Mathematical foundations

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 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 $G$ is a compact Lie group with bi-invariant metric.
• A1. Finite node set $V$, $|V| = n$.
• A2. Discrete time set $\mathbb{T} \subseteq \mathbb{Z}$.
• A3. Fruit threshold $\theta \in (0, 1)$.
• A4. Door threshold $\tau > 0$.
• 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.