UTMF v5.7
Global Consistency

Global Consistency, Empirical Closure, and Falsifiable Predictions

Dustin Beachy • UnifiedFramework.org

Abstract

We present the final closure of the Unified Topological Mass Framework (UTMF), establishing global consistency across mass generation, flavor structure, mixing, CP violation, and empirical observables. Building on operator closure (v5.4), physical observability (v5.5), lepton flavor closure (v5.6), and quark flavor geometry (UTMF-Flavor-II), we demonstrate that all fermionic masses and mixings arise from a single discrete topological mechanism governed by winding number, stability corridors, and gauge-coupled motif geometry.

No Yukawa matrices, arbitrary phases, or continuous flavor parameters are required. We prove global compatibility with quantum mechanics, quantum field theory, and renormalization, derive unified constraints across leptons and quarks, and present a finite set of falsifiable predictions spanning neutrino masses, flavor multiplicity, CP structure, and beyond-Standard-Model exclusions.

UTMF v5.7 completes the framework as a closed, predictive, and testable physical theory.

1. Introduction

The Standard Model successfully describes particle interactions but leaves the origin of mass, flavor, hierarchy, and CP violation unexplained. These features are encoded through externally supplied parameters—Yukawa couplings and complex phases—whose values are not predicted.

The Unified Topological Mass Framework (UTMF) replaces these inputs with a single organizing principle:

Previous papers established the necessary components:

  • v5.4 — Operator closure and stability corridors
  • v5.5 — Physical observability and renormalized mass
  • v5.6 — Lepton flavor closure and hierarchy
  • UTMF-Flavor-II — Quark masses, CKM geometry, and CP violation

This paper completes the program by demonstrating global consistency and empirical closure across all sectors.

2. Unified Topological Postulates

UTMF rests on five postulates:

P1 — Discrete Relational Topology

Physical states are encoded by discrete tuples

P2 — Operator Realization

Observable quantities correspond to self-adjoint operators on .

P3 — Stability Corridors

Admissible dynamics are confined to invariant subsets of tuple space.

P4 — Gauge-Coupled Geometry

Gauge interactions modify effective winding through motif-kernel holonomy.

P5 — Empirical Equivalence

Physically indistinguishable states collapse into equivalence classes under measurement.

3. Unified Mass Operator

The topological mass operator is

From v5.4–v5.5:

  • is essentially self-adjoint
  • Physical mass is the renormalized eigenvalue
4. Global Flavor Structure

4.1 Lepton Sector (Summary)

  • Charged leptons occupy distinct stability corridors
  • Allowed windings:
  • Mass hierarchy arises exponentially
  • Neutrinos emerge at corridor intersections with suppressed effective winding
  • Prediction:

4.2 Quark Sector (Summary)

Gauge-topology coupling introduces fractional winding:

Corridor splitting yields:

Quark masses follow the same exponential law, with running-scale agreement.

5. Unified Mixing Geometry

5.1 Mixing as Corridor Overlap

Mixing matrices arise from overlap integrals:

Consequences:

  • Small charged-lepton mixing
  • Large neutrino mixing
  • Hierarchical CKM structure

5.2 CP Violation as Holonomy

Define the topological connection

The CP phase is geometric:

Result:

  • Single physical CP phase
  • Jarlskog invariant
6. Global Parameter Economy

UTMF requires only three global constants:

ParameterRole
Mass scale
Exponential hierarchy
Gauge-topology coupling

All masses, mixings, and phases emerge without flavor-specific tuning.

7. Global Consistency Theorems

Theorem 1 (Operator Consistency)

All sector-specific operators commute with the global mass operator within stability corridors.

Theorem 2 (Flavor Closure)

No additional fermion flavors are admissible without violating operator stability.

Theorem 3 (Renormalization Compatibility)

All effective masses admit finite renormalization consistent with QFT.

8. Empirical Predictions

UTMF v5.7 makes the following quantitative predictions:

  1. Exactly three charged leptons
  2. Exactly six quark flavors
  3. No sterile neutrinos with unsuppressed mass
  4. Single CP-violating phase
  6. No fourth fermion generation
9. Falsifiability Criteria

UTMF is falsified if any of the following are observed:

  • A fourth charged lepton or seventh quark
  • More than one independent CP-violating phase
  • Non-exponential fermion mass scaling
  • Flavor mixing inconsistent with mass-gap geometry
10. Relation to the Standard Model
Standard ModelUTMF
Yukawa matricesTopological winding
Arbitrary phasesGeometric holonomy
Input massesEmergent spectra
Flavor unexplainedFlavor closed

UTMF does not replace the SM—it explains its parameters.

11. Scope and Limitations

UTMF v5.7 addresses fermionic mass, flavor, and mixing. Gauge dynamics and gravity are compatible but treated separately. Cosmological extensions are under development.

12. Conclusion

We have completed the Unified Topological Mass Framework. Mass, flavor, hierarchy, mixing, and CP violation arise from a single discrete topological mechanism governed by stability and geometry.

The framework is operatorially closed, empirically predictive, and falsifiable. UTMF v5.7 represents a complete, testable alternative to Yukawa-based explanations of fermion structure.

References
  1. Beachy, D., UTMF v5.4: Operator Closure and Stability Corridors
  2. Beachy, D., UTMF v5.5: Physical Observability of Topological Mass
  3. Beachy, D., UTMF v5.6: Flavor Closure and Lepton Mass Structure
  4. Beachy, D., UTMF-Flavor-II: Quark Flavor Structure and CKM Geometry
  5. Particle Data Group, Review of Particle Physics
  6. Weinberg, S., The Quantum Theory of Fields
  7. Jarlskog, C., CP Violation