  The single-ion anisotropy and magnetic interactions in spin-ice systems give
rise to unusual non-collinear spin textures, such as Pauling states and
magnetic monopoles. The effective spin correlation strength ($J_{eff}$)
determines the relative energies of the different spin-ice states. With this
work, we display the capability of capacitive torque magnetometry in
characterizing the magneto-chemical potential associated with monopole
formation. We build a magnetic phase diagram of Ho$_2$Ti$_2$O$_7$, and show
that the magneto-chemical potential depends on the spin-sublattice ($\alpha$ or
$\beta$), i.e., the Pauling state, involved in the transition. Monte-Carlo
simulations using the dipolar-spin-ice Hamiltonian support our findings of a
sublattice-dependent magneto-chemical potential, but the model underestimates
the $J_{eff}$ for the $\beta$-sublattice. Additional simulations, including
next-nearest neighbor interactions ($J_2$), show that long-range exchange terms
in the Hamiltonian are needed to describe the measurements. This demonstrates
that torque magnetometry provides a sensitive test for $J_{eff}$ and the
spin-spin interactions that contribute to it.
