Alchemist's Gambit

• WOTC Official Reminder Textofficial rules
  Cleave — You may cast this spell for its cleave cost. If you do, remove the words in brackets from its text for as long as it remains on the stack or battlefield.

• WWizards of the Coast2021-11-19
  Official Ruling
  If you cast a spell for its cleave cost, that spell doesn't have any of the text in square brackets while it's on the stack.
• WWizards of the Coast2021-11-19
  Official Ruling
  Exiling Alchemist's Gambit happens as part of its resolution. If the spell is countered, it will go to the graveyard as normal.
• WWizards of the Coast2021-11-19
  Official Ruling
  If you somehow skip the extra turn Alchemist's Gambit gives you or skip that turn's end step, the delayed triggered ability never triggers. (If you cast Alchemist's Gambit by paying its cleave cost, the delayed triggered ability is never created at all.)
• WWizards of the Coast2021-11-19
  Official Ruling
  If multiple “extra turn” effects resolve in the same turn, the most recently created turn will be taken first.
• WWizards of the Coast2021-11-19
  Official Ruling
  You can't cast a spell for both its cleave cost and another alternative cost. For example, if an effect gives an Alchemist's Retrieval in your graveyard a flashback cost of , you can't cast it from your graveyard for its cleave cost.
• WWizards of the Coast2021-11-19
  Official Ruling
  If an effect allows you to “cast a spell without paying its mana cost,” you can't cast that spell for its cleave cost.
• WWizards of the Coast2021-11-19
  Official Ruling
  A cleave cost is an alternative cost that's paid instead of the spell's mana cost. Casting a spell for its cleave cost doesn't change the spell's mana value.

Rulings via Scryfall