Let $A$ be Hurwitz and $f$ locally Lipschitz with $f(0) = 0$. Which explicit conditions on $f$ make the origin of $\dot{x}=Ax+f(x)$ globally asymptotically stable? Using quadratic Lyapunov functions $V=\langle Px,x\rangle$, we show that $V$ is a strict certificate exactly when $f$ lies in a pointwise half-space, and that the largest certified sector $\|f(x)\| \le g\|x\|$ is $g < r_C(A)$, the complex stability radius. Our new results concern perturbations not confined to a ball: a direction $h(x)Mx$ is certified for every scalar $h$ if and only if $M$ is skew for some $P$ with $A^{\top}P+PA < 0$, and a gain–dissipation trade-off allows unbounded gain where local dissipation compensates it. We then compare $r_C$ with the real stability radius $r_R$: they coincide for normal matrices, or when the supremum defining $r_C$ is attained at $w=0$, and otherwise, for an explicit $3\times3$ example, nonlinear perturbations destabilize strictly below $r_R$, bracketing the exact threshold in $[0.1671,0.2001]$. We also prove that $r_{NL}$ equals the radius $r_{TV}$ for time-varying perturbations when $n\ge3$, which yields an exact criterion for when the quadratic threshold is the global one, and that for $A_3$ the upper bound can be taken as $0.2000007$. We also give averaged-gain and switching theorems with exact rate functions, which recover the sharp duty cycle in the burst examples, and discuss applications.