Theorem G — Door Stability

Prerequisites: Theorem B (door finiteness), perturbation analysis (Chapter 6).

Theorem G. (Complete proof in Chapter 6, Theorem G.)

Under perturbation $\|\delta W\|_\infty$, setting $\epsilon=|V|\cdot\|\delta W\|_\infty$:

(i) $|b'-b|\le|V|\cdot\|\delta W\|_\infty$. (ii)--(iv) Doors with margin $>\epsilon$ are stable; only the $\epsilon$-boundary layer may change.