Relative mass Proton / Electron

A Surprising Way to Calculate the Proton/Electron Mass Ratio

Posted on April 29, 2025

 

Have you ever wondered why particles in the universe have the masses they do? The proton, a cornerstone of every atom, has a mass about 1836 times that of the electron. But why? Science has measured this ratio with incredible precision, yet no one has fully cracked the code behind it. Today, we’re diving into a fascinating calculation that claims to derive the proton’s mass from the electron’s using only fundamental physical constants.

The Puzzle of Particle Masses

In the world of physics, some things we can calculate with elegant equations, like the orbit of a planet or the energy of a photon. Others, like the mass of a proton, remain stubbornly mysterious. We know a proton is roughly 1836.15267 times heavier than an electron (according to CODATA 2020), but the Standard Model of particle physics doesn’t tell us why. It’s a number we measure, not derive.

 

T. Lockyer developed a theory grounded in a precise geometry of the proton and neutron. My aim here isn’t to lay out his entire framework—on the next page, I’ll share details of his published book, where you can explore it in depth if you’re curious. But even if his theory doesn’t hold up, we’d still need to explain why this iterative calculation nails the proton/electron mass ratio so accurately. That’s the mystery I want to highlight.

The Ingredients: Nature’s Building Blocks

The calculation uses a handful of fundamental constants, the kind you’d find in any physics textbook. Here’s the lineup, all drawn from CODATA 2020 data:

• Electron mass (mₑ): The starting point, about 9.1093837015 × 10⁻³¹ kg.
• Speed of light (c): The cosmic speed limit, 299,792,458 m/s.
• Fine structure constant (α): The strength of electromagnetic interactions, ~0.00729735257.
• Permeability of free space (μ₀): A magnetic constant, 1.256637062 × 10⁻⁶ N/A².
• Electron charge (e): The electric charge, 1.602176634 × 10⁻¹⁹ C.
• Electron magnetic moment (μₑ): Related to the electron’s spin, ~9.2847647 × 10⁻²⁴ J/T.
• Compton wavelength (λₑ): The quantum wavelength of the electron, ~2.42631024 × 10⁻¹² m.
• Planck constant (h): The quantum of action, 6.62607015 × 10⁻³⁴ J·s.

These constants are the alphabet of the universe, and they all play nicely together in SI units, ensuring the math stays consistent. But why do they combine this way to reveal the proton’s mass?

The Magic Number: 1836.15267

According to the theory, these constants feed into an iterative algorithm—a mathematical recipe—that starts with the electron’s mass and builds up to the proton’s. The result? A calculated proton/electron mass ratio of 1836.15219. Compare that to the official CODATA 2020 value: 1836.15267. That’s a match to the first seven significant digits! 

 

Equally intriguing, the method reportedly calculates the neutron/electron mass ratio too (neutrons are about 1838.68366 times the electron’s mass). If it nails both ratios, we’re looking at something potentially groundbreaking—or at least very clever.

How Does It Work?

The details of T. Lockyer’s algorithm are coded in a JavaScript program, a translation of his BASIC script from years ago. Below is the complete code (you can try it yourself - save it as HTML and open it in a browser), which works like this: the program uses an iterative loop. It starts with the electron mass and, step by step, increases it in layers of increasing masses following a geometric progression. Only fundamental constants are used for the calculation.

 

What’s striking is the claim that there’s no “cheating.” No arbitrary numbers are slipped in to force the result. The constants and their relationships do all the work, and the units balance perfectly, as they must in any legit physical calculation. The program’s transparency is part of its appeal—you can check the logic yourself and see there’s no sleight of hand.

Is It Too Good to Be True?

A seven-digit match is no small feat. It’s tempting to call it a coincidence, but the precision suggests something deeper. Could this be a clue to a hidden connection between the electron and proton? In physics, the electron’s mass comes from its interaction with the Higgs field, while the proton’s mass is mostly due to the strong force binding quarks inside it. These are wildly different mechanisms, so a direct link feels like a stretch. Yet, the numbers don’t lie—or do they?

 

Skeptics might argue the algorithm was tuned to hit the known ratio. Iterative methods can sometimes be designed to converge on a target, especially if you know the answer you’re aiming for. But here, that’s impossible—each step in the process contributes to a higher mass, building naturally toward the final result without arbitrary adjustments.

 

Try It Yourself!

The JavaScript code is out there for anyone to test—just copy it into a file (to remove text formatting, copy and paste it into a plain text editor like Notepad on Windows, TextEdit on macOS, gedit on Linux, or a basic note app on iOS/Android - e.g., Notes or Google Keep. This strips all formatting, leaving plain text.), save it as myscript.html, and open it in your browser. You’ll see the calculations unfold, step by step, leading to the ratio 1836.15219. 

 

<html><body>
<style>
table, th, td {
  border: 1px solid black;
  border-collapse: collapse;
  padding: 3px;
  text-align : right;
}
</style>
<script>
// Light speed (m/s)
const lightSpeed = 299792458;            
// fine structure constant - dimensionless
const fineStructureConstant = 0.00729735257
const pi = Math.PI;      
// Magnetic vacuum permeability  
// (Henry per meter : H/m  [N/A^2])
const vacuumPermeability = 4*pi / Math.pow(10,7);  
// vacuum permittivity (8.8541878176 * 10^-12 F/m)
const vacuumPermittivity = 1 / (lightSpeed * lightSpeed * vacuumPermeability);
// electron charge (A*s)  
const electronCharge = 1.602176634 * Math.pow(10,-19);  
// Compton wave length  (m)  
const comptonWavelength = 2.4263102367 * Math.pow(10,-12);
// rationalized Compton wavelength electron (m)  
const rationalizedComptonWavelength = comptonWavelength/(2*pi);
// electron mass (kg)  
const electronMass = 9.1093837015 * Math.pow(10,-31);  
// Relative electron mass;  
const relativeElectronMass = 1;
const sqrt2 = Math.sqrt(2);
const codataProton = 1836.15267343;
const codataNeutron = 1838.683662;

// Rationalized fine structure constant
const rationalizedFineStructure = fineStructureConstant/(2*pi);
const scalingfactor = 2 * (Math.sqrt(1 + rationalizedFineStructure) - 1);
// The radius of each level is slightly smaller
const layerSpacing = (1-(sqrt2/2))/2 - scalingfactor/sqrt2;  
var scallingContribution = Math.pow(electronCharge,2)/(electronMass * 4 * pi * Math.pow(lightSpeed,2) * rationalizedComptonWavelength * layerSpacing * vacuumPermittivity);     

document.write ("<table>");
document.write ("<th colspan=5 style='text-align:center;'>_-_ &nbsp Calculation of the proton/electron mass ratio &nbsp _-_ </th>");
document.write ("<tr><td>Level</td><td>m1</td><td>m2</td><td>m1+m2</td><td>Sum</td></tr>");

var m1 = new Array(); var m2 = new Array();
m1[1] = relativeElectronMass; m2[1] = scallingContribution;
var totalMass=0;
const denominator = 1-sqrt2*scalingfactor;

// Execute calculations and displays
for(i=1;i<=19;i++){
    m1[i+1]= relativeElectronMass * Math.pow(sqrt2,i) / denominator;
    m2[i+1]= scallingContribution * Math.pow(sqrt2,i) / denominator;
    totalMass += m1[i] + m2[i];
    document.write ("<tr><td>" + i + "</td><td>" +
    m1[i].toFixed(6) + "</td><td>" +
    m2[i].toFixed(6) + "</td><td>" +
    eval(m1[i]+m2[i]).toFixed(6) + "</td><td>" +
    totalMass.toFixed(6) + "</td></tr>");
}
document.write ("</table>");
 
document.write ("<br>Calculated proton/electron mass ratio : "+ totalMass.toFixed(8)+ " <br>--- Official proton/electron mass ratio : 1836.15267343 &nbsp(CODATA 2020)");
mr = eval((totalMass-codataProton)/codataProton);
document.write ("<br>Relative error = " + Number.parseFloat(mr).toExponential(2) + " = " + mr.toFixed(8)+ " = " + (100*mr).toFixed(6) + "%");

// Neutron mass is given by proton mass + two first layers mass
document.write ("<br><br>Calculated neutron/electron mass ratio : "+ (totalMass+m1[1]+m2[1]+m1[2]+m2[2]).toFixed(8) + "<br>--- Official neutron/electron mass ratio : 1838.683662 &nbsp(CODATA 2020)");
mr = eval((totalMass+m1[1]+m2[1]+m1[2]+m2[2]-codataNeutron)/codataNeutron);
document.write ("<br>Relative error = " + Number.parseFloat(mr).toExponential(2) + " = " + mr.toFixed(8) + " = " + (100*mr).toFixed(6) + "%");
</script></body></html>

 

What’s Next?

This calculation raises big questions. Is it a genuine insight into the universe’s blueprint, or a clever mathematical trick? Could it point to a new theory linking electromagnetic and strong forces? Or is it a reminder that numbers can align in surprising ways? Whatever the truth, it’s a testament to human curiosity—our drive to find patterns in the cosmos.

 

Disclaimer: The calculations discussed here are based on T. Lockyer’s theory and have not been peer-reviewed or widely accepted in the scientific community. Always approach bold claims with a curious but critical mind.

Next pages :

. Thomas N. Lockyer's theory  🚩

. Deduction of the formulas used in the ratio calculation  🚩