Polarization and the ban on diagonal additivity

The second-vertex map is linear in its quadratic vertex. The diagonal iteration is quadratic, not additive, in $V_{1,g}$.

For two specified quadratic vertices $V,W$, with positive Euler coefficients $L_V,L_W$, its polarization is the bookkeeping formula

It is bilinear in these two coefficients and reduces on the diagonal to twice (eq:iteration). Replacing $V$ by $V+W$ therefore retains the mixed terms. Dropping them under a claim of diagonal additivity would be incorrect.

Formula (eq:polarization) does not construct a hierarchy on arbitrary entries. The causal statements made here are the displayed chain support, linearity or polarization, and vanishing when the vertex supports cannot form a causal chain into the insertion support. In particular, both new maps vanish if $\mathrm{supp}g\cap J^{-}(\mathrm{supp}c)=\varnothing$. No full causal-factorization hierarchy is claimed.