From the Activision Support website for this game:
"To get a 100% progression in the ending result screen or the Manhattan Project achievement/trophy, you must:
Explore every square of the map.
Collect every item throughout the game."
Items include:
Health upgrades (8)
Pizza Box upgrades (2)
Smoke Bomb upgrades (5)
Shuriken upgrades (20)
Chi Upgrades (4)
Ice Cream Kitty collectibles (exp. booster) (10)
After you beat the final boss, you'll be returned to your last save point when re-opening the game. If you pause and open your map it will tell you how many items there are left to collect in that map area (The east and west components of each section are combined in this item tally).
Unfortunately, it will NOT mention how many Ice Cream Kitty collectibles are left, so I would keep a close tally of where you pick them up.
As for exploring every square of the map, you must ensure that the perimeter of all of your maps has a white border. Wherever there is not a white border indicates an unexplored square.
I would also closely compare your maps in the pause menu with these completed maps done by Scharn73 on playstationtrophies here:
https://web.archive.org/web/20150423121014/https://www.plays...It seems like every 2 squares of the map explored counts for 0.1% completion.
Some common unexplored areas include the top of the map you may have to use a vertical smoke bomb dash to get to, 1 square in the left elevator shaft of the TCRI building, 2 squares in the right elevator shaft of the TCRI building (use the climbing claws to climb up the elevator wire to get close), areas behind shock/bomb shuriken doors in previously explored sections, and the top half (2 squares) of the final boss arena (I had to use a vertical smoke bomb dash to get to each square).
Notes:I would highly recommend using Scharn73's complete guide here:
http://www.playstationtrophies.org/forum/teenage-mutant-ninj...Especially to get all the ice cream kitty collectibles and for the under one hour speedrun.
The special conversations (see: The Conversationalist achievement guide) do not have to be completed to achieve 100%.
It is very helpful to explore/collect along the way as much as possible so you don't have to backtrack as much, and then defeating the final boss will help you find the last missing items (minus ice cream kitty) in the pause menu, but the indicator does not tell you where in the map section they are so it can still be frustrating.
The achievement pops as soon as you reach 100% (you don't have to beat the final boss again).
Good luck, this can be quite maddening so keep close track of your ice cream kitties and explore every square of the map.
Edit: updated map link