The cheapest F-type function costs n + 2 gates at n = 4, 5, 6

A Boolean function is F-type if every size-optimal And-Inverter Graph for it contains at least two gates of fan-out at least two, while its minimum formula is exactly one gate larger than its minimum circuit. Let μ(n) be the minimum circuit size of an F-type function with exactly n essential variables. An elementary edge count gives μ(n) ≥ n + 1, and a lift construction gives μ(n) ≤ n + 2 for every n ≥ 4. We record that μ(4) = 6, μ(5) = 7 and μ(6) = 8, so the upper bound is attained at the first three arities, and we state μ(n) = n + 2 as a conjecture. At n = 6 no census of six-variable functions is needed: the equality case of the edge count fixes the shape of any circuit that could meet the lower bound, and the candidates reduce to 321 NPN classes. An exhaustive enumeration that uses no solver settles all of them, and a SAT computation run class by class on the same list agrees on every class. We have no proof for general n, and we say what one would have to use.