Corollary: Presentation degree requires a descent check

A polynomial-degree splitting of presentations does not, merely by being defined on ${\mathcal{P}}_{\mathrm{int}}$, define a splitting on reconstructed kernels. Any proposed degree-based joint operation must satisfy (eq:descent).

Proof. The null presentation (eq:null) relates summands of different polynomial degrees whose reconstructions are nonzero. A rule on presentations that separates them must still respect their sum's zero reconstruction. Proposition prop:descent is exactly the condition ensuring that it does.