Does Trinisphere affect spells that already have an alternative cost that brings them below three mana?

Trinisphere's ability states that as long as it is untapped, each spell cast costs at least three mana to cast. This applies after all other cost modifications — including alternative costs, additional costs, and cost reductions — have been applied. If the total mana cost ends up below {3}, Trinisphere raises it to exactly {3} (CR 601.2f, 601.2h).

The key rule is CR 601.2f, which establishes the order of cost modifications: first apply cost increases, then cost reductions, then apply Trinisphere's floor effect last. This means no alternative cost, no matter how cheap, can circumvent Trinisphere's minimum.

For example, if you try to cast a spell for free using its Suspend trigger (which normally has a mana cost of {0}), Trinisphere still forces you to pay {3} for that spell. Similarly, a spell with Madness discarded for {0} would also be bumped up to {3}.

Note that Trinisphere only affects mana costs — it cannot add mana to pay for non-mana costs such as life payment or sacrificing permanents. If a spell has an alternative cost involving non-mana payments, those are unaffected; only the mana portion is raised to the minimum of {3} (CR 601.2h).

This makes Trinisphere a powerful hate piece against strategies that rely on casting spells for free or very cheaply, such as Force of Will, Affinity cards, or zero-cost artifacts like Mox Opal.