Min rate is easy to determine, due to it being metals-only, just add up all the metallicity for the input and output, then divide the least common multiple by each of them.
I = ☉ = 6
O = 2 * ☿ + ♂ + ☽ = 2 + 3 + 5 = 10
gcd(6, 10) = 30

30 / 6 = 5
30 / 10 = 3

So 5 gold creates 3 amalgams, which means min rate is

(5 * 2) / 3 = 10/3 = 3.333...
which this solve achieves with as much grace as is expected from me.