Mathematical foundations

Current questions — October 2026#

The current mathematical work asks what a declared observation determines about an underlying structure. Exact local rank checks, generic identifiability, and a certificate for an entire solution fiber are distinct claims. Global uniqueness for the particular instance under investigation remains open, and no new complexity-theoretic conclusion is established.

See the learned-structure update for how this connects to formation, realization, and identification. A separate conceptual inquiry asks when selection becomes relevance and evaluation; its proposed distinctions remain provisional.

Earlier foundation — RelationWorld#

The following book overview preserves the earlier mathematical programme. It is not evidence that the newer identifiability or relevance questions have been resolved.

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#

• SCC archive — where this machinery was used historically to ground Soft Cognitive Cohesion.
• ONN archive — where it was intended to certify invariants of learned representations.
• Unified Latent Representation — where active tests type each observable, gauge, intervention, and evidential boundary explicitly.