16.2 An explicit moment certificate
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On the face \(b=1\), every point except \(v=(a=0,z=(1,\ldots,1))\) is negative for \(G_k\). A polynomial of degree at most \(k\) sums to zero over the \((k+1)\)-dimensional cube: each monomial omits some coordinate and contributes an even number of ones. Therefore
\[
v_k(v)=\sum_{u\ne v\text{ on the face}}v_k(u).
\]
This is the certificate from Theorem 15.4, proving that no safe degree-\(k\) polynomial detects \(v\).
The finite four-input computation found \(\max_\ell\alpha_1(f+\ell)=\beta^*(f)\) for every function in that domain. The family above disproves extrapolation of that equality: it already fails at five inputs, with values three and four, and the difference grows without bound.