Can you use a cost reduction effect to make a spell free and still cast it?

Yes, cost reduction effects can legally reduce the total cost of a spell to zero, and you may still cast it. Under CR 601.2f, after determining all cost modifications (increases and reductions), you simply pay whatever the final cost is — even if that cost is nothing at all. Paying zero is a valid payment.

CR 601.2f and 118.5 together establish that costs can be reduced to zero but not below zero. This means a reduction can eliminate the entire mana cost, but excess reduction does not give you mana back or cause any other effect. The spell is still legally cast and goes on the stack normally.

It's important to remember that some spells have additional costs, alternative costs, or mandatory additional costs (like kicker or targeting requirements) that are separate from the base mana cost. A cost reduction only reduces the mana component of the total cost unless it specifically says otherwise. So a spell might still have non-mana costs even if its mana cost is reduced to zero.

Example: You control Goblin Electromancer, which makes instant and sorcery spells cost {1} less. You cast Lightning Bolt, which costs {R}. Since {R} has a generic component of zero, the {1} reduction doesn't apply to bring it below zero, but if you were casting a {1}{R} spell like Shock's more expensive cousin, the cost becomes {R}. If you had a spell costing {2} with Electromancer, it becomes {1}, and with two Electromancers it becomes {0} — free — and you cast it paying nothing.

In summary, casting a spell for free due to cost reduction is completely legal and common in the game. The spell resolves normally, and all its effects apply just as if you had paid its full cost (CR 601.2, 118.5).