The Ce$^{3+}$ pseudospin-$frac{1}{2}$ degrees of freedom in the pyrochlore magnet Ce$_2$Zr$_2$O$_7$ are known to possess dipole-octupole (DO) character, making it a candidate for novel quantum spin liquid (QSL) ground states at low temperatures. We report new heat capacity ($C_p$) measurements on Ce$_2$Zr$_2$O$_7$ which rise sharply at low temperatures, initially plateauing near 0.08 K, before falling off towards a high temperature zero beyond 3 K. Above $sim$0.5 K, the $C_p$ data set can be fit to the results of a quantum numerical linked cluster (NLC) calculation, that allows estimates for the terms in the XYZ Hamiltonian expected for such DO pyrochlore systems. Fits of the same theory to the temperature dependence of neutron scattering and magnetic susceptibility complement this analysis to produce robust estimates of the exchange parameters. Polarized neutron diffraction results show spin flip scattering with pinch points, largely consistent with predictions from the NLC calculation. However, the non-spin flip structure factor shows zone-boundary scattering similar to that observed in the dipolar spin ice Ho$_2$Ti$_2$O$_7$ and likely arising from long-range dipolar interactions. We conclude that Ce$_2$Zr$_2$O$_7$ realizes a U(1)$_pi$ QSL state at low temperatures, and one that resides near the boundary between dipolar and octupolar character. We then interpolate between the NLC-calculated $T$ $>$ $sim$0.5 K $C_p$ and a low temperature form for $C_p$ that is consistent with both the calculated near-neighbour exchange parameters and a $T^3$ $C_p$ appropriate for emergent gapless photons. This low temperature form for $C_p$ allows for $Rln(2)$ entropy to be accounted for up to $T$ = 10 K. Phenomenologically, the entropy recovery above the plateau in $C_p$, near $T$ = 0.08 K, gives $Rln(2)$ less $frac{R}{2}ln(frac{3}{2}$), the missing Pauling, spin ice entropy.
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