1.1 Central Question
"What is a world, and what is existence?"
This theory proposes:
- World: a relational field---a pattern of weighted, group-valued relations among nodes.
- Existence: the gauge-invariant essence of a naturally occurring cohesive cluster (fruit) within the relational field.
1.2 The Sweet-Potato-Vine Metaphor
The global structure of a relational field is captured by the image of a sweet-potato vine:
╭────────╮ ╭──────────╮
│ Fruit 1 │── Stem ──│ Fruit 2 │
╰────────╯ │ ╰──────────╯
│
╭────────╮
│ Fruit 3 │
╰────────╯- Fruit (tuber): a locally cohesive cluster with low conductance---a metastable sub-world.
- Stem (filament): a weak but persistent connection channel bridging fruits.
This is not merely an analogy; it is mathematically precise:
- Fruit subset with Cheeger conductance .
- Stem nodes belonging to no fruit, plus inter-fruit bridge edges.
1.3 Five Philosophical Principles
Principle I: Primacy of Relations
The fundamental unit of the world is a relation, not an individual.
Nodes are mere endpoints of relations. Meaning emerges from the pattern of relations.
Principle II: Cohesion as Existence
Existence is the cohesion of relations. It is discovered, not decreed.
Fruits arise spontaneously from the scalar structure (weights) of the relational field. No external authority decides what exists; when conductance drops below the threshold, a fruit simply is.
Principle III: Gauge Invariance
The essence of existence does not depend on its representation.
The same relational structure, viewed in different coordinate systems (gauges), may look different, but its essence---the gauge equivalence class---is unique.
Principle IV: No-Boundary Principle
The exterior is never defined explicitly; it is detected only as interior singularities.
Modelling the exterior (stem) of a fruit explicitly would introduce arbitrary choices. Instead, contact with the exterior is observed only through doors---energy-concentration singularities inside the fruit. This parallels the philosophy of Uhlenbeck bubbling.
Principle V: Non-Intervention of Interpretation
Interpretation does not alter the world.
Door detection and gauge canonicalisation are acts of observation, not manipulation. Surgery "stabilises the reading", not "repairs the world".
1.4 Correspondence with Existing Theories
| This theory | Classical counterpart |
|---|---|
| Gauge equivalence class of relational field | Yang--Mills gauge theory |
| Fruit (low-conductance set) | Cheeger theory, spectral clustering |
| Door (energy-concentration singularity) | Uhlenbeck compactness / bubbling |
| Existence | Point in Uhlenbeck compactification |
| Metastability | Random-walk mixing time |
| Edge group element | Discrete connection (parallel transport) |
| Loop holonomy | Discrete curvature |
1.5 Structure of the Theory
Axioms A0--A5
│
▼
Relation (Ch 2) ──▶ Relational Field (Ch 3) ──▶ Fruit (Ch 4) + Stem (Ch 5)
│ │
│ ▼
│ Door (Ch 6)
│ │
│ ▼
└──────────────▶ Existence (Ch 7)
│
▼
World (Ch 8)
│
▼
Theorems A--H (Ch 9)Each layer depends only on the preceding ones; there are no circular dependencies.
1.6 Reading Guide
| Part | Chapters | Content |
|---|---|---|
| I. Foundations | 1--8 | Definitions from relation to world |
| II. Theorems | 9--10 | Main theorems with full proofs; worked examples |
| III. Cohomology | 11--12 | Cech cohomology and three computational axes |
| IV. Dynamics | 13--14 | Yang--Mills flow and time evolution |
| V. Applications | 15 | Physics, topology, combinatorics |
| VI. Frontiers | 16 | Open problems and conjectures |
| Appendices | A--C | Notation, prerequisites, discrete--continuous dictionary |