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ZTF SN Ia DR2 follow-up: Exploring the origin of the Type Ia supernova host galaxy step through Si II velocities

The relationship between Type Ia supernovae (SNe Ia) and their host galaxy stellar masses is well documented. In particular, Hubble residuals display a luminosity shift based on host mass, known as the mass step, which is often used as an extra correction in the standardisation of SN Ia luminosities. Here we investigate Hubble residuals and the mass step in the context of Si II $λ6355$ velocities, using 277 near-peak SNe Ia from ZTF DR2. We divide the sample into high-velocity (HV) and normal-velocity (NV) SNe Ia, separated at 12,000 km/s, resulting in 70 HV and 207 NV objects. We then examine links between Si II $λ$6355 velocities, light-curve stretch $x_{1}$, colour $c$, and host properties to explore potential environmental and/or progenitor-related effects. Although we only find a marginal difference between the Hubble residuals of HV and NV SNe Ia, the NV mass step is $0.149 \pm 0.024$ mag ($6.3σ$), while HV SNe Ia show $0.046 \pm 0.041$ mag ($1.1σ$), consistent with zero. The NV-HV mass-step difference is $\sim 2.2σ$. The clearest subtype difference is seen in central regions ($d_{DLR} < 1$), where NV SNe Ia show a strong mass step but HV SNe Ia none, yielding a $3.1-3.6σ$ difference. A host-colour step appears for both: NV $0.142 \pm 0.024$ mag ($5.9σ$) and HV $0.158 \pm 0.042$ mag ($3.8σ$). Overall, NV and HV colour steps are consistent. HV SNe Ia show modest ($\sim 2.5$-$3σ$) steps in outer regions ($d_{DLR} > 1$), while NV SNe show stronger environmental trends. Thus, NV SNe Ia appear more environmentally sensitive, especially in central, likely metal-rich and older regions, while HV SNe Ia show weaker, subset-dependent trends, and applying a universal mass-step correction could introduce biases. Refined classifications or environment-dependent factors may improve future cosmological analyses beyond standard $x_{1}$ and $c$ cuts.