ℓPart 1
Part I · Foundations of RelationWorld
Eight chapters building the core formalism — relations, relational fields, fruits, stems, doors, existence, and the notion of a world.
- §1.1
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.
- §1.2
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.
- §1.3
Relational Field
Gauge group and action on relational fields, scalar invariants, holonomy and discrete curvature, and the completeness of invariants in the connected case.
- §1.4
Fruit
Low-conductance clusters as the basic unit of cohesive existence. Cheeger conductance, the fruit definition with threshold θ, and Theorems A (Energy Isolation) and D (Metastability).
- §1.5
Stem
The connective tissue between fruits. Stem region, bridge edges, and the no-boundary principle that prevents the exterior from being defined explicitly.
- §1.6
Door
Internal singularities arising from anomalous external contact, detectable only within the fruit interior. The intrinsic data axiom A5, door definitions, and Theorems B, C, G.
- §1.7
Existence
Existence as a gauge-invariant triple combining the optimal gauge class, door locus, and residual energy. Flattening energy, optimal gauge, and Theorems E and H.
- §1.8
World
The world as a three-layer hierarchy of relational fields, fruits with existence triples, and their inter-fruit connectivity. Theorem F (Spectral Stability) closes Part I.