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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).
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.
Theorem A — Energy Isolation
Internal edge-energy of a fruit is at least (1-theta) of its total volume. A three-step proof from the conductance bound.
Theorem D — Metastability
The expected escape time from a fruit under the lazy walk is at least 1/(2 theta), via the Cheeger inequality and the Sinclair–Jerrum spectral bound.