This appendix provides a single, comprehensive symbol reference for the entire theory.
| Label | Axiom | Content |
|---|
| A0 | Gauge group | A compact Lie group G with Lie algebra g, identity e, and bi-invariant metric dG |
| A1 | Node set | A finite set V with $n := |
| A2 | Time set | A totally ordered set T |
| A3 | Fruit threshold | θ∈(0,1) |
| A4 | Door threshold | τ>0 |
| A5 | Intrinsic data axiom | The degree dt(i) is included in a fruit's intrinsic data (Axiom 6.1) |
| Symbol | Definition | Meaning |
|---|
| (i,j,wt(i,j),gt(i,j)) | Def 2.1 | A relation: source i, target j, scalar weight wt(i,j)∈R≥0, group transit gt(i,j)∈G |
| Wt(i,j) | Def 2.5 | Symmetrised weight: 21(wt(i,j)+wt(j,i)) |
| dt(i) | Def 2.6 | Degree: ∑j∈VWt(i,j) |
| Wtraw | Def 3.1 | Raw relational field (V,wt,gt) |
| G | Def 3.2 | Gauge group GV |
| gth(i,j) | Def 3.3 | Gauge-transformed transit: h(i)gt(i,j)h(j)−1 |
| Wt | Def 3.6 | Relational field (gauge equivalence class of Wtraw) |
| Holt(γ) | Prop 3.8 | Holonomy of a closed loop γ |
| Ωt(△) | Cor 3.9 | Discrete curvature: holonomy of triangle (i,j,k) |
| ωt(△) | Cor 3.9 | Scalar curvature: dG(Ωt(△),e)2 |
| Symbol | Definition | Meaning |
|---|
| Gt | Def 4.1 | Induced weighted graph (V,Et,Wt) |
| volt(S) | Def 4.2 | Volume: ∑i∈Sdt(i) |
| cutt(S,Sˉ) | Def 4.2 | Cut: ∑i∈S∑j∈/SWt(i,j) |
| ϕt(S) | Def 4.2 | Conductance (Cheeger ratio): cut/min{vol(S),vol(Sˉ)} |
| θ | Ax 4.4 | Fruit threshold |
| F | Def 4.5 | A fruit: volt(F)≤21volt(V) and ϕt(F)≤θ |
| Ft | Def 4.6 | Set of all fruits at time t |
| Lt | Def 4.8 | Normalised graph Laplacian: I−Dt−1/2AtDt−1/2 |
| λk | -- | k-th eigenvalue of Lt |
| ht | Thm 4.9 | Cheeger constant: minSϕt(S) |
| Pt(i,j) | Def 4.13 | Transition matrix: Wt(i,j)/dt(i) |
| P~t | Def 4.13 | Lazy walk: 21(I+Pt) |
| Ft∪ | Def 5.1 | Union of all fruits |
| St | Def 5.2 | Stem region: V∖Ft∪ |
| Etbridge | Def 5.3 | Bridge edges |
| Symbol | Definition | Meaning |
|---|
| D(F,t) | Ax 6.1 | Intrinsic data: ({Wt(i,j)}i,j∈F,{gt(i,j)}i,j∈F,{dt(i)}i∈F) |
| bF,t(i) | Lem 6.3 | Boundary coupling: dt(i)−∑j∈FWt(i,j) |
| rF,t(i) | Def 6.4 | Leakage rate: bF,t(i)/dt(i) |
| ∂VF | Def 6.6 | Fruit boundary: {i∈F:bF,t(i)>0} |
| τ | Ax 6.5 | Door threshold |
| Στ(F,t) | Def 6.7 | Door set (energy-based): {i∈∂VF:bF,t(i)≥τ} |
| εF,thol(i) | Def 6.11 | Holonomy door energy |
| Στhol | -- | Holonomy-based door set |
| e(F,t) | Def 6.15 | Door energy vector: {ep}p∈Σ with ep=bF,t(p) |
| Symbol | Definition | Meaning |
|---|
| F∘ | Def 7.1 | Kernel of fruit: F∖Σ |
| EF∘(h) | Def 7.2 | Flattening energy: ∑Wt(i,j)dG(gth(i,j),e)2 |
| h∗ | Thm 7.4 | Optimal gauge: argminhEF∘(h) |
| Gconst | Def 7.5 | Residual (constant) gauge: {h:h(i)=k∀i}≅G |
| ρF∘(i) | Lem 7.6 | Residual curvature at node i |
| [A∞(F,t)] | Const 7.10 | Canonical connection (gauge class on F∘) |
| Existence(F,t) | Def 7.12 | ([A∞],Σ,e) |
| M(F,Σ) | Def 7.14 | Moduli space of existence |
| Symbol | Definition | Meaning |
|---|
| Wt | Def 8.1 | Instantaneous world: (Wt,Ft,Σt) |
| W | Def 8.2 | Full world: {Wt}t∈T |
| Ext | Def 8.3 | Existence functor |
| Theorem | Name | Core Statement | Proof Status |
|---|
| A | Energy isolation | Internal energy ≥(1−θ)vol(F) | Complete |
| B | Finiteness of doors | $ | \Sigma |
| C | Self-interpretation | Σ,e determined by intrinsic data | Complete |
| D | Metastability | Escape time ≥1/(2θ) | Complete |
| E | Curvature localisation | Residual curvature concentrates near doors | Complete |
| F | Spectral stability | Strong fruits persist under perturbation | Complete |
| G | Door stability | Door structure stable under small perturbation | Complete |
| H | Flow stability | [A∞] is a stable fixed point of discrete YM flow | Complete |
- Numbering: Definition/Theorem/Lemma X.Y means Chapter X, item Y.
- Gauge action is a right action: (gh1)h2=gh2h1.
- dG always denotes the bi-invariant geodesic distance on G.
- Sˉ:=V∖S for any S⊂V.
- All sums over edges are over ordered pairs unless stated otherwise.
- "Sketch" proofs explicitly list unverified assumptions.