A long book-in-progress (working title: RelationWorld Theory) that develops the mathematical machinery the rest of the programme depends on. The object of study is a weighted graph equipped with a discrete gauge structure — every edge carries both a positive weight and a group-valued transition. The question is: what can be said about this object with the tools of algebraic topology and gauge theory?
The answer turns out to be: quite a lot. Yang–Mills gauge theory, Uhlenbeck compactness, and Cheeger spectral geometry all have clean combinatorial analogues in this setting, and together they give a picture where natural aggregates (fruits), internal singularities (doors), and existence itself become gauge-invariant topological invariants computable by finite linear algebra.
Structure
Six parts plus appendices, each building on the previous one.
| Part | Chapters | Focus |
|---|---|---|
| I. Foundations | ch01–ch08 | Relation, relational field, fruit, stem, door, existence, world |
| II. Theorems | ch09–ch10 | Eight theorems A–H with complete proofs; examples |
| III. Cohomology | ch11–ch12 | Čech framework; three axes of topological readout |
| IV. Dynamics | ch13–ch14 | Yang–Mills flow; time evolution and phase transitions |
| V. Applications | ch15 | Physics, topology, combinatorics |
| VI. Frontiers | ch16 | Open problems and conjectures |
Part I is reproduced in full in the notes; Part II is summarised there and Parts III–VI will follow as the chapters settle.
The six axioms
- A0. Structure group is a compact Lie group with bi-invariant metric.
- A1. Finite node set , .
- A2. Discrete time set .
- A3. Fruit threshold .
- A4. Door threshold .
- A5. Intrinsic data axiom — boundary coupling is internally computable.
The eight main theorems
Full proofs live in Part II of the notes.
Papers on this track
- An Axiomatic Framework for Understanding Relations via Gauge-Invariant Cohomology — the published-facing statement of the theory.
- Three LQG manuscripts that deploy the same mathematical machinery in a quantum-gravity setting: Quantum Bounce, Internal Time and GR Emergence, and Scalable LQG Simulations.
Related
- Perception theory — where this machinery is used to ground Soft Cognitive Cohesion.
- Ontology Neural Networks — where it certifies the invariants of learned representations.