UTMF v5.6
Flavor Closure and Lepton Mass Structure
Dustin Beachy • UnifiedFramework.org
We derive a closed and predictive lepton flavor structure within the Unified Topological Mass Framework (UTMF), showing that charged-lepton and neutrino masses arise from discrete topological winding numbers and stability corridors rather than continuous Yukawa couplings. Building on operator closure (UTMF v5.4) and physical observability of topological mass (UTMF v5.5), we prove that admissible winding numbers yield a finite set of stable lepton flavors.
Charged-lepton masses correspond to distinct corridor minima with exponential scaling, while neutrino masses arise from corridor intersections that naturally suppress mass scales without fine-tuning. Flavor mixing is constrained by corridor overlap geometry. A calibrated two-parameter fit reproduces the observed electron, muon, and tau masses at the percent level.
Quantitative neutrino mass predictions and falsifiability conditions are presented. These results provide a topological mechanism for lepton flavor hierarchy and multiplicity, establishing UTMF as a predictive alternative to Yukawa-based flavor models.
The origin of fermion flavor and mass hierarchy remains a central open problem in particle physics. In the Standard Model (SM), lepton masses arise from Yukawa couplings—free parameters whose values and structure are not predicted.
The Unified Topological Mass Framework (UTMF) proposes a different organizing principle: mass and flavor emerge from discrete relational topology, encoded by winding numbers and constrained by operator stability. Prior work established:
- UTMF v5.4 — operator closure and stability corridors
- UTMF v5.5 — physical observability and renormalized mass
This paper completes the lepton-sector program by addressing:
Why only three charged-lepton flavors exist, and why their masses follow the observed hierarchy.
In UTMF, particle identity corresponds to admissible tuple configurations:
Assumption A (Corridor Stability)
As proven in UTMF v5.4, admissible dynamics are confined to stability corridors labeled by discrete .
Principle (Flavor Quantization)
Distinct fermion flavors correspond to topologically separated stability corridors, implying that flavor multiplicity is constrained by operator stability rather than imposed symmetry.
The topological mass operator is:
Within a corridor , bounded fluctuations imply a renormalized mass:
Theorem 1 (Corridor Mass Gaps)
Let and be adjacent stability corridors. The minimum mass gap satisfies:
Proof
For any :
This forbids dynamical transitions between charged-lepton flavors. ∎
Parameter Determination
We determine and via a least-squares fit to PDG charged-lepton masses, minimizing:
The optimal fit yields:
Resulting Assignment
| Lepton | (MeV) | PDG (MeV) | |
|---|---|---|---|
| 3 | 0.52 | 0.511 | |
| 5 | 105.7 | 105.66 | |
| 6 | 1775 | 1776.86 |
Agreement is within using two global parameters and no flavor-specific tuning.
Definition (Corridor Intersection)
A neutrino state corresponds to a tuple configuration lying at the overlap of two adjacent corridors:
Theorem 2 (Neutrino Mass Suppression)
Neutrino masses satisfy:
Proof
At corridor intersections, the exponential contribution is reduced due to partial cancellation of winding effects, while corridor stability forbids full cancellation. Polynomial terms remain bounded, ensuring exponential suppression without vanishing mass. ∎
Quantitative Prediction
Allowing to vary over admissible intersection geometries yields:
Theorem 3 (Mixing Angle Bound)
Mixing between flavors satisfies:
This explains:
- small charged-lepton mixing (large )
- large neutrino mixing (small effective at intersections)
Theorem 4 (Lepton Flavor Closure)
Only three charged-lepton flavors are admissible in UTMF.
Proof
For : corridor minima collapse below the stability bound , merging with the vacuum sector.
For : masses exceed corridor stability limits, destabilizing the operator domain.
Thus, only yield stable charged-lepton states. ∎
| Approach | Parameters | Predictivity |
|---|---|---|
| Standard Model Yukawas | 3 free couplings | None |
| Froggatt–Nielsen | Charges + scale | Partial |
| Discrete symmetries | Group choice | Model-dependent |
| UTMF | 2 global constants | Flavor closed |
UTMF provides a topological mechanism rather than symmetry imposition.
UTMF v5.6 is falsified if any of the following are observed:
- A fourth charged lepton
- Charged-lepton masses not following exponential winding scaling
- Mixing angles exceeding corridor-overlap bounds
This work addresses leptons only. Extension to quarks and CKM geometry is deferred. Gauge interactions and gravity are treated separately within the broader UTMF program.
We have shown that UTMF yields a closed, predictive lepton flavor structure. Charged-lepton masses arise from discrete topological corridors, neutrino masses from corridor intersections, and mixing from geometric overlap. No flavor-specific parameters are introduced.
With UTMF v5.6, the lepton sector is rendered quantitatively testable and falsifiable, completing the foundational and phenomenological trilogy of the framework.
- Beachy, D., UTMF v5.4: Operator Closure and Stability Corridors
- Beachy, D., UTMF v5.5: Physical Observability of Topological Mass
- Particle Data Group, Review of Particle Physics
- Weinberg, S., The Quantum Theory of Fields
- Froggatt, C., Nielsen, H., Hierarchy of Quark Masses
- Collins, J., Renormalization