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 $g_{μν}$
4	Edge group element $g_{t} (i, j) \in G$	Connection 1-form $A \in Ω^{1} (M, g)$
5	Gauge group $G = G^{V}$	Gauge transformation group $Map (M, G)$
6	Gauge action $g_{t}^{h} (i, j) = h (i) g_{t} (i, j) h (j)^{- 1}$	$A^{g} = g A g^{- 1} + g d g^{- 1}$
7	Triangle holonomy $Ω_{t} (△)$	Curvature 2-form $F_{A} = d A + A \land A$
8	Scalar curvature $ω_{t} (△) = d_{G} (Ω_{t}, e)^{2}$	$∥ F_{A} ∥^{2}$
9	Degree $d_{t} (i)$	Volume element $g d^{n} x$
10	Volume $vol_{t} (S)$	Riemannian volume $Vol (U)$
11	Cut $cut_{t} (S, \overset{ˉ}{S})$	Boundary area $Area (\partial U)$
12	Conductance $ϕ_{t} (S)$	Cheeger constant $h (M)$
13	Fruit $F$ ( $ϕ_{t} (F) \leq θ$ )	Metastable sub-manifold
14	Stem $S_{t}$	Thin neck / bridge region
15	Boundary coupling $b_{F, t} (i)$	Normal flux through $\partial U$
16	Door $Σ_{τ} (F, t)$	Uhlenbeck singular set ${p_{1}, \dots, p_{L}}$
17	Door energy $e_{p}$	Bubble energy $lim_{r \to 0} \int_{B_{r} (p)} ∣ F_{A} ∣^{2}$
18	Door threshold $τ$	Energy quantum $ϵ_{0}$
19	Kernel $F^{\circ} = F ∖ Σ$	Punctured manifold $M ∖ {p_{1}, \dots, p_{L}}$
20	Flattening energy $E_{F^{\circ}} (h)$	Yang--Mills energy $\int_{M} ∥ F_{A} ∥^{2} d V$
21	Optimal gauge $h^{*}$	Coulomb gauge / Uhlenbeck gauge
22	Canonical connection $[A_{\infty}]$	Limit connection (gauge class)
23	Existence $([A_{\infty}], Σ, e)$	Point in Uhlenbeck compactification
24	Moduli space $M (F, Σ)$	Moduli space of connections
25	World $W_{t}$	Spacetime with gauge field
26	Normalised Laplacian $L_{t}$	Laplace--Beltrami operator $Δ_{g}$
27	Random walk $P_{t}$	Heat kernel / Brownian motion
28	Escape time $T_{esc}$	Mixing time / first exit time
29	YM gradient flow $\dot{h} = - \nabla E$	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 ( $L \leq E / ϵ_{0}$ )
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 $T$	Time parameter in evolution equations
A3: Fruit threshold $θ$	Cheeger constant threshold
A4: Door threshold $τ$	Energy concentration quantum $ϵ_{0}$
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

Discrete (this theory)

Continuous (gauge theory / Riemannian geometry)

Finite set

V

, $

Edge set

E_{t}

Tangent bundle structure

Symmetric weight

W_{t} (i, j)

Riemannian metric

g_{μν}

Edge group element

g_{t} (i, j) \in G

Connection 1-form

A \in Ω^{1} (M, g)

Gauge group

G = G^{V}

Gauge transformation group

Map (M, G)

Gauge action

g_{t}^{h} (i, j) = h (i) g_{t} (i, j) h (j)^{- 1}

A^{g} = g A g^{- 1} + g d g^{- 1}

Triangle holonomy

Ω_{t} (△)

Curvature 2-form

F_{A} = d A + A \land A

Scalar curvature

ω_{t} (△) = d_{G} (Ω_{t}, e)^{2}

∥ F_{A} ∥^{2}

Degree

d_{t} (i)

Volume element

g d^{n} x

Volume

vol_{t} (S)

Riemannian volume

Vol (U)

Cut

cut_{t} (S, \overset{ˉ}{S})

Boundary area

Area (\partial U)

Conductance

ϕ_{t} (S)

Cheeger constant

h (M)

Fruit

F

(

ϕ_{t} (F) \leq θ

)

Metastable sub-manifold

Stem

S_{t}

Thin neck / bridge region

Boundary coupling

b_{F, t} (i)

Normal flux through

\partial U

Door

Σ_{τ} (F, t)

Uhlenbeck singular set

{p_{1}, \dots, p_{L}}

Door energy

e_{p}

Bubble energy

lim_{r \to 0} \int_{B_{r} (p)} ∣ F_{A} ∣^{2}

Door threshold

τ

Energy quantum

ϵ_{0}

Kernel

F^{\circ} = F ∖ Σ

Punctured manifold

M ∖ {p_{1}, \dots, p_{L}}

Flattening energy

E_{F^{\circ}} (h)

Yang--Mills energy

\int_{M} ∥ F_{A} ∥^{2} d V

Optimal gauge

h^{*}

Coulomb gauge / Uhlenbeck gauge

Canonical connection

[A_{\infty}]

Limit connection (gauge class)

Existence

([A_{\infty}], Σ, e)

Point in Uhlenbeck compactification

Moduli space

M (F, Σ)

Moduli space of connections

World

W_{t}

Spacetime with gauge field

Normalised Laplacian

L_{t}

Laplace--Beltrami operator

Δ_{g}

Random walk

P_{t}

Heat kernel / Brownian motion

Escape time

T_{esc}

Mixing time / first exit time

YM gradient flow

\dot{h} = - \nabla E

Yang--Mills flow

\partial_{t} A = - d_{A}^{*} F_{A}

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 \leq E / ϵ_{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

T

Time parameter in evolution equations

A3: Fruit threshold

θ

Cheeger constant threshold

A4: Door threshold

τ

Energy concentration quantum

ϵ_{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#

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