Theorem: F9a on the complete declared new sector

Every new coefficient family has joint fixed-support test-family continuity into $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$. For degree-four intermediate arguments, this means continuity of

The corresponding degree-two and first-stage families have the same property. F9b, continuity or hypocontinuity on the actual intermediate subspace topologies, is unclaimed.

Proof. The new positive-order degree-four values are finite sums of already established one-edge and two-edge families. Their coefficient kernels and all trace coordinates are continuous by Proposition prop:firstcontinuity, the carrier's selected self-output input, and Proposition prop:F9a2. Classical words have the same fixed-support continuity by their explicit compact coefficient constructions and directional Leibniz formulae. The scalar entries have finite-order distribution bounds on the fixed joint compact support.

These proofs estimate the pulled-back generating families. They never estimate ${p_L}^{-1},{p_R}^{-1},{q_L}^{-1}$, or ${q_R}^{-1}$ in the carrier topology. Algebraic injectivity supplies no such inverse estimates.