ℓPart 2
Part II · Main Theorems & Examples
Complete proofs of Theorems A–H and worked examples in U(1) gauge theory.
Essays
Theorems
- §2.9
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.
- §2.9
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.
- §2.9
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.
- §2.9
Theorem B — Finiteness of Doors
The number of door nodes is bounded by theta times the volume divided by the threshold. A direct energy-budget argument.
- §2.9
Theorem C — Self-Interpretation
Under Axiom A5, the door set and door energies are determined entirely by the fruit's intrinsic data — no exterior information is needed.
- §2.9
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.
- §2.9
Theorem E — Curvature Localisation
Residual curvature under the optimal gauge concentrates exponentially near doors for U(1), and satisfies a contraction bound for general compact G.
- §2.9
Theorem F — Spectral Stability
Small weight perturbations produce bounded conductance changes; strong fruits persist under perturbation with an explicit constant C1 = 2(1+theta).
- §2.9
Theorem G — Door Stability
Under weight perturbation, doors with sufficient margin above the threshold are stable — only the epsilon-boundary layer may change.
- §2.9
Theorem H — Flow Stability
The Yang–Mills gradient flow converges to a stable critical point, with exponential rate when the Lojasiewicz exponent is 1/2, via real-analyticity and compactness.