Theorem: F9a for the completed new families

On

the descended degree-four coefficient families

are jointly continuous into $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$. The corresponding degree-two families and scalar family retain their inherited F9a continuity. The zero new $C_3$-families are continuous.

Proof. For degree-four $j=2$, the assertions are exactly the inherited two-edge test-family continuity.

At degree-four $j=1$, the terms

are separately continuous trilinear maps of the fixed-support tests. This follows from inherited test-family continuity, Lemma lem:Tcontinuity, and separately continuous classical multiplication on the carrier. Lemma lem:Frechet makes each trilinear family jointly continuous. Their prescribed sums are therefore continuous.

For $j=0$, the family $B_f{W}_g{B}_{f^{\prime }}$ is separately continuous in the three tests by repeated classical multiplication. The same Fréchet multilinear argument gives joint continuity.

The degree-two positive coefficients and the scalar family are inherited from IX-Free. Their order-zero families also follow from the same classical-product argument. Explicit zero coefficients require no estimate.