not be possible to recover $s_{e,t}$ from $\psi_{\beta_1,\beta_2}$. Inequivalence can result from the existence of graphons maximizing $s_{e,t}$ for some $(e,t)$ which are not maximizers of $\psi_{\beta_1,\beta_2}$ for any $(\beta_1,\beta_2)$. Specific instances of such loss of information in $\psi_{\beta_1,\beta_2}$ are shown in a future paper [@RS], but one consequence can already be seen in the transitions, studied previously [@PN; @CD; @RY] in the grand canonical ensemble, between independent-edge graphs across a phase transition curve in the phase space; see Figure 2 for the Strauss model. Such ‘free particle’ graphs, with only edge density $e$ as a variable, optimize $\psi_{\beta_1,\beta_2}$ for $(\beta_1,\beta_2)$ throughout the upper half of the grand canonical phase space, so $e$ is a function of $(\beta_1,\beta_2)$ off the transition curve there; see [@RY] for details. These graphs all lie on the curve $t=e^3$ in Figure 1, not a 2-dimensional region in that microcanonical phase space, making it difficult to use singularities of the free energy $\psi_{\beta_1,\beta_2}$ to imply singularities of the entropy $s_{e,t}$. For this reason we have focused here on phase transitions in the lower region of the microcanonical phase space, Figure 1.

.4truein ![The curve of all singularities of $\psi_{\beta_1,\beta_2}$ for $\beta_2>-1/2$[]{data-label="trans"}](transition.eps "fig:"){width="3in"}

We have shown 