_s=\frac{\text{Count}\times 1.0}{L}$

• Interaction: $F=I_{s1}\times I_{s2}$

• Why it's strong: It's the raw stiffness of the substrate itself, operating at full strength. This direct contact explains the immense binding energy in nuclei and its extremely short range.

4.2 The Electromagnetic Geometry (The Linear Lever)

• Geometry factor ($cg$): $\approx 0.034$

• Physical picture: A fixed lever arm. A charge is not a bare knot in the substrate but one attached to a lever of precise length. This universal lever ratio ($1\mathrm{/}29.5$) is a built-in property of the vacuum's geometry around a charge.

• Field: $I_e=\frac{\text{Count}\times 0.034}{L}$

• Interaction: $F=I_{e1}\times I_{e2}$

• Why it's weaker: You're not pushing directly but through a lever that reduces effective force. The familiar inverse-square law emerges from the $1\mathrm{/}r$ projection of this levered intensity.

4.3 The Weak Geometry (The Torsional Spring)

• Geometry factor ($wg$): $\approx {10}^{-6}$

• Physical picture: A wound torsion spring. Interaction occurs through internal twisting stress within a particle transducer. Most energy goes into winding the spring, with only a tiny fraction transmitted.

• Field: $I_w=\frac{\text{Count}\times {10}^{-6}}{L}$

• Interaction: $F=I_{w1}\times I_{w2}$

• Why it's weak and short-ranged: The spring can store limited energy before releasing it (decay) or slipping. This explains beta decay's characteristic feebleness and extreme shortness.

4.4 The Gravitational Geometry (The Sparse Mesh)

• Mass Geometry factor ($mg$): $1.0$is mass in kg / m_P, the m_P is not mystical, it just scalse the kg to a natural ratio.

•  Geometry factor (mg) = mass_in_kg / m_P

• Physical picture: Sparse gear mesh. The geometry is the same 1:1 direct contact as the strong force, but the "gear" is mostly empty—only about ${10}^{-20}$ of its structure is present. Mass is a count, the geometry is a kind of substrate density.

• Field: $I_g=\frac{\text{Count}\times 1.0}{L}$ where $\text{Count}\approx {10}^{-20}$ for a proton or neutron.  Take the mass in kg and divide it by the non reduced planck mass jacobian to get the geometry, tht is the mg.

• Interaction: $F=I_{g1}\times I_{g2}$

• Why it's apparently weak: The interaction law is similar to the rest, but the geometry is a microscopic fraction. Gravity force isn't weak; it's the full-strength interaction of massively diluted geometric substrate.

5. Resolution of the Hierarchy

The enormous differences in perceived force strengths—the "hierarchy problem"—dissolve in this framework. The ${10}^{36}$ gap between electromagnetic and gravitational strengths is simply:

$$\text{<span style="background-color: white;">Ratio</span>}<span\; style="background-color:\; white;">=</span>{<span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">c</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}<span\; style="background-color:\; white;">\times </span>\text{<span style="background-color: white;">(fractional count)</span>}}<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\approx </span>{<span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.034</span>}}{{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">20</span>}}}<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\approx </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">36</span>}}$$

The force engine is always 1:1. The difference lies entirely in how the geometry scales the count before it enters the engine.

6. The Origin of Mass: Why Protons and Neutrons?

If mass is an integer count of geometric substrate density, why does most mass in the universe resonate at proton/neutron values ($\approx 938\text{ MeV}\mathrm{/}{c}^2$)?

The answer lies in geometric resonance. The substrate has natural vibration modes determined by its stiffness ($sg=1$) and density ($mg=1$). The proton/neutron mass represents the fundamental harmonic—the lowest-energy excitation that forms a closed, persistent structure while satisfying all geometric constraints:

1. Coherence across all four geometries: The nucleon is the unique solution where strong, electromagnetic, weak, and gravitational interactions phase-lock into a stable knot.

2. Stability window: The slight mass difference between proton and neutron ($\approx 1.29\text{ MeV}$) matches the weak geometry scale, allowing β-decay while preserving nuclear stability.

3. Ground state dominance: Higher resonances decay rapidly to this state; lower ones cannot support all interactions coherently.

The proton mass isn't arbitrary—it's the geometric necessity for stable existence in our substrate, much as π is the necessity for closure in a circle.

7. Computational Implementation: A Programmer's View

As a programming model, this framework is elegantly simple:

python

def invariant_interaction(I1, I2):
    """The universe's only force law."""
    return I1 * I2

def field_intensity(count, geometry, distance):
    """Project a count into the substrate."""
    return (count * geometry) / distance

# The four geometries as fundamental constants
STRONG_GEOM = 1.0
EM_GEOM = 0.034
WEAK_GEOM = 1e-6
GRAV_GEOM = ~1e-20  

# Example: Calculate gravitational attraction between two protons
mass_count  = 1, GRAV_GEOM = 1.6726e-27 / 2.1765e-8  # Proton mass / Planck mass ≈ 7.7e-20

distance = 1.0  # In natural units

I1 = field_intensity(mass_count, GRAV_GEOM, distance)
I2 = field_intensity(mass_count, GRAV_GEOM, distance)
F_gravity = invariant_interaction(I1, I2)

# The same engine calculates all interactions—only the geometry changes.

8. Conclusion: One Law, Four Geometries

We have shown that physics simplifies dramatically when we recognize:

• One interaction engine: $F=I_1\times I_2$

• Four geometric blueprints: Strong (1:1 gear), Electromagnetic (0.034 lever), Weak (10⁻⁶ torsion spring), Gravitational (1:1 sparse mesh)

• Two count types: Integers of (charge) and (mass)

This framework eliminates action at a distance, arbitrary coupling constants, and the artificial separation of forces. It reveals a universe built from a single invariant interaction processing counts through four concrete geometries.

The task of fundamental physics becomes clear: reverse-engineer the substrate's architecture from these four geometric signatures. Why these specific geometries? Like π for circles, they may be the necessary ratios for stable existence in our reality—the only possible blueprints that allow coherent, persistent structures to form.

Physics is easy because the universe is a single interaction engine processing four different geometric inputs. One Law. Four Geometries. Zero Spooky Action.

*Author's Note: This paper presents a theoretical framework that unifies interactions geometrically. While consistent with observed phenomenology, it makes no new testable predictions: (1) No deviation from the inverse-square law at any scale for any force, (2) All coupling "constants" should be derivable from geometric ratios. Further development is needed to fully derive the geometric factors from first principles.*