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I'm confused about on-summon trigger effects. Consider this scenario:

  1. Opponent has a card on the field, with the effect When your opponent Summons a monster: ... (e.g. Trap Hole or Evolzar Solda. Note the when in the effect.)

  2. I Summon a monster that has the effect If this card is Summoned: ... (e.g. Infinitrack Anchor Drill).

  3. Because I am the turn player I get to activate effects first. I activate my monster's effect.

  4. (The part I'm asking about) My opponent want to activate their effect, but, if I understand correctly, they CAN NOT since my effect started a chain, and the last thing that happened in that chain was that I activated my effect, not that I Summoned a monster.

This is how I understand it. I've gathered this from looking around the internet. One relevant quote from the wiki is this (emphasis mine):

During the Summon response window, due to Fast Effect Timing, first the turn player must activate any compulsory effects that they triggered, such as "Summoning Curse" and "Summoner Monk", then their opponent must do the same. Then, the turn player has the opportunity to activate optional effects like "Torrential Tribute" and "Armageddon Knight"; if the turn player passes, then the opponent also has the opportunity to activate these effects. If at any point either player starts a Chain, standard Chain rules are followed.

Since a Chain is started when I activate my effect, my opponent cannot activate their effect in response to my Summon since it misses timing.

One relevant image is this: game state ordering

I imagine that the order is this:

  1. The game state is open (my Main Phase 1).
  2. Turn player takes an action that does NOT start a Chain (my Summon).
  3. Does this activate a triggered effect? (Yes, my monster's effect).
  4. Chain rules (opponent can't activate effect, since they would be chaining it to my effect activation, not my Summon).

A good answer needs sources to relevant rulings (if any). Preferably directly from Konami, but links to a wiki/trusted source is "good enough" if no Konami quote can be found. If I'm wrong and my opponent can chain to my Summon the answer should explain how and why. Edge cases should be included too, if there are any (e.g. a Summon that starts a chain vs one that does not).

2

You are allowed to activate cards like torrential tribute, as you mention in your own quote:

During the Summon response window, due to Fast Effect Timing, first the turn player must activate any compulsory effects that they triggered, such as "Summoning Curse" and "Summoner Monk", then their opponent must do the same. Then, the turn player has the opportunity to activate optional effects like "Torrential Tribute" and "Armageddon Knight"; if the turn player passes, then the opponent also has the opportunity to activate these effects. If at any point either player starts a Chain, standard Chain rules are followed.

The reason is that, even if another card is activated in response to the summon, that whole chain is treated as reacting to that summon.

Torrential tribute can however miss the timing, as DarkCygnus mentioned. However, this has to do with how the summon happened, and not with the cards that were activated in reaction to the summon afterwards.
Ultimate offering is indeed a good example of this, here chain link 1 resolving would be the last thing to happen.

Another example as given by Konami:

Starlight Road: If you Chain “Starlight Road” to “Heavy Storm” while you control 2 or more Spell or Trap Cards and Special Summon “Stardust Dragon” you cannot activate “Torrential Tribute” afterwards. Even though the effect of “Heavy Storm” is negated, the activation was not, so the card will still resolve without effect and you will miss the timing to activate “Torrential Tribute.”

An example of Torrential tribute being chained to another effect after a summon. (it is listed under the older rulings from the UDE era, but those remain valid until overridden):

"Torrential Tribute" can be activated after a Flip Summon, but it does not negate the Flip Effect (it is chained to the Flip Effect).

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You are correct. Your opponent can't activate such card like Trap Hole in response to an on-summon effect, as it misses timing.

Some relevant quotes and rulings. First from the ruling on Trap Hole, emphasis mine:

If "Ultimate Offering" is chained to an effect so that it Normal Summons a monster at Chain Link 2 or higher, then you cannot activate "Trap Hole", "Cloning", and similar cards that activate when a monster is Normal Summoned. This is because after the monster is Normal Summoned, Chain Link 1 resolves, so you miss the timing.

Also, from the wiki page on Missing the Timing we have a picture of the explanation given on the Official Rulings regarding the difference between the "If...you can" versus "When...you can" (emphasis mine):

  • Cards that say you can do something "When" something happens must activate their effects immediately when that something occurs. If they cannot, or something happens between the time the condition is met and the next time cards or effects can be activated, then you miss your chance to use them.

As we can see, in the scenario you describe, your opponent can't activate Trap Hole in response to the on-summon effect activated, as it missed timing.

Different case would have been with a "If...you can" card, which could have been chained to the on-summon effect of your monster (as the "if" condition was already met). From that same picture of the Official rulings:

  • Cards that say you can do something "If" something happens must activate their effects at the first possible opportunity to activate cards or effects after the "If" condition is met.
  • This isn't really how this scenario plays out. The case where a monster is summoned by ultimate offering as chain link 2 is completely different from a monster activating its own effect when it is summoned. I'll add an answer with more explanation. – kyudosai Jul 9 at 7:19
  • It was an analogy and example with another card, explaining how something can miss timing, the most relevant quote is the Official one about If and When effects... I think you didn't read the whole answer or are taking it too literal – DarkCygnus Jul 9 at 12:54
  • It is irrelevant to the scenario in the question and leads you to a wrong conclusion. You are correct in the scenarios i described above (ultimate offering, stardust road). But the main question was about chaining to effects after the summon of a monster. Wich is allowed like in the example of a flip effect monster. – kyudosai Jul 9 at 13:28

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