At room temperature, bulk La$_{7/8}$Sr$_{1/8}$MnO$_3$ is in the dynamic Jahn--Teller O$^*$ phase, but undergoes a transition to a static, magnetically ordered Jahn--Teller phase at lower temperatures. Here we study a $6$ unit cells thin film of this compound grown on SrTiO$_3$, resulting in small compressive strain due to a lattice mismatch of $lesssim 0.2%$. We combine X-ray absorption spectroscopy with multiplet ligand field theory to study the local electronic and magnetic properties of Mn in the film. We determine the Mn $d_{3z^2-r^2}$ orbital to be $0.13;text{eV}$ lower in energy than the $d_{x^2-y^2}$, which is a disproportionately large splitting given the small degree of compressive strain. We interpret this as resulting from the strain providing a preferential orientation for the MnO$_6$ octahedra, which are strongly susceptible to such a deformation in the vicinity of the phase transition. Hence, they collectively elongate along the $c$ axis into a static Jahn--Teller arrangement. Furthermore, we demonstrate the strongly covalent character of La$_{7/8}$Sr$_{1/8}$MnO$_3$, with a contribution of nearly $50%$ of the one-ligand-hole configuration $d^{5} underline{L}^1$ to the ground state wavefunction. Finally, we find the system to be in a high-spin configuration, with the projection of the local magnetic moment on the quantization axis being about $3.7;mu_{text{B}}/text{Mn}$. We show, however, that the system is close to a high-spin--low-spin transition, which might be triggered by crystal field effects.
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