The Resolution of the Hierarchy Problem: Linear Lever Arms of Fractional Mass-Counts vs. Integer Charge-Counts

J. Rogers, SE Ohio

Abstract

Modern physics identifies a 

${10}^{36}$

 discrepancy between the strength of electromagnetism and gravity, a mystery known as the "Hierarchy Problem." We demonstrate that this is not a fundamental property of the forces themselves, but a coordinate artifact of measurement scaling. By employing non-reduced Planck units (based on 

$h$

) and treating mass and charge as discrete counts within the substrate, we show that the force ratio is simply the ratio between the linear "lever arms" of these counts. We reveal that while charge is an integer interaction (

$e=1$

), the mass of a nucleon is a tiny fraction of the substrate's natural time-rate (

$\approx {10}^{-20}$

). The 

${10}^{36}$

 difference is shown to be precisely the square of this fractional mass-count, scaled by the linear electromagnetic coupling.

---

1. The Principle of Linear Interaction

In a coordinate-free substrate, physical law is expressed as the local intersection of field potentials. A force law 

$F=({\text{source}}_1\times {\text{source}}_2)\mathrm{/}{r}^2$

is not a "spooky action at a distance" but the product of two overlapping 

$1\mathrm{/}r$

 potential gradients.

Critical to this understanding is the distinction between Linear and Rotational geometry. The standard fine structure constant (

$\alpha$

) is an orbital constant; it contains an inherent 

$2\pi$

 because it describes energy levels in a circular geometry (the Bohr atom). However, the force between two points is a linear interaction. To see the true "lever arm" of the electromagnetic machine, we must remove the circular projection.

2. Defining the Natural Substrate (Non-Reduced Planck Units)

We use the original Planck units constructed from 

$h,c,G$

. We specifically reject the reduced Planck constant (

$\mathrm{\hslash }$

) as it introduces a rotational artifact (

$2\pi$

) into the base units of the substrate.

• Planck Mass (

  $m_P$

  ): 

  $\sqrt{hc\mathrm{/}G}\approx 5.455\times {10}^{-8}$

   kg

• Planck Length (

  $l_P$

  ): 

  $\sqrt{hG\mathrm{/}{c}^3}\approx 4.051\times {10}^{-35}$

   m

• Substrate Identity: In this coordinate chart, the natural exchange rate for mass-energy is 

  $1.0$

  .

3. The Lever Arm of Electromagnetism (The Handle)

In our framework, electric charge is a dimensionless counting number (

$e=1$

). The strength of the electromagnetic interaction is determined by the amp_force_natural, which represents the linear "stiffness" of the vacuum.

As previously derived, this constant is the linear version of 

$\alpha$

:

$${\text{<span style="background-color: white;">LeverArm</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">\alpha </span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">0.0011614</span>}$$

This value (

$\approx {10}^{-3}$

) is the "Gear Ratio" of the electromagnetic handle. It represents how a single integer count of charge (

$q=1$

) scales to create a force in the substrate.

4. The Lever Arm of Gravity (The Anchor)

Gravity is the interaction of the "Mass Anchors." To compare it to the "Charge Handle" on equal terms, we must count mass in discrete units—specifically the Neutron.

4.1 The Fractional Mass of the Nucleon

Unlike charge, which is an integer (

$1.0$

) in the substrate, the mass of a neutron (

$m_n$

) is a tiny fraction of the Planck Mass (

$m_P$

). We define the Natural Mass-Count (

$m_{n,nat}$

):

$${\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}}{{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1.6749</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">27</span>}}\text{<span style="background-color: white;"> kg</span>}}{\mathrm{<span\; style="background-color:\; white;">5.455</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">8</span>}}\text{<span style="background-color: white;"> kg</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">3.070</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">20</span>}}$$

This number reveals that a neutron is a "trapped time" pattern that is twenty orders of magnitude "quieter" than the natural 

$1:1$

 scale of the substrate.

4.2 The Gravitational Interaction of the Count

The potential created by a single neutron is 

$\tau =m_{n,nat}\mathrm{/}r$

. When two such potentials intersect, the force is:

$${\text{<span style="background-color: white;">LeverArm</span>}}_{\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">r</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">v</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1.0</span>}<span\; style="background-color:\; white;">\times </span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">t</span>}}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

$${\text{<span style="background-color: white;">LeverArm</span>}}_{\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">r</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">v</span>}}<span\; style="background-color:\; white;">\approx </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">3.070</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">20</span>}}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">9.42</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">40</span>}}$$

5. Resolving the Hierarchy Ratio

We now calculate the ratio between the "Handle Gear" (EM) and the "Anchor Gear" (Gravity) for a single particle interaction:

$$\text{<span style="background-color: white;">Ratio</span>}<span\; style="background-color:\; white;">=</span>\frac{{\text{<span style="background-color: white;">LeverArm</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}}}{{\text{<span style="background-color: white;">LeverArm</span>}}_{\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">r</span>}\mathrm{<span\; style="background-color:\; white;">a</span>}\mathrm{<span\; style="background-color:\; white;">v</span>}}}$$

$$\text{<span style="background-color: white;">Ratio</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.0011614</span>}}{\mathrm{<span\; style="background-color:\; white;">9.425</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">40</span>}}}$$

$$\text{<span style="background-color: white;">Ratio</span>}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">1.23</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">36</span>}}$$

This is precisely the value observed in experimental physics.

6. Discussion: The End of the Mystery

The 

${10}^{36}$

 difference is not a "strength" of nature. It is a Scaling Artifact emerging from two facts:

1. We are comparing an Integer (

   $q=1.0$

   ) to a Fraction (

   $m\approx {10}^{-20}$

   ).

2. Force is a second-order product of these sources.

The "Weakness" of gravity is simply the square of the neutron's fractional existence relative to the substrate. The universe is not asymmetric; it is perfectly consistent. If we were to measure an interaction between two "Planck-mass particles" (where 

$m=1.0$

), gravity would be stronger than electromagnetism (

$1.0$

 vs 

$0.00116$

).

Gravity only appears weak to us because nucleons are incredibly small "ripples" in the time-field, whereas the charge-interface is an integer geometric "handle."

7. Conclusion

Physics is easy when you stop measuring "Forces" and start measuring "Gear Ratios."

• The EM Lever Arm is 

  $\alpha \mathrm{/}2\pi \approx {10}^{-3}$

  .

• The Gravity Lever Arm is 

  $m_{nat}^2\approx {10}^{-40}$

  .

• The Hierarchy is the ratio: 

  ${10}^{37}$

  .

There is no "Hierarchy Problem." There is only the realization that the "Mass Anchor" of a nucleon is twenty orders of magnitude smaller than the "Charge Handle" it carries. The 

${10}^{36}$

 is the geometric consequence of that 

${10}^{20}$

 scale difference.

Two things. Two interactions. One substrate. Linear Lever Arms. 

Appendix A: The Wrong Question

A.1 The Categorical Error in "Force Strength"

For over a century, physicists have asked: "Why is gravity so much weaker than electromagnetism?" This question embeds a fundamental misunderstanding of what gravity is.

Gravity is not a "force" with adjustable "strength." Gravity is the geometric relationship between mass and spacetime curvature, expressed in natural coordinates as:

$$F_{\text{grav}} = \frac{m_1 \times m_2}{r^2}$$

This is pure geometry. Asking "can gravity be stronger or weaker?" is equivalent to asking "can the ratio of a circle's circumference to its diameter be something other than π?" The question is categorical nonsense.

A.2 What G Actually Represents

In SI coordinates, we write:

$$F = G \frac{m_1 m_2}{r^2}$$

where $G \approx 6.674 \times 10^{-11}$ m³/(kg·s²).

Traditional physics treats $G$ as a "fundamental constant" that could, in principle, have been different. This is wrong. $G$ is not a property of gravity. It is a coordinate transformation coefficient (Jacobian) that converts the natural geometric relationship into SI measurement units.

Specifically:

$$G = \frac{l_P^3}{m_P \cdot t_P^2}$$

where $l_P$, $m_P$, and $t_P$ are the non reduced Planck length, mass, and time—the unique solution to the self-consistent system of equations relating $c$, $h$, $G$, and $k_B$.

$G$ encodes how misaligned our SI unit axes are from the natural geometric ratios of the substrate. It is unit-system bookkeeping, not physics.

A.3 The Only Variable That Matters

Given that gravity is pure geometry ($F \propto m_1 m_2 / r^2$ in natural units) and $G$ is merely coordinate scaling, there is only one variable that can affect gravitational interaction strength:

The rest mass values of the particles involved.

In natural units:

• An electron has $m_e / m_P \approx 4.19 \times 10^{-23}$
  
• A proton has $m_p / m_P \approx 7.69 \times 10^{-20}$
  
• A neutron has $m_n / m_P \approx 3.07 \times 10^{-20}$
  

These are fractional counts relative to the substrate's natural mass scale. They are the allowed resonant modes of "trapped time" in our universe's configuration—the rest mass entries in the particle zoo.

A.4 The Meaningful Question

The wrong question:

"Why is gravity weak?"

The correct question:

"Why do the stable massive particles in our universe (nucleons) resonate at approximately $10^{-20}$ of the Planck mass?"

This is a question about particle physics and resonance modes, not about "gravitational strength."

Or equivalently:

"Could the rest masses of nucleons have been different in a universe with different resonance conditions?"

The answer is yes—different vacuum configurations, different symmetry breaking scales, or different fundamental field couplings could produce a particle zoo with nucleons at, say, $10^{-15} m_P$ or $10^{-25} m_P$. In such universes:

• The geometric relationship $F = m_1 m_2 / r^2$ would be unchanged
• The value of $G$ (in SI units) would be unchanged (it's still just $l_P^3 / m_P t_P^2$)
• But the "apparent strength" of gravity would scale with $(m_{\text{nucleon}})^2$
  

A universe with nucleons at $10^{-10} m_P$ instead of $10^{-20} m_P$ would experience gravitational forces $10^{20}$ times stronger (for the same number of nucleons). Not because "gravity got stronger," but because the input mass counts increased.

A.5 Time Variation of G: The Only Physical Possibility

The standard cosmological question "Does $G$ vary with time?" is *almost* the wrong question, but contains a grain of correct physics.

Since $G = l_P^3 / (m_P t_P^2)$ is a ratio of Planck units—themselves defined from $h$, $c$, and $G$—the statement "G varies" is circular unless one clarifies: varies relative to what?

The meaningful physical question is:

"Do the rest masses of particles (in Planck units) vary with time?"

That is: Does $m_{\text{nucleon}} / m_P$ change as the universe evolves?

This could occur if:

1. The vacuum expectation value (VEV) of the Higgs field evolves
2. The strong force coupling changes (affecting nucleon binding)
3. Some other field whose VEV sets particle masses is dynamical

**This** would be genuine physics—a change in the particle zoo's resonance spectrum—not a change in "gravitational strength." The geometry $F = m_1 m_2 / r^2$ remains constant. What changes is the *magnitude of the masses* we input into that geometry.

A.6 Summary: Dissolving the Confusion

Statement	Status
"Gravity is weak"	Meaningless — Gravity is geometry with coupling = 1.0
"G could be different"	Wrong — $G$ is coordinate bookkeeping ($l_P^3 / m_P t_P^2$)
"G varies with time"	Confused — Must specify: relative to which units?
"Nucleon masses could be different"	Correct — This is physics (resonance spectrum)
"Nucleon masses might vary with time"	Meaningful — Tests if vacuum/field VEVs evolve

The hierarchy problem dissolves when we recognize that:

1. Gravity has no "strength parameter"—it's pure geometry
2. $G$ is a unit conversion factor, not a fundamental property
3. The "weakness" is entirely due to nucleons being $\sim 10^{-20} m_P$
   
4. Asking if gravity could be stronger is asking if nucleons could have different masses
5. That is a particle physics question, not a gravitational one

Stop asking if gravity can be weak or strong. Ask instead: Why do nucleons resonate at the mass fraction they do? That is the only question with physical content.

The universe doesn't have a "gravity strength dial." It has a particle mass spectrum. The "hierarchy" is just $(10^{-20})^2 = 10^{-40}$, squared because force is second-order. Geometry working as designed, with fractional inputs.

There is no weakness. There is only geometry acting on the counts you provide.