Here is the full list of all 15 The Spectrum Retreat trophies - there are 13 bronze, 1 silver and 1 gold.
Complete both endings.
Complete the game.
Read the manager's notebook in the observation room.
Complete floor 5 puzzles.
Enter the fifth 4-digit code.
Complete floor 4 puzzles.
Enter the fourth 4-digit code.
Complete floor 3 puzzles.
Teleport for the first time.
Enter the third 4-digit code.
Complete floor 2 puzzles.
Enter the second 4-digit code.
Complete floor 1 puzzles.
Enter the first 4-digit code.
Find the phone.