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曙門界 = Shomonkai, the name of the City. Its kanji mean daybreak/dawn, gate and realm/world in this order, hence why the Kaminoshimobe constructed this theory that it's in fact a 'gate'.
天国 = Tengoku, celestial city, it's the name of the central district where the Tennou's private quarters and the Academy are located.

外環 = Sotowa, the Outer Ring, it consists of all Districts save the Tengoku.
内環 = Uchiwa, the Inner Ring, it only contains the Central District of Tengoku.

区 = ku, the word that means 'district'. The names of the Districts are often accompanied by it in documents. (f.e. Guren-ku, Hakuryuu-ku, etc. )

緋門 = Himon, the scarlet gate, positioned North of Tengoku, facing Gurenku
青門 = Seimon, the azure gate, positioned Southeast of Tengoku, facing Seiaku-ku
白門 = Hakumon, the white gate, positioned in the Northeast of Tengoku, facing Hakuryuu-ku
黎門 = Reimon, the black gate, positioned in the Northwest of Tengoku, facing Kurohyou-ku
翠門 = Suimon, the emerald gate, positioned in the Southwest of Tengoku, facing Ryokugen-ku

白竜 = Hakuryuu, white dragon, the district located in the Northeastern part of the City.
紅蓮 = Guren, red lotus, the district located in the Northern part of the City.
黒豹 = Kurohyou, black panther, the district located in the Northwestern part of the City
青空 = Seiaku, blue sky, the district located in the Southeastern part of the City
緑原 = Ryokugen, green field, the district located in the Southwestern part of the City

穴場 = anaba, a good place known to a few, people refer as such to Kariya's 'Nekonogetsu'
猫の穴 = Nekonogetsu, cat's hole, name given to Kariya's place in the rundown area of Ryokugen

神の僕 = Kaminoshimobe, servants of god, name of the group that wants to destroy the City from the inside.
巫女 = Miko, holy maiden, (literally: shaman woman), the Kaminoshimobe regard her greatly, her identity unknown, supposedly plays a grand role on their scheme.

天皇 = Tennou, the Emperor of the Shomonkai who has sworn to protect its people. He is looked upon as a gentle and kind figure who treats everyone as equals.

修験道 = Shugendo, translated loosely it can mean the 'way of the power attained by training in solitude', it is one of the two schools of the Academy, centered more in training the spirit and Reiki than physical strength and weapons.
修験者 = Shugenja, person who practices the Shugendo.
武士道 = Bushido, the 'way of the warrior', it is on of the two schools of the Academy, centered more on raising the physical strength and overall constitution of its trainees rather than training their spirit further. Uses of various weapons are also taught here.
武士 = Bushi, a warrior.

元帥 = Gensui, the Captain-general, this is the title given to the Captain of the First Division only.
探偵 = Tantei, investigator/detective, most commonly referred to as 'soldier investigator', it's the title given to the soldiers of the fifth Division.
忍者 = Ninja, people of the seventh Division are informally called that.

歩霊 = Horyou, Ghost Step
風乗 = Kazenoru, Wind Ride

縛の術 = Shibaru no jutsu, Binding Art (literally; binding technique)
絆 = Kizuna, bonds, the effect of Shibaru no jutsu

霊気 = Reiki, Soul/Spirit Energy
第六感 = dairokkan, Sixth Sense
霊軌 = Reiki, Spirit Track
霊刷 = Reisatsu, Soul Print
蔽 = Hei, Spirit Cloak (literally; cover, conceal)

声輪 = Seirin, Voice Link

霊斥 = Reiseki, Spirit Repel

霊尽 = Reijin, Soul Feeding (literally; soul consumption)

ゲイシュ = Geishu, Japanese pronunciation of the Irish 'Geis'
地獄の鈴 = Jigokunorin, hell's bell, the artifact needed to perform the ritual required in order to receive Geishu.

霊剣 = Reigen, Soul Arm (literally; soul sword)