sambrookjm: …

Alm888: I believe it is as follows:
1) "Falcon Punch" perk was there before the devs revamped the HtH combat mechanic, so, for quite some time (I believe, till v1.07 inclusive) it was the only "upgrade" for Unarmed skill and it costed 7AP;
2) Then, the devs have introduced enhanced HtH skill system: from v1.08 onward HtH performance depends on the skill itself, character's STR (6 or 7 and above being mandatory, AFAIK) and (maybe) a character's level. Now, reaching certain thresholds in the skill adds base MAX and MIN damage (with a step of 20) and attaining "Sport's Master Candidate" (Unarmed >=120) and "Sport's Master" (Unarmed >=150) grants new moves;
3) These moves are superseded by "Falcon Punch", so it would be kinda useless to invest skill points to Unarmed past the level of "guaranteed hit" provided one has decided to get that perk, so the devs have thrown another benefit for the high Unarmed: passing certain thresholds decreases AP requirements for all Unarmed moves (don't know exact mechanics, but a fresh Lv.1 Cadet spends 3AP for punch and 4AP for kicks but after training the "prices" drop to 2 and 3AP respectively). I believe the reason "Falcon Punch" move costs only 5AP is due to these "High Unarmed Skill Buffs".

It is just that this HtH mechanic lacks proper in-game explanation.

Alm888: Sorry, I did not bother to look how these special traits are called in English version and just used their translated original namings (after all, a "Sensei"? Seriously? Are we in Japan? I doubt in 1986 there was even a word like this in the USSR!).
Under "Unarmed" I mean the skill that can be risen with skill points (via levelling up).
Also, a side note: every combat skill starts to gain % to critical chance after rising to 100. I believe it raises for 1% with every 10 skill points for normal attacks and 10% for aimed attacks/shots. The actual formulas are murky, unfortunately. It could be something like this (5% base crit. value and 180 Skill value with 40% boost for targeting head):

(5% * (1 + (180 - 100) / 100 ) ) * 140% = 12.6%

Sadly, there is no in-game description how all of this works. :(

TheSeaPeoples: I swear, I acquired Falcon Punch and it's just a regular attack. Doesn't even do more damage. Am I missing something? It does cost 6 AP, though.

sambrookjm: I was curious how the Falcon Punch ability would work with the Martial Arts character that I'm currently running, so I saved and respeced the character with the Oblivion Ampoule to try it out. I selected:

Way of the Warrior
Master of Dodging
First Blood
Falcon Punch

and set Marial Arts to 199, gaining the Martial Artist, Sensei, and Fatality Master distinctions.

The Falcon Punch description says it's a new unarmed technique worth 7 AP that always gives a critical effect, but only with the bare fist. I tried it out. It looks like there's a Falcon Punch ability that's 4AP for punching, and another Falcon Punch that's 5 AP with kicking. Does the Sensei ability gained at 150 Martial Arts override the 7 AP cost of the Falcon Punch? The Falcon Punch replaced the "Hammer" punch in the unarmed attack at 4 AP, and replaced the 5AP kick as well.

When you do hit with a Falcon Punch, the critical abilities weren't critical hits, but rather "The enemy is blinded" or "The enemy dropped their weapon." Is a critical hit one of the possible critical abilities from the Falcon Punch? I was hoping that the Unlucky Optimist (-5 Critical chance, but much higher critical damage), combined with an automatic critical hit from Falcon Punch would one-shot just about anybody. Sadly, that combination didn't work like I thought it would.