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In Terraria, there are five different "types" of damage (similar to elemental damage in other games). These types are melee, ranged, magic, summon, and throwing damage. Various armors buffs the total damage output of a certain damage type - this is good if you're trying to build a character that (for example) uses only bows - you can spec them out with armor and accessories that boost their "ranged" damage stat.

There are some armors/accessories in the game which automatically damage enemies if they get too close. For example, A full set of Chlorophyte armor spawns a leaf crystal above your head, which will shoot at any enemies it can see. Another example is the Spore Sac, which spawns small spores around you which will home in and damage enemies that wander too close. There are also some equips that reflect damage back at the attacker, such as the turtle armor.

Are these "extra" damage sources affected by damage-boosting stats? Which damage type do they fall under?

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These "extra" damage sources that you talk about are not affected by any damage increasing buffs. This is as they all do a set amount of damage, that does not change.

According to the official Terraria Wiki, each of the "extra" damage types have a specific amount of damage that they deal.

For the Chlorophyte armour's leaf:

The projectiles can cause 100 damage each.

For the spores for the spore sac:

The spores explode on contact, dealing about 400 damage over 1 second and inflict the Poisoned debuf

For the flower petals from the Orichalcum armour set:

The flower petal deals around 30 damage

I can also say from a lot of personal experience playing this game, that the buffs granted to you by the different armour sets always deal the same amount of damage.

However, I've never played through expert mode, as I normally use the Xbox version of the game. So I've never used a spore sac before. It is an accessory that you equip, but there is a chance that it does deal some sort of damage type, so it may be affected by things that boost whatever it may be. But as I said, I don't know, and I don't know what type it would be if any.

But I can say that for armour sets the damage is constant.

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