  Microscopic models of realistic thermodynamic systems usually involve a
number of parameters, not all of equal macroscopic relevance. We examine a
decorated $(1+3)$ Ising spin chain containing two microscopic parameters: a
"stiff" $K$ mediating the long-range interactions, and a "sloppy" $J$ operating
within local spin groups. $K$ dominates the macroscopic behavior, and varying
$J$ has weak effect except in regions where $J$ brings about transitions
between phases through its conditioning of the local spin groups with which $K$
interacts. We calculate the heat capacity $C_H$, the magnetic susceptibility
$\chi_T$, and the thermodynamic curvature $R$. For large $|J/K|$, we identify
four magnetic phases: ferromagnetic, antiferromagnetic, and two ferrimagnetic
ones, according to the signs of $K$ and $J$. We argue that for characterizing
these phases, the strongest picture is offered by the thermodynamic geometric
invariant $R$, proportional to the correlation length $\xi$. This picture has
correspondences to other cases, such as fluids.
