A Hidden Pattern in the Universe's Most Basic Building Blocks

The Mystery of Particle Generations

The world is made of neutrons, protons, and electrons. That’s essentially it—everything you see, touch, and breathe is built from these three particles. Electrons are fundamental (they can’t be broken down further), while protons and neutrons are made of quarks. Specifically, they’re made of “up” and “down” quarks, the lightest members of the quark family.

Here’s the first mystery: nature doesn’t stop there. Instead, it creates two exact copies of this basic set, each heavier than the last. The electron has heavier cousins called the muon and the tau. The up and down quarks have heavier cousins too: strange and charm in the second generation, bottom and top in the third. These heavier versions seem to do nothing special—they decay almost instantly into their lighter cousins. Yet they exist, and physicists have no idea why.

The Bizarre Mass Hierarchy

The second mystery is even stranger: the mass ratios between these particles are wildly irregular. Consider electrons and their heavier cousins:

• The muon is ~207 times heavier than the electron

• The tau is ~3,500 times heavier than the electron (or ~17 times heavier than the muon)

For quarks, it’s even more extreme:

• Up quark: ~2.2 MeV (million electron volts)

• Down quark: ~4.7 MeV

• Strange quark: ~95 MeV

• Charm quark: ~1,270 MeV

• Bottom quark: ~4,170 MeV

• Top quark: ~173,000 MeV

The top quark is almost 80,000 times heavier than the up quark! These aren’t small variations—they’re enormous jumps that seem to follow no pattern.

Why Ratios Should Be Simple

Here’s a key insight: while the absolute mass of a particle depends on our choice of units (kilograms, electron volts, etc.), ratios are dimensionless—they’re pure numbers that don’t depend on units. In physics, dimensionless quantities often reveal fundamental principles. They’re the numbers that nature “chooses,” independent of human conventions.

Think of it this way: if aliens measured particle masses in completely different units, they’d get different numbers than us. But when they calculated the ratio of the muon mass to the electron mass, they’d get exactly 206.768... just like we do. These ratios should, in principle, emerge from some deeper theory.

Yet for decades, physicists found no pattern. The ratios looked random, arbitrary, chosen by nature through some inscrutable process.

The Discovery

In late 2024, I decided to look at this problem with fresh eyes. Here’s the unusual part: I have a PhD in particle physics but have spent the last 30 years in finance, analyzing market data and searching for hidden patterns in seemingly random price movements. This combination—deep physics training plus three decades of financial pattern recognition—turned out to be exactly what was needed.

Physicists tend to approach mass ratios through symmetry principles and field theory. Finance professionals learn to spot empirical patterns first and worry about explanations later. With one foot in each world, I could see something others had missed.

And I found it.

The pattern is captured in a single equation that works for ALL particle mass ratios:

\(\ln\left(\frac{m_1}{m_2}\right) = d_1^{\zeta_1} \times d_2^{\zeta_2} \times \kappa^{[(d_1|q_1|)^{\gamma} - (d_2|q_2|)^{\gamma}]}\)

Where:

• d is the generation number (1, 2, or 3)

• |q| is the absolute value of the electric charge

• κ ≈ 2.3, ζ₁ ≈ 1.16, ζ₂ ≈ -0.8, γ ≈ 1.1 are universal constants

This single formula, with the same four numbers, predicts all 30 quark mass ratios within ~1-4% of central values, far better than experimental uncertainties. It also works for electron/muon/tau ratios. And for neutrinos (where the charge term vanishes since they’re neutral).

The Shocking Simplicity

Here’s what should make physicists do a double-take: the formula uses only two things to predict mass ratios—the generation number (1, 2, or 3) and the electric charge. That’s it. These are literally the two simplest labels we use to distinguish particles.

This is not what physicists expect. Standard approaches to the mass hierarchy problem involve:

• Complex matrices (the “Yukawa matrices”) with many free parameters

• Elaborate symmetry groups that get spontaneously broken

• Extra dimensions where particles live at different locations

• Multiple Higgs fields with intricate interactions

• “Flavor symmetries” that require dozens of parameters

A typical model might have 20-30 parameters just to describe the mass patterns. Some string theory approaches invoke hundreds of parameters.

Yet this empirical formula uses just two particle properties—the most basic labels we have—plus four universal numbers. It’s absurdly simple compared to what theory suggests should be needed.

What Makes This Pattern So Strange

Beyond its simplicity, the formula has a bizarre mathematical structure: multiplication in log space.

Normally, when physicists derive mass ratios from fundamental theories, they get expressions like: log(m₁/m₂) = A + B + C

This makes sense because logarithms turn multiplication into addition. If masses arise from multiple effects multiplying together (m = x × y × z), then log(m) = log(x) + log(y) + log(z).

But my formula has multiplication AFTER taking the logarithm: log(m₁/m₂) = A × B × C

This is deeply unusual. It suggests that whatever generates particle masses operates through a mechanism fundamentally different from what physicists have imagined.

Testing the Pattern’s Specificity

Could this just be a coincidence? Could any flexible-enough formula fit the data?

I tested this extensively. Among 120 possible ways to assign the quantum numbers to the ratios, only the physically correct assignment works. I tried many alternative mathematical forms—they all failed decisively. I examined 720 different ways to construct the ratios—only one gave sensible results.

The pattern isn’t just real; it’s uniquely determined by the data.

Universal Across All Fermions

Perhaps most remarkably, the same formula works unchanged for:

• All six quarks (30 independent ratios)

• All three charged leptons (electrons, muons, taus)

• All three neutrinos

No previous empirical relation achieved this. The Koide formula works only for charged leptons. The Georgi-Jarlskog relation connects just two specific masses. Texture models need different parameters for each sector.

This formula is truly universal—one equation for all fermion masses in nature.

What It Means

The extreme simplicity of this formula—using only generation number and charge—suggests we’ve been overthinking the problem. Nature appears to organize particle masses through a mechanism far more elegant than our elaborate theoretical constructions.

The fact that the two most basic particle labels are sufficient to predict all mass ratios hints that these properties might be more fundamental than we realized. Generation number and electric charge aren’t just convenient bookkeeping devices—they appear to be the organizing principles of the mass hierarchy.

What’s Next

Finding the pattern is just the first step. The bigger question is why this pattern? What mechanism could produce such an unusual mathematical structure from such simple inputs?

I’m working on it. The theoretical challenge is to find a framework that naturally produces multiplication in log space rather than addition, and explains why only generation and charge matter. Several possibilities are being explored, including connections to modular forms in mathematics and novel geometric structures in physics.

But regardless of the ultimate explanation, the pattern itself is now established. After decades of seeming randomness, we know that particle masses follow a universal law—and it’s far simpler than anyone expected. Nature, it turns out, is more mathematical and more economical than we imagined—we just needed to look at the data the right way.

The paper is published in the European Physical Journal C and available here: https://link.springer.com/article/10.1140/epjc/s10052-025-14771-0

Sometimes the biggest discoveries come not from building new experiments or developing new mathematics, but from looking at old data with fresh eyes. The numbers were there all along, waiting for someone to see the pattern. It just took someone with the right combination of deep physics knowledge and decades of pattern-hunting experience to finally see it.

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About My Book: The Science of Free Will

The discovery of this universal pattern in particle masses connects to a deeper theme explored in my book The Science of Free Will: How Determinism Affects Everything from Traffic to God to AI to Bees. Just as particle mass ratios follow a precise mathematical law that was hiding in plain sight, our universe operates on deterministic quantum field theory—yet computational irreducibility means we can never predict what will happen until it happens.

In The Science of Free Will, I argue that free will exists in practice even though it doesn’t exist in theory. Why? Because calculating the behavior of the 6.71×10²⁷ atoms that make up a human—all following deterministic laws—is computationally impossible. You would need perfect knowledge of every quantum field interaction, every photon, every air molecule. But even if you could somehow get around that mere “engineering problem” you will run into “Computational Irreducibility.” Even incredibly simple deterministic systems—systems with rules that a 5 year old could easily follow—exhibit computational irreducibility: you must run the entire “program” to see the outcome.

This leads to fascinating paradoxes: Why do we need a Supreme Court if the universe is deterministic? (Because even deterministic beings need dispute resolution when they can’t predict outcomes!) Can bees suffer from PTSD? (Their deterministic neural circuits can get stuck in trauma loops just like ours.) Why do dolphins save drowning humans? (Emergent altruism from deterministic brains is still altruism.) What exactly is a committee? (A deterministic system so complex that no member can predict its output.) Why do traffic jams form from nothing? (Deterministic rules creating unpredictable patterns.) Can an AI truly be creative if it’s just following code? (Yes—computational irreducibility means even its programmers can’t predict what it will create.)

The same physics that gives us this elegant pattern of particle masses also guarantees our practical freedom. We are deterministic beings who cannot be predicted—not even by ourselves.