Did some playing around with the middle numbers to work my way down.
4 Zaps
Turn 1, SSCC gets you a standard Stonemason, Procession, and Nomad Camp. Turn 2, NcSSC plus the Coin token gets Watchtower and Catacombs. Turn 3, play Necropolis, then Stonemason on the Catacombs to pick up a Fortress and two more Processions. Watchtower puts the Fortress in your hand and the two Processions on your deck, then use your first Procession on the Fortress to draw the second and third and then get a Catacombs which you turn into the fourth. Now you've got a hand of Procession, Procession, Fortress, Watchtower with a Procession on your deck and you're ready to go boom.
3 Zaps
Turn 1, SSSCC picks up a Stonemason, two Border Villages, Catacombs, and Procession. Turn 2, CC plus the Coin token gets a Watchtower. Turn 3, draw any five of the six gained cards and use one Border Village to draw the sixth. Stonemason on Catacombs for Fortress and two more Processions, then play the other Border Village to draw one of the Processions. Back at the setup for the Procession explosion.
2 Zaps
Turn 1, SSCCC and the Coin token picks up Stonemason, two Border Villages, Catacombs, and Watchtower. Turn 2, CC gets Hamlet. Turn 3, draw five, play Border Village, draw sixth. Play Stonemason on Catacombs for Fortress and two Processions. Draw them with Hamlet and the other Border Village. No Procession on deck yet but that gets fixed quickly with the first Procession wave.
So it's possible with two Zaps, but any less and you can't get make it to $8 to double-buy Border Villages with Stonemason. This removes a lot of price flexibility.