I understand why Trinity Seven because there are seven deadly sins represented by a girl which is needed to be controlled by Arata. But I don't understand why Trinity? Which is referred to as the Trinity? Does Arata's magic have three forms? Does he need to do three things to control these seven sins? Are there three magus who can control these seven sins? What does the Trinity represent?
This is what I currently know about the term Trinity Seven and Trinity in the context of this manga.
The accuracy of this post depends largely on the fan translation of the manga series.
When writing this answer, I confirmed all the facts by checking back on the manga. However, I didn't check out the whole series from start to end. I only picked out the chapters which I remembered to have the relevant content and cited from it.
Trinity Seven refers to the 7 strongest mage in a world (there can be many parallel worlds).
In Arata's world, the Trinity Seven are the 7 girls: Asami Lilith, Kannazuki Arin, Kazama Levi, Yamana Mira, Fudou Akio, Kurata Yui and Lieselotte Sherlock.
According to Liese in chapter 17, to be a member of the Trinity Seven means that
one has acquired the Lost Technica in one's archive. In Liese's case, it was "Last Crest".
As explained in chapter 36 of the manga, the magic king is closely related to the Trinity Seven. Below is the summarized definition, based on the fan translation by VISCANS:
There exists an infinite number of worlds, with each world connecting to each other by causality. Worlds begin and end, are born and destroyed over and over again. Magic king exists for such system, and his mission, his fate is to reduce his world to nothingness. A magic king is born human, a baby with the secret of unlimited magic power. Regardless of the circumstances of the magic king, he will encounter the Trinity Seven and begin his transformation into a proper magic king.
In chapter 36 of the manga,
... after Arata copied Lieselotte's Logos Arts in Acedia archive, along with Lilith's Outer Alchemy in Luxuria archive and Akio's Matra Enchantment in Gula archive, he was confirmed by the sword Judecca to have attained Trinity.
However, the condition for Trinity is quite ambiguous. At the beginning of chapter 35,
... Arata found the Black Imperial Sword Judecca. The sword identified Arata's Magic King element and confirmed his Superbia archive and Impero thema, then it went on to recognize Outer Alchemy in Luxuria archive and Matra Enchantment in Gula archive. However, the sword concluded that Arata did not meet the Trinity requirement, and it started to initiate Magic King Berserk Mode.
It was quite clear from chapter 5 of the manga that there are many thema (research topic) in an archive. From my personal impression, it seems that different mages working on the same thema may study and master different magics.
And along that line of reasoning, I speculate that Arata had yet to come up with or mastered any magic under Imperium thema, so it was not counted towards the Trinity requirement.
Apart from the likely direction of Arata eventually copying the magic from all members of Trinity Seven, and thus in control of magics from all 7 archives named after the 7 deadly sins, it is not clear how the Trinity requirement works. We know "trinity", from Christian doctrine, has tri- meaning three in it. However, it is not clear three of what (archive alone is incorrect, as seen above), and whether it requires the mage to have certain level of mastery.
To make the matter more confusing, according to chapter 37 of the manga,
Arata attained power of a Trinity when Akio, Liese and Arata himself connected to their archives.
And at the end of the chapter,
Abyss Trinity, Lilith's father and antagonist in the arc, was surprised by the fact that Arata exceeded Trinity after Lilith reconnected to her archive, which means Arata had 4 archives executed at the same time.
So there seems to be a further distinction between achieving the Trinity status and actually performing magic in Trinity mode.
The Trinity might refer to the mastery of three thema because in episode 10 of the anime, we see Hijiri using a mode ,which is in a way, similar to Arata's Astral Trinity form. Hijiri says that this form is stronger than Magus Mode and she also adds that this is the quintessence of her archive and is the result of her mastering three thema proper to the Ira archive. However in the film Heavens library & Crimson Lord, Lilith is said to possess the quintessence of Luxuria archive in her body without achieving Trinity through the mastery of three thema.
But for Arata, the Trinity form is more difficult to achieve because he has some some conditions to clear in order to do that. Still in the film Heavens Library, Liese explains that there is three conditions to fulfill if Arata wants to copy another magic or thema coming from Archives other than his own :
-First condition: he needs to understand what is the magic he is using (the name) and how it must be activated.
-Second condition : he needs to know the origin and the creator of the magic
-Third condition : he needs to decompose the magic
Arata already achieved Trinity in the whole sense of the term when he used Mentra Enchanted from the Gula Archive. This means the conditions are different for candidates and Demon Lord or it depends on the Archive they chose or Arata did not use conventional means to achieve Trinity. Judging by the actions of Judecca in chapter 35, it seems Arata must master three themas from three differents Archives excluding Superbia which leads to : Gula Archive : Fides, Acedia Archive: Stagna , Luxuria Archive: Abies needed for Arata to be recognized by Judecca. In Heavens Library, a clue may be introduced to us as to why Judecca wanted him to go berserk when Arata says he controls 4 four archives and four themas. Mastering and Controlling are two differents things, one is achieved through efforts and the other through sheer force.
This leeds me to believe that Arata will be different from the other Demon Lords by the theme he chosed. The other Demon Lords always wanted to destroy the world even Abyss Trinity even though he went against his own Archive to give birth to Lilith, he still followed the mission given to the Demon Lords which was to destroy the world for it to be reborn. Arata chose to follow another route and decided to control the world to stop its destruction in other words went against his mission.