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Enchanting membrane (from wiki):
Increase Enchanting Dust generation by 5^R/100.
On activation, grants 5^R Enchanting Dust.
R = 1 + 0.25 × Tier
Tiers:
Common = 0
Uncommon = 1
Rare = 2
Epic = 3
Legendary = 4
Result:
Common (R=1), increase 5% (multiplicative), activation 5 (base).
Legendary (R=2), increase 25% (multiplicative), activation 25 (base).
EDIT: I must've missed the line for the passive, the post above is correct. (And the math checks out, the Uncommon value, and the progression seems accurate for what I'd expected). 5/7/11/16/25, only variation here was I expected a 23.# rounded to 24, 25 is a pleasant surprise.
I noticed this number doesn't seem to include the gathering bonus, which is supposed to increase enchanting dust in v0.81. It only is affected by Enchanting Membrane, trial bonuses, weekly bonus, and probably paragon 40 (I don't have that yet so cant tell).
The bonus also increases the Enchanting Membrane's active ability by the same amount. The number shown by the active ability is also not affected by crafting or gathering levels.
Make sure to wear the Resonator Ring if you want to max enchanting dust, since it give 4 more enchant levels.
I think this bonus is the global bonus affected to ALL enchanting dust sources.
Gathering is specifically used for passive generation, and Crafting for experiment. That's why Gathering doesn't affect this value.
P40 does, as you guessed, affect this value, however.
By the way, if you're wondering where Archmag found the information, each weapon and offhand has its own page on the wiki (you can use the research bar to find it, and I think you can access it from the item page by clicking on the name of the item?)
Yes, you can access items directly from the Items page, both listings. The first listing ('reveal' the massive walls of items) shows costs and per-level bonus for non-weapon/off-hand (those are more complex, you have to go to their own pages for the formulas), and the second list is "Stats at Legendary" only. Mousing over the set-name in either list will show you what items are in the set, but there is (of yet) no way of clicking them from this (not sure anybody is even working on this, so don't expect it).
https://idle-wizard.fandom.com/wiki/Items
PS: I wasn't correcting YOU specifically, I was providing the information for anybody who might not have been to the page (thus continuing the discussion), and providing a quick link.