A Sparse Geometric Proposal Concerning the Proton Scale and the Mass Ratio

An invitational note for working physicists and STEM professionals

Abstract

A minimal geometric construction is presented that recovers the measured proton charge radius and the proton-to-electron mass ratio from two elementary relations: a quantized circulation condition with winding number 4 and a mass-radius product equality between the proton and the electron. The construction introduces no new free parameters beyond the topological integer and the observed Bohr radius. It is offered not as a replacement for quantum chromodynamics or the Standard Model, but as a compact infrared constraint that may be useful to examine. The note invites critical scrutiny of the arithmetic and of the underlying assumptions.

1. Motivation

Precision measurements of the proton charge radius have converged near $(0.84,\text{fm})$. The proton-to-electron mass ratio remains one of the most accurately known dimensionless numbers in physics. Both quantities are ordinarily treated as outputs of complicated dynamics or as input parameters. It is legitimate to ask whether either of them admits a simpler geometric origin that is still consistent with existing high-energy theory. The present note explores one such possibility and asks only that the resulting expressions be checked against data and against internal consistency.

2. Two geometric relations

Circulation condition.
Require that a stable, finite-size topological configuration of winding number (Q=4) circulating at speed (c) satisfy $$ \oint\mathbf{v}\cdot d\mathbf{l}=\frac{4\hbar}{m}. $$ For a circular equatorial loop this immediately yields $$ r_p=\frac{4\hbar}{m_pc}\approx0.841,\text{fm}. $$ The numerical value lies within the present experimental band for the proton charge radius.

Mass-radius product equality.
Impose the elementary balance $$ M_pr_p=M_eR_e, $$ where $(R_e)$ is identified with the Bohr radius $(a_0=\hbar/(m_ec\alpha))$. Substitution of the geometric $(r_p)$ recovers the observed proton-to-electron mass ratio to high accuracy without additional parameters.

These two statements constitute the entire core of the proposal.

Here is one of the expressions for the mass ratio:

$$m_p/m_e = \alpha^2/(\pi r_p R_{\infty})$$

$$ = 1836.15267344$$

3. What is not claimed

• The construction does not replace the quark-gluon description of the proton’s internal structure.
• It does not assert that quantum chromodynamics is incorrect at short distances.
• It does not claim to have solved the full set of open problems in particle physics or cosmology.
• It does not introduce new dynamical fields beyond the geometric and topological elements already stated.

The relations are offered strictly as infrared geometric constraints that any successful microscopic theory might be expected to respect or to derive.

4. Immediate checks that can be performed

1. Verify the arithmetic that converts $(4\hbar/(m_pc))$ into the numerical radius $(0.841,\text{fm}).$
2. Insert the same radius into the product equality and confirm that the resulting mass ratio matches the CODATA value within the stated precision.
3. Examine whether a topological soliton of Hopf charge 4 in a suitable continuum model can saturate the circulation condition while remaining consistent with known QCD sum rules or lattice results for the charge radius.
4. Test whether the same geometric scale appears, even approximately, in other precision observables that are sensitive to the proton’s overall size.

All four checks are within the competence of any working physicist who has access to standard constants and to existing literature on topological solitons or proton structure.

5. Invitation

The value of a sparse proposal lies in its vulnerability. If the geometric radius or the product equality fails under closer examination, the construction can be discarded with little cost. If both relations continue to hold, they may supply a useful boundary condition or an organizing principle for the infrared sector of the theory of ordinary matter.

Critical examination is therefore requested. Detailed refutations, alternative geometric derivations, or demonstrations that the same results already follow from QCD without additional assumptions would all be constructive. The present note asks only that the arithmetic and the underlying topological premise be looked at directly.

6. Closing

Science advances by the continual testing of simple statements against precise data. The two geometric relations given above are offered in that spirit. They are short enough to be checked in an afternoon and definite enough to be falsified. That combination, rather than any claim of completeness, is the reason they may be worth a few hours of professional attention.

Respectfully submitted for scrutiny.

$$\alpha\Omega$$

$$\vec{\Omega}$$