• TRUMP's Oh Shit It's A Planet-Sized Bomb module has determined that only certain hexagonal regions of the sphere are explosive.
• TRUMP's gaming module has determined that this appears to be a game of Minesweeper.
• TRUMP's deja vu module has determined that the explosive and non-explosive regions surrounding the colored numbers represent ones and zeroes.

Solution

The sphere is primed to explode because this is a game of Minesweeper. The mines are all in the hexagons, and only the pentagons are numbered. The orange 1 next to the blue 4 makes it possible to fill in three mines in the three regions that are adjacent to the 4 but not the 1, and the puzzle falls logically from there (Ian says: "which is actually a small miracle, because I set up the desired answer configuration and then checked to see if the puzzle was solvable."):

Once all the mines are in place, it's time to use the fact that the colored numbers are all oriented toward one of the surrounding hexagons; these orientations provide starting points for clockwise examinations of the surrounding mines. Around the red 2, there are three hexagons with no mines followed by two hexagons with mines. This gives a binary 00011, which is C (binary should have been fresh in everyone's mind after SENSES.) These letters, in rainbow order, spell CARBON, which is a satisfying ending answer for several reasons (sixth element, C60 looks like a soccerball, the ball of evil in the movie did basically turn into a ball of carbon). This is Ian's favorite puzzle of the Game. (Once again, though, the answer was too appropriate, and Briny Deep guessed it in 7 minutes!)

We were going to make you catch the falling soccerball using either your "Silly Putty" or the real thing, but we decided not to, out of respect for Eudemonia's carpet.

It's intentional that the start codes for the last two puzzles have six letters; we did this to distinguish them from the metapuzzle components and to mark them as part of the brief "sixth element" phase of the game.