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So I got the Shield of the Just and equipped it. But even when I'm not using it I have more HP than Gladio. Is that because I have it equipped or because it's a Royal Arm? Are all the stats, even STR effected when I'm not using it? Does having all 13 make Noct a one man army by default?
The stat boost in the stat box of each weapon is a permanent boost while equipped which can be tested by just looking at your HP even when using one of the other three weapons instead of the shield - as you already have noticed, but that is not because it is a Royal Arm, all weapons that show their boost in that respective equip -> boost box have it as "even while not active" boost for your characters. The Strength that stands next to the weapon itself is of course a "now active" value. And no, the weapons do not make Noctis a one-man army. The weapons are good for stat boosts and because you can deal high damage easily with some of them but you cannot connect a warp strike during a partner command with one of them being your active weapon as well as does the usage cost you HP. Of course that can still be worth it, I mainly used Royal Arms during my first run and I still use them for stat boosts and even combining stuff like the HP drain with the effect of the Apocalypse broadsword. Really good weapons later are weapons like Balmung, Zwill Crossblades and Apocalypse but the stat boost of the royal arms can even then be very good. I have super accessories, am Level 99 and all and have like 1-2 royal arms equipped nonetheless.