Closure under absorption of half-factors

For every universe-level-0 pair of types $X,A$, every finite enumeration of $A$, every function $h:X→ℝ$, and every split-weight datum $D$ on $X$ indexed by $A$, let the absorbed datum keep $D$'s coefficients and replace each feature $D.feature(a,x)$ by $h(x)D.feature(a,x)$. Then all three of the following hold: for every $l,r∈X$, its associated weight is exactly $h(l)h(r)D.weight(l,r)$; if $D.coefficient(a)≥0$ for every $a∈A$, then every coefficient of the absorbed datum is nonnegative; and if $h(x)≥0$ for every $x∈X$ and $D.weight(l,r)≥0$ for every $l,r∈X$, then the absorbed datum's associated weight is nonnegative for every $l,r∈X$. No finiteness assumption is made on $X$, and the third assertion has no coefficient-sign hypothesis. If $A$ is empty, coefficient quantifiers are vacuous; if $X$ is empty, every pointwise statement over $X$ is vacuous.