A New DEF Paper: Why the Muon Is 207 Times Heavier Than the Electron
Differential Expansion Framework – The Electron, The Muon and the Tau particles.
One of the most mysterious numerical patterns in physics is the charged-lepton mass ladder:
- Electron
- Muon (≈ 207× heavier)
- Tau (≈ 3477× heavier)
In the Standard Model these numbers are simply inserted by hand through Yukawa couplings.
They work — but they are not explained.
In a new paper, I explore whether this hierarchy can arise from geometry and causality alone inside the Differential Expansion Framework (DEF).
The result is striking:
A single causal closure index, fixed by the fine-structure constant, appears to generate the entire lepton family.
The core idea
In DEF, particles are not point objects.
They are non-radiating circulations of a universal expanding field.
For charged leptons:
- Phase flows around a toroidal loop.
- Closure requires a 4π (spinor) topology.
- Boundary matching quantises which circulations are stable.
- One tiny mismatch channel becomes electromagnetism — fixing the fine-structure constant α.
From this same boundary condition, a fundamental integer “stitch count” emerges:
That number is not chosen.
It follows directly from α.
Miss the off-ramp → become a muon
The paper then asks a simple mechanical question:
What if the circulating phase misses the first closure window?
Because the rotor is spinorial, the next allowed alignment occurs after a half-integer recurrence.
Mathematically this gives:
which lands almost exactly on the observed muon–electron mass ratio.
The tau then appears as a much higher recurrence of the same closure condition, lying close to an 8π multiple of the same base index.
In this picture:
- Electron = ground closure mode
- Muon = first missed-closure recurrence
- Tau = higher π-structured recurrence
No new particles.
No arbitrary Yukawa parameters.
Just geometry, topology, and causality.
What the paper contains
The new manuscript develops:
- A recap of DEF and causal phase rotors
- Toroidal–poloidal phase locking
- Boundary monodromy and the origin of α
- The π-based closure ladder
- Muon and tau derivations
- Stability and decay arguments
- Experimental predictions and falsifiability
It is written as a self-contained technical paper, suitable as a standalone DEF sub-work.
Predictions and tests
The framework makes concrete claims:
- Tiny departures from strict lepton universality
- Structure in decay spectra tied to boundary phase
- The possibility of higher lepton recurrences at extreme energies
These provide ways the idea could be tested — or ruled out — by future precision experiments.
Why this matters
If correct, this work would replace one of the Standard Model’s most opaque inputs with:
a single geometric rule.
Mass ratios would no longer be arbitrary.
They would be consequences of causal phase closure in an expanding field.
Read the paper
The full paper is now available: https://imsn.co.uk/wp-content/uploads/2026/02/Muon_Tau.pdf
Lepton Mass Hierarchy from Causal Phase-Closure in the Differential Expansion Framework
I welcome technical criticism, questions, and independent checks — especially on the closure equation and the π-multiples.