>[!warning]
>This content has not been peer reviewed.
# Graph Size Hierarchy (RST)
**Summary:** The entire universe — quantum forces, matter, the Periodic Table, and cosmology — is mapped by one topological concept: **Graph Size ($N$)**. As the graph grows from 2 nodes to infinity, the structural information possibilities generate the Standard Model, baryons, chemistry, and General Relativity. This resolves the **"Origin of gauge groups"** open question: $SU(2)$ and $SU(3)$ are not abstract symmetries; they are the information-flow matrices for $K_2$ and $K_3$ cliques.
---
## The Unification
```mermaid
flowchart TB
subgraph Forces [Quantum Forces]
K2[K_2 Edge]
K3[K_3 Triangle]
K2 --> Pauli[3 Pauli SU2]
K3 --> GellMann[8 Gell-Mann SU3]
end
subgraph Matter [Matter]
K4[K_4 Tetrahedron]
K4 --> Baryons[Baryons 3 quarks]
end
subgraph Chem [Chemistry]
Pack[Packing K_4s]
Pack --> PT[Periodic Table]
end
subgraph Cosmo [Cosmology]
Inf[N to Infinity]
Inf --> MOND[MOND D_f n]
end
Forces --> Matter --> Chem --> Cosmo
```
---
## Level 1: Quantum Forces (Sub-graph Symmetries)
In graph theory, the number of ways information can flow through a completely connected sub-graph of $N$ nodes is defined by its traceless Hermitian matrices:
$\text{Generator count} = N^2 - 1$
| $N$ | Graph | Matrices | Lie group | Physical manifestation |
|:---|:---|:---|:---|:---|
| **1** | $K_1$ (single node) | 1 (phase) | $U(1)$ | EM, photon |
| **2** | $K_2$ (edge) | $2^2 - 1 = 3$ Pauli | $SU(2)$ | Weak force, spin-1/2 fermions |
| **3** | $K_3$ (triangle) | $3^2 - 1 = 8$ Gell-Mann | $SU(3)$ | 8 Gluons, Strong force |
Color charge (Red, Green, Blue) = the 3 nodes of $K_3$. For $N=1$, $N^2 - 1 = 0$; $U(1)$ is the special case (phase-only). See [[N=1 — Electromagnetism and U(1) (RST)]].
**Sub-pages:** [[N=1 — Electromagnetism and U(1) (RST)]], [[N=2 — Weak Force and Pauli (RST)]], [[N=3 — Strong Force and Gell-Mann (RST)]]
---
## Level 2: Birth of Matter ($K_4$)
A triangle (3 nodes) is flat. **4 nodes** is the minimum required to enclose a 3-dimensional volume. The moment the graph hits $K_4$, it transitions from "Force" into **Matter** (a 3D structural knot).
- **Baryons** (protons, neutrons) = 3 quarks + binding energy = 4 vertices of the $K_4$ tetrahedron
- [[Nuclear Fusion - Code|rst_nuclear_fusion.py]] correctly sets the Nucleon Motif to $K_4$
**Sub-page:** [[N=4 — Baryons and Tetrahedron (RST)]]
---
## Level 3: Periodic Table (Packing the $K_4$s)
Once the universe discovers $K_4$, it packs tetrahedra together. Tetrahedra cannot pack perfectly without gaps (geometric frustration).
- **Periodic Table** = catalogue of every mathematically stable packing of $K_4$s within the Landauer budget
- **Magic numbers** (He, O, Ca, Pb) = integer counts where $K_4$s form closed Platonic shapes
- **Radioactive decay** = asymmetrical packing; Shannon cost exceeds $\log_2$ budget; graph snaps a piece off (Alpha Decay)
**Sub-page:** [[N=5 to 300 — Periodic Table (RST)]]
---
## Level 4: Cosmology ($N \to \infty$)
Packing $K_4$s by the billions: microscopic geometric gaps accumulate. This **lacunarity** is the fractal sponge's deficit. At macroscopic scale ([[Cosmic Visualizer (RST)|rst_master_visualizer.py]]):
- Accumulated deficit bends the spatial metric into $D_f \approx 1.86$
- Fidelity function becomes the MOND curve $n \approx 1.24$
**Sub-page:** [[N to Infinity — Cosmology (RST)]]
---
## Summary Table
| $N$ | Domain | Manifestation |
|:---|:---|:---|
| **1** | EM | Photon / $U(1)$ |
| **2** | Weak force | Pauli / Electrons |
| **3** | Strong force | Gluons / Quarks |
| **4** | Baryons | Protons / Neutrons |
| **5 → 300** | Chemistry | Periodic Table |
| **$N \to \infty$** | Cosmology | GR & MOND |
---
## Links
| Document | Purpose |
|:---|:---|
| **[[Derivation Chain Overview]]** | Resolves "Origin of gauge groups" — now **DERIVED** |
| **[[The Sovereign Chain]]** | 10 SM constants from $n$; complements graph hierarchy |
| **[[The Spectrum of Relational Topologies]]** | Dimensional Ladder; $n = \dim(\mathfrak{g})$ from graph |
| **[[Standard Model (RST)]]** | Gauge generators; graph-theoretic origin |
| **[[Relational Liquid Drop Model]]** | $K_4$ motif; nucleon structure |
| **[[Periodic Table (RST)]]** | Packing application |
| **[[Cosmology and GR (RST)]]** | $N \to \infty$ limit |