Mycroft is still in town, delighting certain recappers… by which I mean me. He spends the first half of the episode trying to get Sherlock to come have dinner at the Diogenes, and the second half trying to get Sherlock to leave New York and come back to London. His claim is that their father is going to cut off his trust fund and access to the brownstone if Sherlock doesn't come home. Sherlock decides not to do it, but he doesn't learn that Mycroft was lying about everything. He's got some kind of scam going on, apparently.
Meanwhile, a woman falls from a balcony onto a delivery truck. The only reason she hit the truck is so Sherlock can show off how good he is at deducing where the truck was when she hit it. She was stabbed in the stomach before she fell, so it's clearly murder. And she's a poor person who was suddenly moved to a big apartment by a billionaire software genius named Ian Gale, so the immediate suspicion is that he killed her.
The first stop is Ian Gale's lawyer, who claims that Gale is in Kuala Lumpur. But it's just a lookalike who is two inches shorter than the man he's imitating, and Sherlock dismisses this immediately. After a bit more research, Holmes locates Gale hiding out at a hotel where he's reserved an entire floor under a fairly obvious pseudonym. And he's trying to recover from a heart transplant, so he probably wasn't out stabbing ladies and throwing them off balconies. In fact, the victim wasn't even his mistress; she was his illegitimate daughter, who he brought to town in hopes of getting some blood. He's got a special condition where only his relatives can donate blood.
Watson checks out the corpse and finds out the stabbing was done with surgical precision, so she accuses Ian Gale's wife who used to be a pediatric surgeon. After all, Ian Gale changed his will so some of his money stopped going to his wife and started going to his daughter. But that money didn't revert to the wife even after the daughter died, so this goes nowhere. Holmes and Watson go back to the apartment and Holmes finds evidence on the DVR that whoever was in the apartment is probably out betting on horses. The police bring in the victim's ex-boyfriend, but he's got a good alibi: he was out buying medicine for the flu his girlfriend seemed to have.
So that brings up the question of why she was donating blood while she had a flu. This doesn't really get explained, but it leads to the solution: Ian Gale's wife wanted to kill him because she'd get a lot more money that way than if she'd just divorced him. So she stole some tissue from his new heart and injected it into his daughter, see, and then she started generating antibodies, which she took and injected into her husband. Whatever; it's a bunch of medical mumbo-jumbo. The point is that the wife did it.
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Holmes and someone do some single-stick sparring on the roof. It's very picturesque. They have eggs on their heads, and when Holmes smashes his opponent's we see that it's Mycroft. Sherlock didn't go to the opening of the Diogenes, so Mycroft wants him to come to dinner there before he goes home to London. Holmes was hoping that hitting each other would be enough bonding. They put their masks back on and have at it. A delivery man brings a package out of his truck and asks a bystander for the word of the day: "borborygmus." That's the sound of a stomach rumbling. If you spell it "Borborygmos," it's also the name of a legendary cyclops in Magic: The Gathering. There's a loud clang from the direction of the delivery truck, but nothing seems to have changed. When the man drives away, there's a woman with a stomach wound on the roof of the truck. The usual gang of cops and consultants are gathered around the roof. She was stabbed and the driver doesn't know when she hit the roof. Holmes goes inside the truck, looks at the way the roof is indented, and notes that the shelf marked A.1-178 is lower than the shelf marked A.1-179. The driver is called to come in, and he confirms that everything is loaded in delivery order. Holmes wants to know where box A.1-178 was delivered, and the driver asks why. Bell tries to get out of this without a Sherlockian explanation, but it's too late for that. There are paint chips all over the inside of the truck, but some of them show that boxes were there during the impact.
Bell, Holmes, and Watson look up at the tall buildings in the area where the woman presumably hit the truck. Watson suggests starting with the third balcony up, because it's the only balcony that doesn't have the same tree as all the other ones. Maybe it got knocked over when a woman fell off. Sure, that's worth a try.
Now they're in the apartment. There are blood drops on the carpet and signs of a struggle. Holmes compliments Watson on a job well done. Bell says the superintendent doesn't know the name of their victim because everything is very private here. And there are no cameras. He goes off to call the management company. Holmes investigates shoes and jewelry while he and Watson discuss his relationship with Mycroft. Watson finds a purse and reports that the victim's name was Haley Tyler, age 26. From the closet, Holmes guesses that the address on the drivers' license isn't this apartment, and he's correct: she's from Mineral Wells, Texas. Someone had old, cheap clothes and new, expensive clothes. So he thinks she was a kept woman. Bell reports that the apartment is owned by "Ian Gale," who invented an expensive phone and half the computers in the country.