#proof

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.

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 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.

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.

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.

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.

Theorem F — Spectral Stability

Small weight perturbations produce bounded conductance changes; strong fruits persist under perturbation with an explicit constant C1 = 2(1+theta).

Theorem G — Door Stability

Under weight perturbation, doors with sufficient margin above the threshold are stable — only the epsilon-boundary layer may change.

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.