I was taught vertical fist as the main chu'an (拳 "fist") in tai chi chu'an. It seems to come from the Yang style, although I have it through Fu, as Yang and Fu "traded ideas" (pushed hands with each other:)
Here's a video of a Yang stylist that shows the general combination, although the practitioner here is too tight and doesn't leave the open "intercept" hand extended, which would be controlling the opponents arm or shoulder so the punch can coil in. The deflect fist doesn't have to be that downward, and I find it more effective to keep it largely horizontal. (Fu style tends to be more extended and circular than Yang because Fu was mainly known for Bagua, but Yang Chengfu was the real deal.)
This is, in my experience, the best punch. Punching like hsing yi typically means your tai chi technique isn't yet good enough (which is partly why you learn hsing yi!), and even backfist chops in tai chi aren't close to the best strikes.
If used as a true tai chi punch, where there is no focus until after the fist contacts the torso, it's ideal because the combination leaves you inside in a rooted position, so you can use your waist to focus (put force directly into the opponent's body.) That's very difficult to do, but less difficult with this technique.
Ideal target is the kidney, and the with the possibility of rupturing it and ending combat.
(Most people probably cannot rupture organs with a tai chi punch—I've never had a chance to validate it in actual combat, only sparring, where the combination overall is incredibly effective, because the setup fully neutralizes the opponent.)
But this punch can be used as powerful push also, or a less powerful strike to severely disincentivize without trying to do serious damage. It is certainly not pleasant, even with very little focus, if you're focusing into the top two knuckles.
You also see this in Hsing-yi Beng Quan, which is the most linear hsing-yi strike, described as an "arrow" that has smashing power. From the wiki:
The strikes or blows are driven up in a diagonal from the muscles of the rear foot, through the muscles of the torso and out the striking arm. This all combines to give Beng Quan a simple and straightforward power which utilises the entire body's momentum and mass.
You also see it in Pao Quan (cannon fist), but there it is typically less powerful.
For me the ultimate advantage of horizontal fist for internal arts specifically is:
- Horizontal allows you to put forward and then downward, directly into an opponents body, via the first two knuckles, by rolling the top of the fist downward after contact.
This works even if you only focus after contact with the body, and it does not take much power to damage the tissue when it's focused with the down-and-in via the first two knuckles.
Even where it's not a KO move, it is highly disincentivizing, which can be validated by doing to gently against a sparring partner, or have it done to you. The biggest "ouch" factor seems to come from compacting the soft tissue into the ribs.