Time to come out of the rabbit hole. To begin with, I believe that most of us have consented to the two claims: 1. we are not Peach and 2. we are not the jailer. In fact, if a false claim were to be made in this game, then the last roles that mafia would consider would be Peach and the jailer. Consider this, Peach is Mario's #1 playmate. In any game that involves Mario characters, she is in it. Consider Mario Kart, Smash Brothers, and the Mario series. If mafia were to claim a fake character then they would choose Wario, Waluigi, or some other obscure character to prevent a clash from picking popular favorites like Mario, Luigi, Toad, so on and so forth.
Second, let us consider the notion of probability. There are five town roles in this game, two power roles, and two mafia roles to make a game of nine. That means that if mafia were to make a claim on D1, then the best claim would be to stick to a town role. Picking a town role means that even though other people may claim town, they cannot be sure if the other members of town are town or not. Therefore adhering to a vanilla town role would cause mayhem and mishap, a goal of mafia. Claiming a power role on D1 can be compared to suicide. There is already someone else who is guaranteed to have that role and lynch from a pure probability standpoint comes down to a 50-50 chance. That is not a smart move on the part of mafia to play such a big sacrifice on their first move.
Terywj did not only claim role, but also claimed character. It is quite possibly one of the stupidest moves for a doctor-based role to take on the first day of the game. Leading up to his role reveal, it is obvious that Terywj was lapsing in and out of phases of annoyance / agitation. These are possibly techniques used by the mafia to wear down interest in the overall game and to coax out a response from weak players who can't keep their own emotions in check. Terywj needed only to be offered a bait in his weakened mental state and a fisher to cast the line in order to draw him out from the town pool.
Now let us consider this, why does this retrospective statement have any weight? Consider the role of a mafia members and a jailer. Both sides want to fit in with the rest of the town. Standing out may result in them getting lynched and therefore both try to stay under the radar and sneak quietly by without leaving a trace. Jailers, or at least newbie doctors tend to play similar to scum as they often have the same motivations as mafia members. They both want to play under the radar and avoid getting too much attention. Jailers want to freeze the options of the mafia whereas mafia want to play off the suspicions of the people. Both play a similar game of cat and mice seeking to weed out the other. Terywj was not only very bad at blending in with town, he stated that he had his own play style and made virtually no effort to play for the community in place of his individual "my rules, my way" attitude which led to him changing his tone from one of "community" to the community serving him as an "individual."
Let us not stray from the most important factor, who would notice that Terywj was acting like scum and suppose that he was a doctor from the previous information given? Why! Our good friends the mafia of course would be the first ones to key in on this behavior. By having someone as suspicious as Terywj and a possible jailer suspect, it became quickly obvious that he would become a priority target for mafia somewhere along the line. Let us consider these possibilities. Even though we may try to hide them in the back of our minds they must be brought to light for the good of town.