What determines when research can be completed instantly?

Sometimes after certain research is completed, the game tells me other research can be completed instantly.

• Which research can be completed instantly, and when? (so far it's only been autopsies, so my guess is that it has to do with number of corpses? If so, how many?)
• Does it differ by difficulty?

Solution 1:

There are a number of factors that determine when autopsy's can be completed instantly.

The first is simply a question of corpse numbers. You need at least this many corpses to receive instant autopsies, and you will need to spend that many corpses to research it at instant speed. If an autoposy is not listed below, it can never be completed instantly (e.g. Advent Officer):

• Sectoid - 6
• Viper - 5
• Muton - 5
• Berserker - 5
• Archon - 3
• Gatekeeper - 3
• Andromedon - 4
• Faceless - 3
• Chyrssalid - 15 (yes, really)
• Advent Trooper - 10
• Advent Stunlancer - 4
• Advent Shieldbearer - 4
• Advent MEC - 3
• Advent Turret - 3
• Sectopod - 3

The second is simpler -- the opportunity for instant research only pops up when you finish a research project. If you Acquire 10 Muton corpses, Instant Muton Autopsies will become available when you next complete a research task (whatever it is).