/r/YuGiOh Basic Q&A and Ruling Megathread - October 01, 2018

First thing to address is that "targeting" is a specific function in the game that is not a "cost". Targeting occurs when a card or effect is activated, but has not resolved yet. "Costs" must be paid when an card or effect is activated, but they are still distinct from "targeting".

Does the monster still get the boost? Yes. If you look at Sieger's quick effect:

"You can target 1 Machine monster you control with 2100 or more ATK; for the rest of this turn, that monster gains 2100 ATK/DEF..."

Problem-Solving Card Text (PSCT) dictates that all the text after the semi-colon or colon is applied on resolution of a card or effect. A card may not resolve properly due to various circumstances for instance your opponent activating a card or effect. For this effect of Sieger however, there is no mention upon resolution that the monster you targeted must maintain 2100 or more ATK. The effect only says "that monster", not "that monster with 2100 or more attack".

The only time you need to check whether your monster has 2100 or more ATK is when you attempting to activate Sieger's effect. In order for you to activate its effect, you must have a monster with 2100 or more ATK. If Sieger is already below 2100 attack, or none of your monsters have 2100 or more attack, then you cannot activate Sieger's effect. Once you activate Sieger's effect because you have a monster with 2100 or more ATK, you no longer have to worry about the ATK of that monster on the resolution of the effect as it will gain the 2100 attack.

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