Descent and the degree-four diagnostic

For $F_d=\mathsf{Rec}(P_{5,\pm ,d})$, the quotient class

is nevertheless well-defined: the discarded terms are already in $F_{d-1}$. This algebraic associated-graded statement selects no representative or joint symbol.

A continuous section of a specified quotient $q:B\to S$ requires a complemented kernel. Indeed a section gives the projection $I-sq$; conversely a continuous projection $P$ onto $\ker q$ gives $s(qb)=(I-P)b$. No such actual joint quotient and section are constructed here.