Reincarnation is a system for improving the ranks of puppets, allowing them to be viable for competitive play. To access it, visit Patchouli in the Scarlet Devil Mansion's library after the credits roll.
Most puppets that you catch in-game will have less than satisfactory ranks for serious play; indeed, most serious play expects you to have S in every relevant stat.
Fortunately, I have on hand two very competent puppets to help fix this Ruukoto.
There are five slots on the reincarnation screen.
The first slot is the mon you are trying to improve. Be aware that reincarnation will clear its movepool and reset it to level 1, so don't reincarnate your best mons right away.
The second and third slots are the mons whose rankings are used. They are not changed by the process; use mons with good ranks here.
The fourth slot is for marks; these let you change the mark (basically the nature) of the puppet. Using a mark does not consume it. You get all five marks from the reincarnation table when you first use it, and the game will not let you sell or discard them, so don't worry about losing them. The stats they provide are as follows:
Red Mark: +10% Focus Attack
Blue Mark: +10% Focus Defense
Black Mark: +10% Spread Attack
White Mark: +10% Spread Defense
Green Mark: +10% Speed
The fifth slot is for magic stones. There are eight of these, and what you use here determines how ranks are inherited. As with the marks, these are not consumed through use and the game will not let you discard or sell them.
Nothing: Three ranks are chosen at random and replaced by the ranks from the other two puppets. The rest are rerolled.
Omega Magic Stone: All ranks are derived from the first puppet, ignoring the other two. Use this if you just want to change a puppet's style or mark without changing its ranks.
Gold Magic Stone: The first puppet's HP rank is sustained. Three other ranks are chosen at random and replaced by the ranks from the other two puppets; the rest are rerolled.
Red Magic Stone: The first puppet's Focus Attack rank is sustained. Three other ranks are chosen at random and replaced by the ranks from the other two puppets; the rest are rerolled.
Blue Magic Stone: The first puppet's Focus Defense rank is sustained. Three other ranks are chosen at random and replaced by the ranks from the other two puppets; the rest are rerolled.
Black Magic Stone: The first puppet's Spread Attack rank is sustained. Three other ranks are chosen at random and replaced by the ranks from the other two puppets; the rest are rerolled.
White Magic Stone: The first puppet's Spread Defense rank is sustained. Three other ranks are chosen at random and replaced by the ranks from the other two puppets; the rest are rerolled.
Green Magic Stone: The first puppet's Speed rank is sustained. Three other ranks are chosen at random and replaced by the ranks from the other two puppets; the rest are rerolled.
Rainbow Magic Stone: One rank is chosen at random and rerolled; the rest are replaced by ranks from the second and third puppets.
The Omega Magic Stone is given to you when you first use the reincarnation table. The six colored stones are found in the Misty Lake Cave. The Rainbow Magic Stone, which is by far the most useful, is obtained from Kasen by finishing her sidequest; the procedure for obtaining it is below.
Spoiler
First, go to the foggy area on the Youkai Mountain's back path. If you haven't seen the credits yet, it loops endlessly; after you have you can explore the area. Towards the back you will find Kasen; after talking to her, four items appear in Gensokyo (and a number of unrelated events activate, such as Suwako).
The first item is in the foggy area Kasen resides in. Search for a map filled with frogs; one of the frogs near the north section gives it to you when you talk to it.
The second item is in Muenzuka (the area filled with roses behind the Forest of Magic).
The third item is hidden in the well behind the Hakurei Shrine. It is not obtainable until after you beat Shinmyoumaru, who appears in the room with the Akyuu Doll.
The fourth item is hidden in the cave behind the Myouren Temple. Be aware that picking it up starts a battle.
Once you have all four items, take them to Kasen and she will give you the Rainbow Magic Stone.
An example of reincarnation using the puppets shown above with no magic stone:
Focus Attack and Focus Defense were inherited from Kogasa, while Spread Defense was inherited from Sariel. The rest were randomly determined.
If we try using the Rainbow Magic Stone:
HP, Focus Defense, Spread Attack, and Spread Defense were inherited from Kogasa, while Speed was inherited from Sariel. The Focus Attack was randomly determined.
Some useful probabilities:
The chance of getting a mon with 6 S-Ranks, using two parents who have 5 matching S-Ranks and a colored magic stone for the sixth stat: 1/256
The chance of getting a mon with 6 S-Ranks, using two parents who have 5 matching S-Ranks and a Rainbow Magic Stone: 1/96
The chance of getting a mon with 6 S-Ranks, using two parents who have 5 differing S-Ranks and a Rainbow Magic Stone: 1/48
The chance of getting a mon with 6 S-Ranks, using two parents who have 6 S-Ranks and a Rainbow Magic Stone: 1/16
Finally, every use of reincarnation consumes one Magical Fragment. As such, it's best to minimize the number of times you reincarnate each puppet to preserve fragments. You can get more by fighting wilds, or by playing netplay. If you want to farm magical fragments and have nobody to help you, you can copy the game's folder and run it twice, then netbattle yourself and repeatedly forfeit to gain 6 fragments per battle.
