  We calculate the specific heat of a weakly interacting dilute system of
bosons on a lattice and show that it is consistent with the measured electronic
specific heat in the superconducting state of underdoped cuprates with boson
concentration $\rho \sim x/2$, where $x$ is the hole (dopant) concentration. As
usual, the $T^3$ term is due to Goldstone phonons. The zero-point energy,
through its dependence on the condensate density $\rho_0(T)$, accounts for the
anomalous $T$-linear term. These results support the split-pairing mechanism,
in which spinons (pure spin) are paired at $T^*$ and holons (pure charge) form
real-space pairs at $T_p < T^*$, creating a gauge-coupled physical pair of
charge $+2e$ and concentration $x/2$ which Bose condenses below $T_c$,
accounting for the observed phases.
