So, the judges I talked about this with prior to posting here on the forums arrived at three solutions:

1) HCE - Thoughtseize-fix the card drawn, shuffle away the other two.
2) HCE - Thoughtseize-fix the card drawn, put the other two on top of the library to scry.
3) GRV - Backup by putting the cards randomly on top of the library, resolve Serum Visions

(1) was the easiest to arrive at: one card is supposed to be drawn, and the other two are “excess cards” to be shuffled away. (2) feels better to apply, as it doesn't give AP a free shuffle. We can argue that the 2nd and 3rd cards were looked at “prematurely”: NAP chooses two cards to set aside, AP draws the card remaining, then AP scries the two set-aside cards.

However, taking a literal interpretation of the definition of “set” in the IPG (which Scott has confirmed was not intended) suggested that because the player created the “set”, rather than a game rule or effect, it was not clear that this could be HCE. Hence solution (3): treating this as a GRV and performing a simple backup, which actually seems to preserve the game state the best with the least disruption, and still with little-to-no advantage possibly gained for AP.

LEC was ruled out due to the recent philosophy of it applying only to “dexterity or rules errors”. However, now I'm wondering if this is coherent philosophically. What if an effect says: “At the beginning of your upkeep, scry 3 if {condition} is true; if {condition} is not true, draw a card.” If a player misevaluates and incorrectly scries 3, will this be LEC since this is now a rules error? Is the difference in intent here enough for us philosophically to rule these differently? Should reading a card and carrying out its instructions incorrectly (which is equivalent to our Ponder vs. Serum Visions scenario) be considered violating a “game rule”?

Scott's example of paying the wrong mana to cast a spell makes it clear that the intended use of “game rule” is when an opponent has an opportunity to catch the error before the infraction happens, so maybe the IPG should be worded that way? Otherwise we have a lot of “game rules” to which this doesn't apply (e.g., replacement effects, triggers where the result doesn't have to be announced until acted upon), and it might be hard to codify which “rules” are being referenced.

Or maybe I'm being too semantic, and the common sense reading is more obvious than I'm making it out to be?

Edited David Poon (March 2, 2017 11:54:18 PM)

Originally posted by David Poon:

Andi casts Serum Visions and then looks at the top 3 cards of their library. When a judge asks why they did that, they respond, “I mostly play Legacy; I guess my hands assumed I was casting Ponder…”

Originally posted by David Poon:

1) HCE - Thoughtseize-fix the card drawn, shuffle away the other two.
2) HCE - Thoughtseize-fix the card drawn, put the other two on top of the library to scry.
3) GRV - Backup by putting the cards randomly on top of the library, resolve Serum Visions

(1) was the easiest to arrive at: one card is supposed to be drawn, and the other two are “excess cards” to be shuffled away. (2) feels better to apply, as it doesn't give AP a free shuffle. We can argue that the 2nd and 3rd cards were looked at “prematurely”: NAP chooses two cards to set aside, AP draws the card remaining, then AP scries the two set-aside cards.

If I'm reading the initial scenario correctly, the player picked up the top 3 of their library, but those 3 haven't made contact with the hand. Assuming that's true, I strongly dislike any fix that involves shuffling – we may not know exactly which of those 3 cards is supposed to be in hand or subject to the scrying part of Visions, but we know that those 3 cards as a whole should be the ones “seen” by Visions. If we do anything involving putting some of those cards back or shuffling, we are moving away from the most correct game state. If we do anything involving anything but those 3 cards, we're adding potential for the game to end up radically different from how it “should” be.

Given that, I would say solution 1 is the worst of the suggested fixes. AP is supposed to be performing a scry on two of those three cards – I don't see a good reason to either a) deny them the opportunity to do that b) allow them to scry on two totally different cards.

The other two proposed fixes do a better job following the spirit of HCE:

Originally posted by IPG:

Though the game state cannot be reversed to the ‘correct’ state, this error can be mitigated by giving the opponent sufficient knowledge and ability to offset the error so that it is less likely to generate advantage

That said, I would want to go with option 2 – giving the opponent the choice is the standard fix for HCE, and I'd hold to that rather than doing it all randomly (a la option 3). I would have the AP reveal the three cards, have NAP pick one to be the drawn card and move that to AP's hand, then have AP finish resolving their Visions on the remaining two cards.

Here's a pretty similar scenario that Toby Elliot replied to, where he said

Note that the “ponder set” is a perfectly legitimate place to return cards to

, which backs up going with option 2 here.

https://apps.magicjudges.org/forum/topic/34104/?page=1#post-215202