Thermal expansion and magnetoresistance measurements as a function of temperature (T) and magnetic field (B), respectively, were carried out on Os-substituted ${\mathrm{URu}}_2{\mathrm{Si}}_2$ $\left({\mathrm{URu}}_{2-x}{\mathrm{Os}}_x{\mathrm{Si}}_2\right)$ single crystals with $x=0$ to 0.28. Mean-field-like peaks at temperature $T_0$ in the linear thermal expansion coefficient vs T curves in both the hidden order (HO) phase and the large-moment antiferromagnetic (LMAFM) phase of ${\mathrm{URu}}_{2-x}{\mathrm{Os}}_x{\mathrm{Si}}_2$ exhibit a striking resemblance and a discontinuity in $T_0\left(x\right)$ at $x_{\mathrm{c}}=0.15$ delineating the boundary between the two phases. Anomalies at $x_{\mathrm{c}}$ were also seen in both the calculated effective Grüneisen parameter and the energy gap associated with the HO or LMAFM phase. The hydrostatic pressure derivative of $T_0$ calculated from thermal expansion and specific heat [D. L. Kunwar et al., Phys. Rev. B 105, L041106 (2022)] data agree with the experimental results [C. T. Wolowiec, Proc. Natl Acad. Sci. USA 118, e2026591118 (2021)], whereas the calculated uniaxial pressure derivatives in different crystallographic directions are anisotropic, as shown in the anisotropic linear thermal expansion. This anisotropy was further reflected in the radically different responses of $T_0$ to B applied in different directions: a few percent change in $T_0$ is induced for B parallel to the $c$ axis, whereas $T_0$ barely changes for B applied in the $ab\mathrm{plane}$ of the ${\mathrm{URu}}_{2-x}{\mathrm{Os}}_x{\mathrm{Si}}_2$ single crystals, regardless of whether the crystals are in the HO or LMAFM phase.

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• Electrical conductivity
• Exotic phases of matter
• Magnetic anisotropy
• Magnetic order
• Magnetotransport
• Thermal expansion