Sorry, I know I'm running a bit late.
Corruption
Project - $2P
Directly after playing an Action, you may set aside a Treasure that isn't a Copper to play it again. Discard set aside Potions, and trash all other set aside Treasures when that Action leaves play.
I think "make Potions do stuff" is a good direction to take this challenge. This Royal-Carriagification seems pretty fun. I dislike having two exceptional clauses on the card, but that is minor. More problematic is the interaction with Treasure gainers. This + JOAT can instapile Silvers. This + Sculptor can instapile Potions and Silvers for virtually infinite Villagers. The concept seems fun, but is currently broken and needs tweaking.
Education
Project - 10D
At the start of your turn, +$2.
At the end of your Buy phase, if you have any Debt remaining, take 2 Debt.
Neat design. It cancels its profit until you fully pay it off. Most of the time that probably takes 2 turns, and afterwards you start making profit. The interaction with other debt costs is mildly inconvenient - Education + Capital doesn't work, no wonder our universities are full of Communists! It's hard for me to tell how good this will be, it may need cost tweaking, but I think the design definitely works.
Patent
Event - $7*
Gain a card costing up to $5 to the top of your deck.
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This costs $3 less per empty supply pile, but not less than $0.
With no piles empty, this is like Travelling Fair + $5 cost. As more piles empty, it gets cheaper and the topdecking becomes more of a penalty. I think that is pretty neat. One worry are those games where one pile drains quickly and then you can gain $5s for $4, it would need some testing to see how fun that is.
Pirate Training
Event - 8D
Move your Sifting token to a Kingdom supply pile. (When you play a card from that pile, you first get: discard any number of cards, then draw that many.)
I think if token Events were done today, they would be Projects, then you don't really need the token. Anyway, the problem with this for me is that it attaches a decision to playing a card that most of the time you won't care all that much about. It's like a Scrying-Poolification. That increases playtime for not enough benefit I think. Also I don't feel like the debt cost adds all that much to the card.
Walpurgisnacht
Event - $P+
When you buy this, you may overpay up to $6P for it. If you do, gain 2 cards each costing the amount you overpaid.
Potions to Charms! I feel like something could have been done with that theme. Anyway this is a nice enough way to make Potions useful. I just would prefer if it copied Charm's differently named condition rather than give a price limit. The way it is, gaining 2 copies of a strong card with this can still be overpowered.
Discovery
Types: Event
Cost: $2
Discard a Victory card costing at least $4. If you do, remove this from the Supply and look at 4 random Kingdom cards not in the Supply. Add one's Supply pile to the Kingdom and gain up to 2 cards from it.
This is a very compelling idea. But as with many novel ideas, I think tweaking it to make it work is difficult. This could be too swingy if one player rushes for it, I think it should at least give everyone an option to use it somehow. I also think that the special cost is somewhat random and does not tie in mechanically. Maybe require that you need to discard a card from an empty supply pile instead?
Bioengineering
Project - 6D
During cleanup, when you would discard an Action card on play, instead return it to the supply and gain an Action card costing exactly $1 more.
This is another one of those with an intriguing idea that I think is not quite fully realized yet. I like the idea of a massive game-warping Project, but this is not it; I think its main use right now is cute pileout tricks where you buy it to end the game. I find it hard to imagine a board where it is viable to buy this when there is more than one other turn to play. I think it either needs to be less punishing or offer a stronger benefit.
Research
Event - $7
Once per turn: Each other player draws until they have 6 cards in hand then discards a card. Choose a card discarded this way. You may trash any number of copies of it from play and your hand, or gain a copy of it if it isn't a Victory card.
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This costs $1 less per card discarded (by you or any other player) this turn.
I like that this gives you some interesting player interaction. It creates an uncomfortable dynamic in multiplayer though, the first player to discard something is under pressure to discard good stuff and the last player has all the info to know if they can safely discard junk. It creates asymmetry where player 1 gets to trash all of their junk cards of a type and then player two can never trash them because player 1 does not have them to discard. Finally the price fluctuates too wildly; a single Minion play can drop this all the way to $0.
Smeltery
Project - $4*
At the start of Clean-up, you may return a Treasure card to the supply and gain a treasure card costing up to $3 more than it
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This costs extra $1 for each two Coppers in play, rounded up
As pointed out, this needs clarification of where the exchanged Treausre comes from. I'm going to assume from play as otherwise it's really weak. I read through Project outtakes before this and there is a similar cut Project; according to Donald it makes the game too muhc about base treasures. I think something like this will always walk the line of making things boring or being irrelevant.
Commonwealth
Event - 2D
Gain a non-Victory card costing up to $6. Take D equal to its cost in $.
Such a natural idea. So probably broken. I don't think it will lead to very fun games, those high-cost cards have that cost for a reason. Plus there is no real reason for this to have debt cost itself, so it's not quite in the spirit of this challenge.
Democracy
Project - $16*
When you buy this, set aside a non-Duration Action card from your hand. At the start of your turn, play it, then set it aside again.
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This costs $1 less for each differently costed card you have in play, but not less than $0.
Does it need "but not less than $0"? I suppose you could have a Black Market game with lots of Potion/Debt costs in the Black Market. Anyway... it has been pointed out that this would replay the card even if it leaves play. That is a bit of a problem with something like Pillage, but could probably be fixed. I think this overall works, but is kind of similar to Citadel. Also the cost change seems kind of random and not very impactful; since this was the core of this challenge, I need to subtract points for that.
Observatory
Project - $3*
During your Action phase, Silvers are also Actions which give +2 Cards, +1 Actions instead of +2$.
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You can't buy this if you have any Treasures in play.
I see this has changed from Lost Cities to Labs. That is probably for the best. I still don't think Lab is the best choice of effect here, it is just good in basically any deck, I think I would prefer something more situational, maybe terminal so you have to manage your Silvers. I'm also not sure about the cost, there are plenty of kingdom cards that give +$3 or more and with those, it is pretty easy to buy. I think it might be better just having a normal, but high, cost, or maybe have it cost $2 more per Silver in play or something.
Royal Taxation
Event - $0
Reveal and discard up to 4 Victory cards from your hand. If you discarded 4, gain a Gold.
This is just a worse Quest, and Quest isn't a particularly good Event.
Winner:
Education by Tejayes
Runner-up:
Patent by King Leon
Special Creativity award:
Discovery by Fragasnap