Theorem: The central-second-jet defect starts at total order three

On ${\mathcal{R}}_2$, ${\mathcal{V}}_{\xi }$ is a derivation of the retained first-order product. The obstruction in Theorem thm:unfiltered first occurs at total order $\hslash s^3$.

In particular, the unfiltered centrality obstruction cannot by itself exclude a quantum jet comparison claimed only through common total order $s^2$.

Proof. A nonzero term in (eq:Leibniz) requires a $\zeta$-derivative of one coefficient and a $z$-derivative of the other. The former coefficient must have weighted degree at least two and thus explicit order at least $s^2$. The latter must have weighted degree at least one and explicit order at least $s$. Their product is therefore divisible by $s^3$. Every Leibniz defect vanishes in ${\mathcal{R}}_2$.

For the witness pair, use $s^2{M}_a$ and $s{Z}_g$. The nonzero coefficient of Theorem thm:unfiltered is then multiplied by $s^3$, proving the exact order statement.