Pointwise nonnegativity does not imply reflection positivity

On the fixed two-element type $\operatorname{Fin}(2)=\{0,1\}$, define $W_{ij}=1$ when $i=j$ and $W_{ij}=2$ otherwise, regard $W$ also as a weight on the four ordered pairs, and let $\delta_i(x)$ be $1$ when $x=i$ and $0$ otherwise. The theorem asserts the conjunction that every $W_{ij}$ is nonnegative; $W_{ij}=W_{ji}$ for every $i,j$; its partition sum $Z=\sum_{i,j}W_{ij}=6$ is strictly positive; the normalized pair-weight $W_{ij}/6$ is everywhere nonnegative and sums to exactly $1$; the matrix $W=\begin{pmatrix}1&2\\2&1\end{pmatrix}$ is not positive semidefinite; the reflected matrix with entries $\sum_{x,y}\delta_i(x)W_{xy}\delta_j(y)$ (which is again $W$) is not positive semidefinite; and the reflected matrix with entries $\sum_{x,y}\delta_i(x)(W_{xy}/6)\delta_j(y)$ (which is $W/6$) is not positive semidefinite. Here positive semidefinite means that every real quadratic form of the matrix is nonnegative. All indices range over exactly two elements, so none of these universal assertions is vacuous and no additional hypotheses are imposed.