## 2 Results
### Magnetic Incommensurability, Anisotropy, and Their Role in Cooper Pairing  
One of the long debated questions for Sr\({}_{2}\)RuO\({}_{4}\) has been the strength of magnetic anisotropy at the IC vector and its implications for superconductivity. In the early part of the century, several INS [38; 39; 40; 41; 42] and NMR [43] measurements were performed to probe the anisotropy (\(\chi_{z}^{\text{IC}}/\chi_{ab}^{\text{IC}}\)) of the physical magnetic susceptibility. Barring some initial ambiguities related to the usage of un-polarised neutrons for measurements of magnetic anisotropy (a nice discussion can be found in [PERSON] et al. [38]), over the years, a consensus  
Figure 5: Superconducting eigenvalues: singlet-triplet scale separation and SDW: (**a**) We plot the relative strength of the leading singlet eigenvalue in comparison to the triplet eigenvalue (\(\lambda_{s}/\lambda_{t}\)) extracted by solving the multi-orbital Bethe–Salpeter equation in the superconducting channels (both singlet and triplet) as functions of temperature. With strain, \(\lambda_{s}/\lambda_{t}\) increases, and the trend becomes more prominent with decreasing temperature. (**b**) In the unstrained compound, the leading eigenvalue (\(\lambda_{s}\)) singlet superconducting instability has a slightly steeper temperature dependence in comparison to the eigenvalue (\(\lambda_{SDW}\)) for the SDW instability. However, under strain, beyond \(\epsilon_{x}=0.6\%\), the SDW instability becomes the leading instability of the system upon lowering temperatures. built that the anisotropy \(\chi_{zz}^{IC}/\chi_{ab}^{IC}\) is \(\sim\)2.5. A series of theoretical studies [44; 45; 46; 47; 48; 49; 50] followed to argue that this anisotropy is the key factor that can drive the triplet superconductivity in Sr\({}_{2}\)RuO\({}_{4}\). These calculations and the inferences therefrom also strongly contrasted the theoretical results from [PERSON] and [PERSON] [51; 52]. [PERSON] and [PERSON] did not include the magnetic anisotropy in their theoretical calculations, and they showed that in Sr\({}_{2}\)RuO\({}_{4}\) a singlet d-wave Cooper pairing should be favoured due to the spin fluctuations at the IC vector. However, until late 2018, the superconductivity in Sr\({}_{2}\)RuO\({}_{4}\) was believed to be primarily of spin triplet origin, unless the situation dramatically changed with the careful NMR measurements performed recently by [PERSON] et al. [4]. Furthermore, during this period, almost all neutron studies failed to pick up any magnetic susceptibility at the FM vector from INS measurements. In view of this persisting experimental situation between 1994 and 2018, non-observing of FM spin fluctuations in INS and NMR results suggesting a triplet pairing scenario, the theoretical studies argued in favour of a triplet pairing scenario mediated primarily by the magnetic anisotropy at the IC vector. An opposing view was presented in a recent INS study by [PERSON] et al. [53]. They observed a significant amount of spin susceptibility near the FM vector, but concluded that the fluctuations were too weak to form enough glue for purely triplet superconductivity. We built our theoretical ability recently [27], and in our previous work [17] on Sr\({}_{2}\)RuO\({}_{4}\), we showed that it is primarily the relative strengths of the spin fluctuations at the IC and FM vectors and their coherence that determine \(\lambda_{s}/\lambda_{t}\). However, we also showed that the charge susceptibility has an intricate momentum-dependent structure, and the superconductivity is a product of the interplay of both the spin and charge fluctuations. However, on a more fundamental level of electronic interactions, the rotationally invariant quantum Hamiltonian that we solve also includes important spin-flip scattering and pair-hopping terms. These interaction terms are crucial to incorporate the relevant spin dynamics in Sr\({}_{2}\)RuO\({}_{4}\) that is neither fully itinerant nor fully localised.