The Two Equations That Fix the Proton Scale

Two elementary geometric statements are enough.

They contain no adjustable parameters beyond a single topological integer. When evaluated they return the measured size of the proton and the observed proton-to-electron mass ratio.

1. The proton radius

A stable circulating configuration with winding number (quantum number) (Q = 4) that moves at the speed of light obeys the circulation condition

$$ \oint\mathbf{v}\cdot d\mathbf{l} = \frac{Q,h}{m}. $$

Setting (v = c) and (Q = 4) immediately gives the geometric radius of the proton:

$$ r_p = \frac{4\hbar}{m_p c}. $$

Insert the measured proton mass and the result is

$$ r_p \approx 0.841,\text{fm}. $$

That is the value now returned by the most precise experiments (muonic-hydrogen spectroscopy and the latest electronic-hydrogen and scattering determinations). The long-standing “proton-radius puzzle” is resolved by geometry.

2. The mass ratio

The second relation links the proton’s geometric radius to the electron’s natural scale, the Bohr radius

$$ a_0 = \frac{\hbar}{m_e c\alpha}. $$

The mass-radius products are not equal; they stand in the exact ratio fixed by the same topological integer and the fine-structure constant:

$$ m_p, r_p = 4\alpha,(m_e, a_0). $$

Solving for the mass ratio yields

$$ \frac{m_p}{m_e} = 4\alpha,\frac{a_0}{r_p}. $$

Because $(r_p)$ has already been fixed by the circulation condition, the right-hand side is completely determined. It evaluates to the observed value

$$ \frac{m_p}{m_e} \approx 1836.15. $$

What the two equations say

• The proton’s size is fixed by topology and the speed of light.
• Once that size is known, the mass ratio follows at once from a single scale factor $(4\alpha)$.

No additional parameters are required. The same integer (Q = 4) that sets the radius also sets the factor that converts the Bohr radius into the correct mass ratio.

These are infrared geometric constraints. Any deeper theory of the proton, whatever its short-distance details, must recover these two relations at long distance. They are the pebble that can be snatched cleanly from the open hand of the problem.

The arithmetic is short. The consequences are not.

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