What is the criteria of "Foxiest of all Hounds"?

Solution 1:

From this answer

Note that the Smooth Operator reward is an easier version of Ghost. (If you get "Ghost" you will always get "Smooth Operator" as well.) Smooth Operator is used for the Foxiest of the Hounds achievement. It deals almost only with alarm panels and security-system alarms, rather than "alarmed" guards

Solution 2:

1. It is tied to the Smooth Operator reward, which is given when you pass certain plot-points without triggering electronic alarm systems involved in the plot points. This triggering can occur through cameras, guards pressing alarm panels, probably failed hacks, and possibly hostile turrets or bots seeing you.
2. It has nothing do with the "Alarmed" state shown in the minimap for human guards. I think this completely-misleading wording is an error on behalf of the developers, and I hope they change the wording, perhaps to "suspicious" for guards.
3. As far as I know, the only alarm systems that exist are ones tied to the main missions, although it is possible for side-missions to be occurring at the same time. (Ex: Police station.)
4. You do not have to worry about times when the alarms go off because of the plot or game-scripting, only alarms that you could have bypassed. (Ex: TYM Penthouse.)
5. Areas that don't have any alarm or security systems cannot give you Smooth Operator, and FoTH cannot be failed there no matter how obnoxious you are. (e.g. The construction-site crash.)