Wow either that neighbor didn't bother to knock or that door magically opened on its own.
The timeline for Satoru killing his mother is wonky as fuck. The only way it reasonably makes sense is if the killer killed his mother roughly 10 to 15 minutes before Satoru came home and promptly called the cops, thus to frame him. The anime presents this in a different fashion in that Satoru was just leaving the store when his mother was killed. Now the show doesn't bother establishing how far his house is away so I guess it is possible even though a bit improbable. I'm also assuming the murderer didn't call the cops from his cell phone so he must have ran to a public phone, called them and then ran back? 911 calls in the US are always recorded, I have to assume something similar is in place in Japan. All of this doesn't really work if we're talking a time frame of roughly 15-20 minutes. Honestly it doesn't really make sense that the murderer would come back at all but we gotta have dramatic flair. Oh and Satoru never touched the murder weapon. Of course the killer wore gloves so there would have been no prints on the murder weapon. However it also possibly eliminates Satoru due to the time window. He would have had to murder his mother, dispose of the gloves, and then come back. Now if he still had the gloves on him, why would he take them off? If he wanted to stage it as a break-in, why would he call the cops and then run away?
Then we have the reporter. Surely the police interviewed the reporter right? The phone call the mother made to him would have been logged. He would have explained that she was looking into an old case and possibly found a new suspect. Then less than 24 hours later she was murdered. Unless they didn't bother checking her calls and the reporter never brought it up to the police, despite it seemingly being highly relevant to his job as well.
How? How? How? My mind is full of fuck.
Why would Satoru send Airi a text from his mother's phone? Was it to avoid potentially turning his phone on (yes I know Airi snapped it but I'm talking about the police perspective)? However they would also be monitoring his mother's phone so that doesn't make any sense. In fact it seems like they did as they suspected Airi. But why would they suspect her of helping? There was literally nothing she could have done to help him murder his mother.
Literally nothing as she was at the store when he left and she was presumably nowhere near the crime scene. Not to mention there were apparently people that saw Satoru run into the building after the fire had started, so how the fuck does that make a lick of sense? He tried to silence her but then regretted it and tried to save her life and the risk of his own? wut
