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ℓPart 1
Eight chapters building the core formalism — relations, relational fields, fruits, stems, doors, existence, and the notion of a world.
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.
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).
Stem
The connective tissue between fruits. Stem region, bridge edges, and the no-boundary principle that prevents the exterior from being defined explicitly.
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.
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.
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.