Reflection positivity does not imply a positive normalizer

For every type $X$ (with no finiteness assumption in the first two clauses), form the split datum $D_0$ indexed by the one-element type $\operatorname{Fin}(1)$ whose sole coefficient is $0$ and whose feature has value $1$ at every point. The theorem asserts the conjunction of the following: for every type $X$ and every $a\in\operatorname{Fin}(1)$, the coefficient $(D_0)_a$ is nonnegative; for every type $X$ and every $x,y\in X$, the associated weight $w_{D_0}(x,y)$ is nonnegative; for every finite type $X$, the partition sum $\sum_{(x,y)\in X\times X}w_{D_0}(x,y)$ equals $0$; for every finite types $X$ and $I$ and every observable family $O_i:X\to\mathbb R$, the reflected matrix with entries $\sum_{x,y\in X}O_i(x)w_{D_0}(x,y)O_j(y)$ is positive semidefinite; and for every finite type $X$, the normalized pair-weight $w_{D_0}(x,y)/0$ is not a probability weight (that is, it does not simultaneously have nonnegative values and total sum $1$). In fact the datum's associated weight and every displayed reflected matrix are identically zero; real division is total, so $0/0=0$, and the normalized weight therefore has total sum $0$, including when $X$ is empty. Empty $X$ makes the second universal clause vacuous and empty $I$ makes the reflected-matrix assertion zero-dimensional, but the final non-probability assertion still applies to every finite $X$, empty or not.