Qaether Theory [v2.0] 목차

Qaether Theory

Discrete Topological Gauge Geometry on the FCC Tetra–Octa Complex

---

Part I. Foundations of the Discrete Geometry

Chapter 1. Motivation and Conceptual Framework

1.1 From Continuum Gauge Theory to Discrete Topological Models
1.2 Why FCC Tetra–Octa Geometry?
1.3 Flux, Closure, and Matter as Geometric Completion
1.4 Overview of the Confinement Program

---

Chapter 2. The FCC Tetra–Octa Cellular Complex

2.1 Definition of the Mixed Tetrahedral–Octahedral 3-Complex
2.2 Orientation Conventions and Chain–Cochain Formalism
2.3 Incidence Structure and Local Boundedness
2.4 Existence of the Cellular Dual Complex

---

Chapter 3. Discrete (\mathbb Z_{12}) Link Variables

3.1 Link Cochains and Phase Quantization
3.2 Plaquette Curvature (Q=\delta k)
3.3 Closure and Bianchi Identity
3.4 Gauge Redundancies and Representative Choice

---

Part II. Flux, Energy, and Dual Defect Geometry

Chapter 4. Soft Flux Energy with Uniform Gap

4.1 Hard vs Soft Flux Constraints
4.2 Definition of the Soft Flux Functional
4.3 Defect Support and Lower Energy Bounds
4.4 Physical Interpretation of (\epsilon_{\min})

---

Chapter 5. Dual Complex and Geometric Bianchi Identity

5.1 Why Cochains Are Not Chains
5.2 The Duality Map (\star) and Degree Matching
5.3 Source-Free Closure: Closed Dual Loops and Worldsheets
5.4 Topological Sectors and Cohomology Remarks

---

Chapter 6. External Sources via Dirac Sheets

6.1 Spacetime Complex (\mathcal X = X\times{0,\dots,T})
6.2 Static Worldlines and Source 3-Cochains
6.3 Dirac Sheet Construction
6.4 Sourced Bianchi Identity (\delta Q=J)
6.5 Dirac-Sheet Gauge Invariance
6.6 Cohomology Sector Dependence

---

Chapter 7. Area Law and Linear Confinement

7.1 Slice-by-Slice Cut Argument
7.2 Incidence Constant (\eta)
7.3 Worldsheet Area Lower Bound
7.4 Linear Static Potential
7.5 Nucleation Gap and Minimal Closed Bubbles
7.6 Sector Dependence: Tetra vs Octa

---

Part III. Matter as Geometric Completion

Chapter 8. Square Plaquettes and Matter Patterns

8.1 Quark-Type Patterns (All Four Distinct)
8.2 Lepton-Type Patterns (Label Degeneracy)
8.3 Closure Condition (a+b+c+d\equiv0)
8.4 Enumeration of Allowed Patterns

---

Chapter 9. Octahedral Completion and Baryons

9.1 Orthogonal Square Coupling
9.2 Eight Triangular Faces
9.3 Full Boundary Closure Condition
9.4 Minimal Completion Energy

---

Part IV. SU(3) Structure from Triangle Geometry

Chapter 10. Triangle Patterns and Dipole Channels

10.1 Triangle Flux Triples
10.2 Classification of Allowed Triples
10.3 Dipole Patterns ((0,k,-k))
10.4 Junction Patterns ((a,b,c))

---

Chapter 11. Cartan Projection and Root Geometry

11.1 Projection to ((T_3,T_8))
11.2 Emergence of the (A_2) Root Lattice
11.3 Off-Diagonal Channels and Roots
11.4 Cartan Subalgebra from Junction Modes

---

Chapter 12. Quark-Type Octahedra and SU(3) Closure

12.1 Three Independent Dipole Channels
12.2 Nonabelian Closure of Modes
12.3 Geometric Origin of Eight Gluon Modes
12.4 Color as Octahedral Boundary Structure

---

Part V. Lepton-Type Degeneracy and Singlet Collapse

Chapter 13. Lepton-Type Octahedra: Degenerate Boundary Geometry

13.1 Repeated Labels (K={0,a,a,b})
13.2 Forced Triangle Types
13.3 Counting Argument via Incidence
13.4 Dipole Channel Degeneracy Theorem

---

Chapter 14. Collapse of SU(3) Structure

14.1 Loss of Independent Root Directions
14.2 Cartan Dimension Reduction
14.3 Failure of Nonabelian Closure
14.4 Singlet Theorem

---

Chapter 15. Geometric Interpretation of Color Singlets

15.1 Stabilizers under (C_4) and Charge Conjugation
15.2 Why Leptons Do Not Couple to Color
15.3 Comparison with QCD Singlets
15.4 Sector Stability and Energy Suppression

---

Part VI. Dynamical Extensions

Chapter 16. Time Evolution and Phase Transport

16.1 Discrete Parallel Transport
16.2 Comparison with Lattice Gauge Theory
16.3 Emergent Yang–Mills Correspondence
16.4 Limits of the Analogy

---

Chapter 17. Spin Sector and SU(2) Coupling

17.1 Frame Variables (q_i\in SU(2))
17.2 Link Connection Variables (h_{ij})
17.3 Gauge-Invariant Observables
17.4 Interaction Between Spin and Flux

---

Chapter 18. Toward Continuum and Lorentz Recovery

18.1 Scaling Limits
18.2 Effective Field Theory Approximation
18.3 Emergent Metric and Causal Structure
18.4 Open Problems

---

Appendices

Appendix A. Detailed Combinatorial Proofs

• Triangle Type Classification
• Incidence Counting Lemmas
• Proof of Degeneracy Theorem

Appendix B. Dual Complex Construction for Tetra–Octa FCC

Appendix C. Cohomology Sectors and Dirac Sheet Ambiguities

Appendix D. Computational Enumeration Tables