χTopic tag
#gauge-theory
9 documents indexed here. Mathematical objects, methods, systems, and recurring technical questions.
ℓNotes
7- Notes
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.
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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.
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Relational Field
Gauge group and action on relational fields, scalar invariants, holonomy and discrete curvature, and the completeness of invariants in the connected case.
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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.
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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.
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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.
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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.