Corollary: Which relations can contribute to a natural defect

The null defect of $F_{T_0}$ vanishes on $H_{\mathrm{flat}}$, and hence factors algebraically through $H^{\circ }/H_{\mathrm{flat}}$. Locally, its coefficient defects depend on finite transverse jets of the functions $f_{\gamma }(a)$.

These categories form a filtration, not a canonical direct-sum decomposition. A potentially jet-detected relation need not have a nonzero defect.

Proof. Universal relations vanish for every generator-compatible lift by (eq:naturaldefect). Flat-annihilator relations vanish locally by Lemma lem:jetdefect, hence globally by locality. The factorization follows. The same lemma proves finite-jet dependence. No argument asserts nonvanishing for every remaining relation.