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If a red mana you add has certain restrictions or riders associated with it (for example, if it was produced by Base Camp), they'll apply to that mana no matter when you spend it.

Leyline Tyrant's last ability goes on the stack without a target. While that ability is resolving, you may pay any amount of red mana, including from the unspent mana you have. If you do, a second ability triggers and you pick one target to be dealt that much damage. This is different from abilities that say "If you do . . ." in that players may cast spells and activate abilities after mana is paid but before the damage is dealt.

Once Leyline Tyrant leaves the battlefield, you have until the end of the current step or phase to spend your unspent red mana before it is lost as normal. There is no penalty associated with this other than the loss of the mana.

You can keep unspent red mana indefinitely while Leyline Tyrant is on the battlefield. That means if you add a red mana during one step or phase, you can spend it during a later step or phase, or even a later turn. Other types of mana will continue to be lost as normal as each step and phase ends.