= Exponentially narrow gluing-map cusp
{title2=$|\mu|<C(\delta),\quad C(1-\eta)\sim\eta A^{1/\eta}/e$}
= Gluing-map cusp
{synonym}
Fix $0<A<1$ in the <signed gluing-map reduction> and write $\delta=1-\eta$ with $0<\eta<1$. The <saddle-node curves of a signed power gluing map> occur at parameter values $\mu=\pm C$, where
$$
C=\frac{\eta}{1-\eta}[A(1-\eta)]^{1/\eta}.
$$
Thus $C$ is the half-width of the three-branch interval, and $2C$ is its total width.
To see the scale, use $(1-\eta)^{1/\eta}\to e^{-1}$ and $(1-\eta)^{-1}\to1$. Then
$$
C\sim\frac{\eta}{e}A^{1/\eta}\qquad(\eta\downarrow0).
$$
Since $A^{1/\eta}=\exp(\log A/\eta)$ and $\log A<0$, the width decays faster than every positive power of $\eta$. A parameter scan with an algebraic step size can therefore miss the coexistence interval entirely.
This is an essential exponential scaling, rather than the power-law fold separation in the ordinary smooth cubic cusp. The three branches counted here are scalar return-map branches. Their physical lifts differ: a negative scalar fixed point represents two symmetry-related cycles, while a positive one represents one cycle visiting both lobes. The singular zero value is a separatrix limit with infinite return time.
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