There are spoilers within this. Please leave if you have not played Persona's 3, 4, and 5 (including Fes, Golden, and Royal). If you would like a partial amount of the games spoiled for you, you may continue reading this. I do not claim any responsibility for any mental damage spoilers cause.
You read the title correctly. I'm going to be using evidence given throughout Persona 3 to Persona 5 Royal to connect each game together.
Many before me have used the one thing present in the whole series, the Velvet Room. However, the Room itself is not the only connecting point between these games, as many things show that they are within the whole same universe as each other, not separate timelines.
These three games are the most easily connected besides Persona 2: Innocent Sin/Eternal Punishment which are directly based after each other. Persona 1 doesn't count because, well, Persona 1 wasn't exactly the best, and I haven't the time to bother with it.
Let's start with some of the larger evidence.
The biggest connecting point between 3 and 4 may just be the fact that 4 is set a year after the events of 3. This is also proven via Chihiro Fushimi, a second year at Yasogami who greets Yu Narukami, Yosuke Hanumura, and Chie Satonaka at the start of the year. Furthermore, she claims later that the former student council president from her first year helped her write the speech, implying the existence of Mitsuru Kirijo (a major character from Persona 3). Chihiro's first year was at Gekkoukan High School, where Mitsuru was the student council president.
Another piece of partial connectivity is the implication of Rise Kujikawa within the Lovers arcana boss battle during the events of Persona 3. There isn't a ton for me to work with here, but it's still a connecting point!
A big connector is Tanaka’s Amazing/Shady Commodities. He sells on the TV in P3 and P4, and on the broke laptop in P5.
The next point I would like to bring up is the mention of Wild-duck Burger in Persona 5. This small cameo appears within the train news headlines. For those who don't remember, Persona 3 had Wild-duck Burger, a location where you can increase (I think) your Guts stats. This is indefinitely minor, but it is still a piece of connectivity between the three, implying 3 and 4 are already together.
I am going to throw a small bit of the Velvet Room into this. Though I rarely condone the Velvet Room as a source of connectivity, these are the small pieces of evidence that do make sense. Although it has taken the forms of an Elevator, Limousine, and a Prison, there are some small bits within.
Firstly, Lavenza (the fusion of Caroline and Justine) proves the existence of the other Velvet Room residents by mention several times while talking to Ren Amamiya. This is small, but that's how we can piece together that Marie, Elizabeth, and Margaret exist within the series together at separate intervals.
Secondly, if you were like me and bought the DLC (because it was cheaper than the game itself if I bought the bundle), you can actually fuse Orpheus, Izanagi, Messiah, Magatsu Izanagi, Izanagi-No-Okami, and other Persona from the other games and their appropriate spin-offs.
Thus, through small bits and pieces scattered throughout the series itself, the final three Persona games are connected to each other.
If you know of any other evidence, PM or comment it and I will get to sorting it through this page. The ultimate goal is assuring that the Persona games are connected within actual evidence, and not through various fan theroies.
Note: Persona Q/Q2 will not be placed in here, as they are time-traveling via the Velvet Room to the other dimensions, though none of the teams have any recollection of each other.
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