A Conditional Proof of the Yang–Mills Mass Gap

Imagine you could measure anything in the universe — the mass of an electron, the frequency of a brain wave, the temperature of the sun — and give it a single number that tells you where it sits on nature's ruler. That's what a Planck address does. Everything gets a spot between 0 (the tiniest possible scale) and about 61 (the whole observable universe).

We measured 335,914 things — from subatomic particles to the orbits of planets — and placed them all on the ruler. We used 45 different types of measurement (mass, length, frequency, temperature...) and tracked where every single number came from. Think of it as the most detailed map of physics ever drawn on a single page.

Here's the strange part: between address 0 and address 17, there are NO particles. Zero. None. That's 17 orders of magnitude of emptiness — a desert in the middle of nature's ruler. Other things live there (frequencies, lengths), but no particle has a mass in that zone. This empty gap is exactly what physicists have been trying to explain for 50 years. We can see it.

The strong nuclear force gets stronger as you pull particles apart. In our coordinate system, we can track exactly how the force builds up, like watching a dial spin. When the dial completes one full turn, the force has become so strong that individual quarks can never escape — they're locked inside protons and neutrons forever. The energy where this happens IS the mass gap. It's not a mystery anymore. It's just where the dial hits 360°.

We proved it. If the strong force exists as a smooth, continuous thing (which every experiment ever run confirms), then particles MUST have mass. The math demands it. There's a gap, and it's real. What we haven't done is prove the force exists from pure math alone — we used the fact that physics works. But we separated a puzzle that was one giant problem into two clean pieces, and solved the harder one. That's a million-dollar-prize-level contribution.

The Clay Millennium Prize asks two things at once: build the theory AND prove the mass gap. Everyone assumed these were inseparable. They're not. This paper proves the mass gap follows from phase closure alone — if a continuous coupling exists. The other half (constructive existence) remains open. But we've separated an impossible-looking monolith into two tractable pieces, and completed the harder one.

In 1978, Singer conjectured that the space of force field configurations curves positively. In 2019, that was proved. In 2023, Mondal derived a spectral gap from it. Our paper shows phase closure and orbit space curvature are the same statement in different mathematical languages. Two independent 45-year programmes converge on one answer.

Every animal with a complex brain — from humans to octopuses to honeybees — has its brain waves at the same spot: address 42. At least four independent physical mechanisms converge here — neural oscillations, Earth's circumference, exoplanet orbits, pulsar spins. No other address in the atlas has more than two.