is extremely unlikely.

The behavior of thermodynamic functions close to the transition is the following. Writing $\alpha=e(1+\varepsilon)$, for small negative $\varepsilon$, $$\begin{aligned}
 \lefteqn{ A(\alpha) = B(\alpha) =}  \\
 && 1+\frac{1}{2}\varepsilon-\frac{3}{16}\varepsilon^2  
           +\frac{19}{192}\varepsilon^3-\frac{185}{3072}\varepsilon^4
           +\frac{2437}{61440}\varepsilon^5+O(\varepsilon^6)\end{aligned}$$ while for small positive $\varepsilon$, $$\begin{aligned}
 A(\alpha) &\!=\!& 1-6^{1/2}\varepsilon ^{1/2}+2\varepsilon-\frac{6^{1/2}}{20}
 \varepsilon ^{3/2}-\frac{3}{5}\varepsilon ^2 +O(\varepsilon^{3/2}), \\
 B(\alpha) &\!=\!& 1+6^{1/2}\varepsilon ^{1/2}+2\varepsilon+\frac{6^{1/2}}{20}
 \varepsilon ^{3/2}-\frac{3}{5}\varepsilon ^2 +O(\varepsilon^{3/2}).\end{aligned}$$ For $i(\alpha)$, this implies that there is a jump only in the second derivative of at the transition, with $$i(\alpha) = \frac{3-e}{e}-\frac{1}{e}\varepsilon+
 \left\{ \begin{array}{ll} 
   \frac{1}{2e}\varepsilon^2+O(\varepsilon^3) & 
   \mathrm{for} \; \varepsilon < 0 \\ 
   & \\
   \frac{2}{e}\varepsilon^2+O(\varepsilon^3) & 
   \mathrm{for} \; \varepsilon > 0 
 \end{array} \right.$$ while $c(\alpha)$ and $l(\alpha)$ have a jump in the first derivative at the transition, with $$c(\alpha)=\left\{ \begin{array}{ll} 0 & \mathrm{for} \; \varepsilon < 0 \\
 \frac{12}{e}\varepsilon-\frac{4(6)^{1/2}}{e}\varepsilon
 ^{3/2}-\frac{54}{5e}\varepsilon^2 +O(\varepsilon^{5/2}) & \mathrm{for} \;
 \varepsilon > 0 \end{array} 
 \right.$$ and $$l(\alpha)=\left\{ \begin{array}{ll} 0 & \mathrm{for} \; \varepsilon < 0 \\
 \frac{12}{e}\varepsilon-\frac{54}{5e}\varepsilon^2 +O(\varepsilon ^3)
 & \mathrm{for} \; \varepsilon > 0 . \end{array} 
 \right.$$ The expansion for $l$ contains only integral powers of $\varepsilon$, and this may be related to the fact (see section \[sec:numsim\]) that the finite size corrections for the number of edges in the core $L_c$ are slightly nicer than the ones for the 