...Halfway through the fall it disappears.

What. It's not supposed to do that.

This interplanar portal does not really care what it's "supposed" to do.

Well, that sucks! He's not going to be less bored if he doesn't even know what happened to his notebook!

It lands outside an office building.

He's working late in the prosecutor's building; it's past dark by the time he leaves.

He doesn't see the notebook until he nearly trips over it.

One of his colleagues has a very strange sense of humor.

He flips it open to try to figure out which one. He can return it to them tomorrow.

• The human whose name is written in this note shall die.
  
• This note will not take effect unless the writer has the person's face in their mind when writing his/her name. Therefore, people sharing the same name will not be affected.
  
• If the cause of death is written within the next 40 seconds of writing the person's name, it will happen.
  
• If the cause of death is not specified, the person will simply die of a heart attack.
  
• After writing the cause of death, details of the death should be written in the next 6 minutes and 40 seconds.
  
• If the person using this Death Note fails to consecutively write names of people to be killed within 13 days of each other, then the user will die.
  
• If this Death Note is made unusable by tearing it up or burning it, all the humans who have touched the Death Note till then will die.
  

...One of his colleagues has a very, very strange sense of humor.

It's obvious nonsense, of course. He pockets it regardless; there's no need to worry any of his more credulous coworkers.

He drives home. 

There's no point in thinking about ridiculous ideas like magical notebooks that can kill people.

...Thinking about how it's pointless to think about things like that is not really much of an improvement, but apparently it's the best he's going to get.

The justice system is mostly efficient at meting out justice. Nearly everyone is tried the day after their arrest; the conviction rate is 99%. Almost no one gets away with serious crimes.

Almost no one is not, in fact, no one.

In his entire career as a prosecutor, he's only ever failed to win one trial. Formally speaking, it's a draw. Formally speaking, it doesn't count against him. No one (almost no one) would ever dream of telling him that the defendant was clearly going to be acquitted, if he hadn't taken poison on the witness stand.

Still, if their court system is good for anything at all, it surely ought to have managed to arrest Miyanagi Chinami and convict her of the murder that Onamida had been accused of. Instead, she hadn't even been properly questioned until seven months later, in August, and when the defense attorney (???) who had inexplicably been the one assigned to questioning her (????) had been literally poisoned, she had just walked free.

It's been a year since the initial incident.

What is even the point of a justice system that can't convict someone like that?

It would obviously be absurd to test this so-called "Death Note." It's not going to work, and if it did work then he would presumably either die or have to keep killing people.

(and how many more people is she going to kill before they finally manage to convict her of something?)

He looks up her name. Hardly any recent results — the cases were mostly kept out of the press — but he does find a result for the Yūmei University literature department's news page.

Halfway down the page, there's a short article about a cross-department event with the art students. She's in one of the photos, holding hands with another student with dark spiky hair and a pink sweatshirt.

Nothing is going to happen. He's going to write her name down and she'll survive and he'll feel very silly for even considering the possibility that any of this could be real.

美柳Miyanagi ちなみChinami — Accidental death, alone, at 5:00 AM on March 29.

He never sleeps very well. It's not surprising that tonight is no exception.

Not only are there no reports of her death the next day, someone actually uploads a picture of her to the literature department's Instagram page!

Obviously it was just superstition, or more likely a prank or inside joke. He never really expected it to work.

The day after that, her face is on every news station in the city.

Apparently she ... had an identical twin that no one knew about? Who found her dead body and immediately confessed to being complicit in several major crimes?

As a prosecutor, he can, of course, access the autopsy report.

Of course.

She appears to have died of blunt force trauma after slipping on the floor. The time of death is a little uncertain given how long it was before her body was discovered, but it's estimated as being between 4:00 and 6:00 AM on March 29. No foul play is suspected at this time.

He just killed someone.

She was an awful person and he absolutely does not regret it and he just killed someone.