ℓPart 8· Appendix A

Appendix A — Unified Notation Table

877 words3 min read

This appendix provides a single, comprehensive symbol reference for the entire theory.

A.1 Standing Assumptions (Axioms A0--A5)

Label	Axiom	Content
A0	Gauge group	A compact Lie group $G$ with Lie algebra $\mathfrak{g}$ , identity $e$ , and bi-invariant metric $d_G$
A1	Node set	A finite set $V$ with $n :=
A2	Time set	A totally ordered set $\mathbb{T}$
A3	Fruit threshold	$\theta \in (0,1)$
A4	Door threshold	$\tau > 0$
A5	Intrinsic data axiom	The degree $d_t(i)$ is included in a fruit's intrinsic data (Axiom 6.1)

A.2 Core Symbols

Relations and Fields (Chapters 2--3)

Symbol	Definition	Meaning
$(i,j,w_t(i,j),g_t(i,j))$	Def 2.1	A relation: source $i$ , target $j$ , scalar weight $w_t(i,j)\in\mathbb{R}_{\ge0}$ , group transit $g_t(i,j)\in G$
$W_t(i,j)$	Def 2.5	Symmetrised weight: $\tfrac{1}{2}(w_t(i,j)+w_t(j,i))$
$d_t(i)$	Def 2.6	Degree: $\sum_{j\in V}W_t(i,j)$
$\mathcal{W}_t^{\mathrm{raw}}$	Def 3.1	Raw relational field $(V,w_t,g_t)$
$\mathcal{G}$	Def 3.2	Gauge group $G^V$
$g_t^h(i,j)$	Def 3.3	Gauge-transformed transit: $h(i)\,g_t(i,j)\,h(j)^{-1}$
$\mathcal{W}_t$	Def 3.6	Relational field (gauge equivalence class of $\mathcal{W}_t^{\mathrm{raw}}$ )
$\mathrm{Hol}_t(\gamma)$	Prop 3.8	Holonomy of a closed loop $\gamma$
$\Omega_t(\triangle)$	Cor 3.9	Discrete curvature: holonomy of triangle $(i,j,k)$
$\omega_t(\triangle)$	Cor 3.9	Scalar curvature: $d_G(\Omega_t(\triangle),e)^2$

Fruits and Stems (Chapters 4--5)

Symbol	Definition	Meaning
$\mathcal{G}_t$	Def 4.1	Induced weighted graph $(V,E_t,W_t)$
$\mathrm{vol}_t(S)$	Def 4.2	Volume: $\sum_{i\in S}d_t(i)$
$\mathrm{cut}_t(S,\bar S)$	Def 4.2	Cut: $\sum_{i\in S}\sum_{j\notin S}W_t(i,j)$
$\phi_t(S)$	Def 4.2	Conductance (Cheeger ratio): $\mathrm{cut}/\min\{\mathrm{vol}(S),\mathrm{vol}(\bar S)\}$
$\theta$	Ax 4.4	Fruit threshold
$F$	Def 4.5	A fruit: $\mathrm{vol}_t(F)\le\tfrac12\mathrm{vol}_t(V)$ and $\phi_t(F)\le\theta$
$\mathfrak{F}_t$	Def 4.6	Set of all fruits at time $t$
$\mathcal{L}_t$	Def 4.8	Normalised graph Laplacian: $I - D_t^{-1/2}A_tD_t^{-1/2}$
$\lambda_k$	--	$k$ -th eigenvalue of $\mathcal{L}_t$
$h_t$	Thm 4.9	Cheeger constant: $\min_S\phi_t(S)$
$P_t(i,j)$	Def 4.13	Transition matrix: $W_t(i,j)/d_t(i)$
$\tilde P_t$	Def 4.13	Lazy walk: $\tfrac12(I+P_t)$
$\mathcal{F}_t^{\cup}$	Def 5.1	Union of all fruits
$\mathcal{S}_t$	Def 5.2	Stem region: $V\setminus\mathcal{F}_t^{\cup}$
$E_t^{\mathrm{bridge}}$	Def 5.3	Bridge edges

Doors (Chapter 6)

Symbol	Definition	Meaning
$\mathcal{D}(F,t)$	Ax 6.1	Intrinsic data: $(\{W_t(i,j)\}_{i,j\in F},\{g_t(i,j)\}_{i,j\in F},\{d_t(i)\}_{i\in F})$
$b_{F,t}(i)$	Lem 6.3	Boundary coupling: $d_t(i)-\sum_{j\in F}W_t(i,j)$
$r_{F,t}(i)$	Def 6.4	Leakage rate: $b_{F,t}(i)/d_t(i)$
$\partial_V F$	Def 6.6	Fruit boundary: $\{i\in F:b_{F,t}(i)>0\}$
$\tau$	Ax 6.5	Door threshold
$\Sigma_\tau(F,t)$	Def 6.7	Door set (energy-based): $\{i\in\partial_V F:b_{F,t}(i)\ge\tau\}$
$\varepsilon^{\mathrm{hol}}_{F,t}(i)$	Def 6.11	Holonomy door energy
$\Sigma^{\mathrm{hol}}_\tau$	--	Holonomy-based door set
$\mathbf{e}(F,t)$	Def 6.15	Door energy vector: $\{e_p\}_{p\in\Sigma}$ with $e_p=b_{F,t}(p)$

Existence (Chapter 7)

Symbol	Definition	Meaning
$F^\circ$	Def 7.1	Kernel of fruit: $F\setminus\Sigma$
$\mathcal{E}_{F^\circ}(h)$	Def 7.2	Flattening energy: $\sum W_t(i,j)\,d_G(g_t^h(i,j),e)^2$
$h^*$	Thm 7.4	Optimal gauge: $\arg\min_h\mathcal{E}_{F^\circ}(h)$
$\mathcal{G}_{\mathrm{const}}$	Def 7.5	Residual (constant) gauge: $\{h:h(i)=k\;\forall i\}\cong G$
$\rho_{F^\circ}(i)$	Lem 7.6	Residual curvature at node $i$
$[A_\infty(F,t)]$	Const 7.10	Canonical connection (gauge class on $F^\circ$ )
$\mathrm{Existence}(F,t)$	Def 7.12	$([A_\infty],\Sigma,\mathbf{e})$
$\mathcal{M}(F,\Sigma)$	Def 7.14	Moduli space of existence

World (Chapter 8)

Symbol	Definition	Meaning
$\mathfrak{W}_t$	Def 8.1	Instantaneous world: $(\mathcal{W}_t,\mathfrak{F}_t,\Sigma_t)$
$\mathfrak{W}$	Def 8.2	Full world: $\{\mathfrak{W}_t\}_{t\in\mathbb{T}}$
$\mathrm{Ex}_t$	Def 8.3	Existence functor

A.3 Theorems at a Glance

Theorem	Name	Core Statement	Proof Status
A	Energy isolation	Internal energy $\ge(1-\theta)\,\mathrm{vol}(F)$	Complete
B	Finiteness of doors	$	\Sigma
C	Self-interpretation	$\Sigma,\mathbf{e}$ determined by intrinsic data	Complete
D	Metastability	Escape time $\ge 1/(2\theta)$	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	$[A_\infty]$ is a stable fixed point of discrete YM flow	Complete

A.4 Conventions

Numbering: Definition/Theorem/Lemma X.Y means Chapter X, item Y.
Gauge action is a right action: $(g^{h_1})^{h_2}=g^{h_2 h_1}$ .
$d_G$ always denotes the bi-invariant geodesic distance on $G$ .
$\bar S := V\setminus S$ for any $S\subset V$ .
All sums over edges are over ordered pairs unless stated otherwise.
"Sketch" proofs explicitly list unverified assumptions.