Appendix C — Discrete–Continuous Correspondence Dictionary

This appendix provides a unified dictionary relating the discrete constructions of the RelationWorld theory to their counterparts in continuous differential geometry and gauge theory.

C.1 Main Dictionary

#	Discrete (this theory)	Continuous (gauge theory / Riemannian geometry)
1	Finite set $V$ , $	V
2	Edge set $E_t$	Tangent bundle structure
3	Symmetric weight $W_t(i,j)$	Riemannian metric $g_{\mu\nu}$
4	Edge group element $g_t(i,j)\in G$	Connection 1-form $A\in\Omega^1(M,\mathfrak{g})$
5	Gauge group $\mathcal{G}=G^V$	Gauge transformation group $\mathrm{Map}(M,G)$
6	Gauge action $g_t^h(i,j)=h(i)g_t(i,j)h(j)^{-1}$	$A^g=gAg^{-1}+g\,dg^{-1}$
7	Triangle holonomy $\Omega_t(\triangle)$	Curvature 2-form $F_A=dA+A\wedge A$
8	Scalar curvature $\omega_t(\triangle)=d_G(\Omega_t,e)^2$	$\\|F_A\\|^2$
9	Degree $d_t(i)$	Volume element $\sqrt{g}\,d^nx$
10	Volume $\mathrm{vol}_t(S)$	Riemannian volume $\mathrm{Vol}(U)$
11	Cut $\mathrm{cut}_t(S,\bar S)$	Boundary area $\mathrm{Area}(\partial U)$
12	Conductance $\phi_t(S)$	Cheeger constant $h(M)$
13	Fruit $F$ ( $\phi_t(F)\le\theta$ )	Metastable sub-manifold
14	Stem $\mathcal{S}_t$	Thin neck / bridge region
15	Boundary coupling $b_{F,t}(i)$	Normal flux through $\partial U$
16	Door $\Sigma_\tau(F,t)$	Uhlenbeck singular set $\{p_1,\ldots,p_L\}$
17	Door energy $e_p$	Bubble energy $\lim_{r\to0}\int_{B_r(p)}\lvert F_A\rvert^2$
18	Door threshold $\tau$	Energy quantum $\epsilon_0$
19	Kernel $F^\circ=F\setminus\Sigma$	Punctured manifold $M\setminus\{p_1,\ldots,p_L\}$
20	Flattening energy $\mathcal{E}_{F^\circ}(h)$	Yang--Mills energy $\int_M\\|F_A\\|^2\,dV$
21	Optimal gauge $h^*$	Coulomb gauge / Uhlenbeck gauge
22	Canonical connection $[A_\infty]$	Limit connection (gauge class)
23	Existence $([A_\infty],\Sigma,\mathbf{e})$	Point in Uhlenbeck compactification
24	Moduli space $\mathcal{M}(F,\Sigma)$	Moduli space of connections
25	World $\mathfrak{W}_t$	Spacetime with gauge field
26	Normalised Laplacian $\mathcal{L}_t$	Laplace--Beltrami operator $\Delta_g$
27	Random walk $P_t$	Heat kernel / Brownian motion
28	Escape time $T_{\mathrm{esc}}$	Mixing time / first exit time
29	YM gradient flow $\dot h=-\nabla\mathcal{E}$	Yang--Mills flow $\partial_t A=-d_A^*F_A$
30	Lojasiewicz convergence	Simon's convergence theorem

C.2 Correspondence of Theorems

Discrete Theorem	Continuous Analogue
A (Energy isolation)	Cheeger inequality
B (Finiteness of doors)	Uhlenbeck compactness ( $L\le E/\epsilon_0$ )
C (Self-interpretation)	Internal boundary data determines singular set
D (Metastability)	Spectral gap $\Leftrightarrow$ mixing time
E (Curvature localisation)	Removable singularity / regularity away from bubbles
F (Spectral stability)	Stability of Cheeger constant under metric perturbation
G (Door stability)	Stability of singular set under connection perturbation
H (Flow stability)	Lojasiewicz--Simon gradient inequality $\Rightarrow$ convergence

C.3 Axiom Correspondence

Discrete Axiom	Continuous Counterpart
A0: Compact Lie group $G$	Structure group of principal bundle
A1: Finite node set $V$	Compact base manifold $M$
A2: Time set $\mathbb{T}$	Time parameter in evolution equations
A3: Fruit threshold $\theta$	Cheeger constant threshold
A4: Door threshold $\tau$	Energy concentration quantum $\epsilon_0$
A5: Intrinsic data axiom	Boundary trace theorem (Sobolev)

C.4 Key Differences

The discrete theory is not merely a discretisation of the continuous theory. Key structural differences:

Finite-dimensionality: all spaces are finite, so existence and compactness results are immediate (no functional analysis needed).
No regularity issues: there are no Sobolev spaces, no weak solutions, no regularity theory. All functions are finitely specified.
Exact computations: cohomology is computed by finite matrix algebra, not by solving PDEs.
Combinatorial conductance: replaces the geometric Cheeger constant, but retains the same spectral relationship (discrete Cheeger inequality).
Doors vs. bubbles: discrete doors are individual nodes; continuous bubbles are idealised points with concentration measures. The discrete version is inherently regularised.

C.1 Main Dictionary#

C.2 Correspondence of Theorems#

C.3 Axiom Correspondence#

C.4 Key Differences#

C.1 Main Dictionary

C.2 Correspondence of Theorems

C.3 Axiom Correspondence

C.4 Key Differences