= Saddle-node curves of a signed power gluing map
{title2=$r=(A\delta)^{-1/(\delta-1)},\quad\mu=\pm\frac{\delta-1}{\delta}r$}
For $f(z)=\mu+A\operatorname{sgn}(z)|z|^\delta$, a nonzero <fixed point> satisfies
$$
\mu=z-A\operatorname{sgn}(z)|z|^\delta.
$$
Assume $A>0$, $\delta>0$ and $\delta\ne1$. A <saddle-node bifurcation> has multiplier $f'(z)=A\delta|z|^{\delta-1}=1$, so it occurs at $z=\pm r$ with
$$
r=(A\delta)^{-1/(\delta-1)},\qquad
\mu=\pm\frac{\delta-1}{\delta}r.
$$
At either nonzero point, $f''$ is nonzero and the parameter enters additively, giving the ordinary nondegenerate fold conditions.
When $0<\delta<1$, define $C=(1-\delta)r/\delta$. The scalar extension has three fixed-point branches for $|\mu|<C$: two stable outer branches and an unstable inner branch. For nonzero fixed points, their derivatives are below and above one because $A\delta|z|^{\delta-1}$ decreases with $|z|$. At $\mu=0$, the inner branch lies at the singular separatrix value; the <signed gluing-map reduction> explains why it is excluded as a finite-time flow return.
For fixed $0<A<1$ and $\delta\uparrow1$ from below, the fold magnitude has asymptotic value $C\sim(1-\delta)A^{1/(1-\delta)}/e$. This is the half-width; the total parameter interval has width $2C$. The <exponentially narrow gluing-map cusp> gives the exact rescaling. For $\delta>1$ approaching one, the formal fold coordinate instead tends to infinity, outside the local return-map region.
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