9 ing in the classical Heisenberg model tends to increase with lowering the dimensionality of the spins and reaches its largest value in the Ising model. This implies that the r.h.s. part of the phase diagram should generically exhibit a higher specific heat. In view of the RFOT-derived con- nection between the fragility index and the heat capacity jump, i.e. the mentioned m ≃ 34 . 7Δ c$_{p}$ relation, we may expect that substances with a larger J$_{compr}$/J$_{shear}$ ratio will generically be more fragile, but under several, rather restrictive circumstances, as we discuss next. First, we reiterate the present approach is a minimalist way to explicitly account for the interactions that give rise to (high frequency) shear resistance in supercooled liquids. Already in this minimal model, the entropy and heat capacity per bead depend on the bead size a , which is, formally, the ultraviolet cut-off in the theory. In ad- dition to the purely volumetric affect - one vector occu- pies a volume a 3 - the bead size will affect the precise, self-consistently determined value of the order parameter g in the glass phase. Further, the isotropic assumption for local elastic response, from Eq.(11), is an approxi- mation. Contributions other than purely elastic terms in Eq.(6) will be generally present as well. One source of such contribution is reconfiguration-induced electric dipole moment. Earlier estimates 39 suggest that even though the local polarization resulting from bead move- ments contributes only about a percent to local elastic constants, the resulting elemental electric dipoles inter- act comparably strongly to the elastic dipoles of the type we have considered. This effect should be especially sig- nificant in ionic glasses. 16 Interestingly, because of the mentioned antiferromagnetism of the electric dipoles, the electric and elastic interactions may be mutually frustrat- ing. A careful treatment of polymeric materials, on the other hand, should include the effects of chain rigidity and other types of local anisotropy that could not be ac- counted for by only two elastic constants and the bead size a , which are the parameters of our ansatz. Novikov and Sokolov have argued electronic contributions must be considered in metallic glasses. 40 According to Fig.4, we should expect that actual sub- stances will exhibit a continuous range of local stress dis- tributions and heat capacities. Indeed, to distinct points on the phase diagram in Fig.4, there correspond very dif- ferent values of the heat capacity. Furthermore, the heat capacity of an anisotropic Heisenberg model on a peri- odic lattice should exhibit a non-monotonic behavior as a function of anisotropy, namely a spike at a transition that would occur at J$_{compr}$ = J$_{shear}$ in the mean-field limit. J$_{compr}$ = J$_{shear}$ corresponds to the following value of the ratio of the longitudinal and transverse velocities: c$_{l}$/c$_{t}$ ≃ 1 . 63. This is roughly consistent with a very broad range of fragilities observed for substances from a rela- tively narrow range of the c$_{l}$/c$_{t}$ ratio centered at 1 . 75 or so. 41 We point out that the correlation between the Pois- son ratio and the fragility has been a subject of debate for some time. 36,37,40–43 Our results suggest that a corre- lation between the fragility and the Poisson ratio might in fact be expected for non-metallic substances that are far from the transition region J$_{compr}$ = J$_{shear}$ . Yet for substances near the “critical region” J$_{compr}$ = J$_{shear}$ , lit- tle correlation is expected. Note that the view of the activated regime in a fragile substance as a random Ising model below symmetry breaking is consistent with re- cent results of Stevenson at el. 44 who have mapped the localization transition in fiducial liquid structures of a Lennard-Jones mixture onto a replica-symmetry break- ing transition in a random Ising model. Regardless of the detailed value of the heat capacity, our results indicate that supercooled liquids exhibit local stress distribution ranging from frozen-in compression to frozen-in shear. Furthermore, the present findings that the activated liquid regime arises self-consistently from a mechanically stable reference state merge nicely with the self-consistent phonon view of the emergence of the aperi- odic crystal state from the uniform liquid . 1,19 This notion suggests that a unified, quantitative treatment of the liq- uid, aperiodic-crystal, and periodic-crystal regimes is in sight. We conclude by reiterating that despite system- specific variations in direct interactions, accounted for, semi-empirically, in the variation of the elastic constants in Fig.4, the underlying mechanism of activated transport in non-polymeric liquids is system-independent, consis- tent with the conclusions of the RFOT theory. Acknowledgments : The authors gratefully ac- knowledge the Arnold and Mabel Beckman Foundation Beckman Young Investigator Award, the Donors of the American Chemical Society Petroleum Research Fund, and the Small Grant and GEAR Programs at the University of Houston, for partial support of this research. It is sometimes convenient to present the six indepen- dent entries of a symmetric tensor u$_{ij}$ as a six-component vector: Appendix A: Voigt-Mandel notation { u$_{ij}$ | i, j = 1 , 2 , 3 } → ⃗ t ≡     u$_{11}$ u$_{22}$ u$_{33}$ √$_{2}$$_{u}$$_{23}$ √$_{2}$$_{u}$$_{31}$ √$_{2}$$_{u}$$_{12}$    $^{,}$ (A1)   Λ 1111 Λ 1122 Λ 1133 √$_{2Λ}$$_{1123}$ √$_{2Λ}$$_{1131}$ √$_{2Λ}$$_{1112}$ Λ 2211 Λ 2222 Λ 2233 √$_{2Λ}$$_{2223}$ √$_{2Λ}$$_{2231}$ √$_{2Λ}$$_{2212}$ Λ 3333 Λ 3322 Λ 3333 √$_{2Λ}$$_{3323}$ √$_{2Λ}$$_{3331}$ √$_{2Λ}$$_{3312}$ √$_{2Λ}$$_{2311}$ √$_{2Λ}$$_{2322}$ √$_{2Λ}$$_{2333}$ 2Λ 2323 2Λ 2331 2Λ 2312 √$_{2Λ}$$_{3111}$ √$_{2Λ}$$_{3122}$ √$_{2Λ}$$_{3133}$ 2Λ 3123 2Λ 3131 2Λ 3112 √$_{2Λ}$$_{1211}$ √$_{2Λ}$$_{1222}$ √$_{2Λ}$$_{1233}$ 2Λ 1223 2Λ 1231 2Λ 1212   . (A2) so that the convolution u$_{ij}$ Λ$_{ijkl}$ u$_{kl}$ is expressed as bilin- ear form ∑ 6 α,β $_{=1}$t$_{α}$ Λ$_{αβ}$ t$_{β}$ , where Λ$_{αβ}$ is a square 6 × 6 matrix: In the isotropic case, see Eq.(11), this matrix is rather sparse and easily shown to have one eigenvalue 3 K and