One safe packet need not cover every accepting point
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For \(k=3\) consider
The packet \(ab\) is safe and has degree two. It detects every teal point below. It misses three additional positives, which require the other packet \(z_1z_2z_3\).
The one-packet parameter \(\alpha_1\) asks for any nonzero safe polynomial. The complete parameter \(\beta\) asks for safe detection of every positive:
For this monotone function, its minimal positive supports have sizes two and three. Thus \(\alpha_1(F_3)=2\) while \(\beta(F_3)=3\). Selecting one easy accepting region loses the pointwise obstruction at the other region.
All 32 points and the two parameter values were freshly recomputed; the general monotone minterm characterization is a written proof for nonzero functions.