#gauge-theory

Overview — The Sweet-Potato-Vine Worldview

Philosophy and roadmap of RelationWorld Theory. Five foundational principles, the central sweet-potato-vine metaphor, and a correspondence dictionary between the discrete formalism and its classical counterparts.

Relation

The fundamental unit of RelationWorld — a relation tuple (i, j, w, g) carrying scalar intensity and group-valued transit, with symmetrised weight, degree, and their basic properties.

Relational Field

Gauge group and action on relational fields, scalar invariants, holonomy and discrete curvature, and the completeness of invariants in the connected case.

Main Theorems A–H

Complete statements and full proofs of the eight core theorems of RelationWorld Theory — energy isolation, door finiteness, self-interpretation, metastability, curvature localisation, spectral stability, door stability, and flow stability.

Part II · Main Theorems A–H (summary)

A condensed statement of the eight main theorems of RelationWorld Theory with one-paragraph proof ideas and their logical dependencies. Full proofs live in the research archive.

Chapter 11 — Čech Cohomology Framework

Čech cohomology as the natural language of the theory — what holds exactly in the discrete setting, what requires additional topological hypotheses, and how the cocycle condition encodes the closure of relational transit.

Appendix C — Discrete–Continuous Correspondence Dictionary

A thirty-row dictionary mapping discrete RelationWorld constructions to their continuous differential-geometry counterparts, plus theorem and axiom correspondence tables.

An Axiomatic Framework for Understanding Relations via Gauge-Invariant Cohomology

What does it mean for something to exist within a web of relations? We answer this question by developing a rigorous mathematical framework in which existence e

RelationWorld theory brainstorm -- five cycles on state spaces, gauge, and events

Five brainstorm cycles probing the RelationWorld state-space question: candidate constructions, gauge commitments, red-team objections, and a machine-realisable runtime schema.