Theorem: Cocycle obstruction for the frozen class

For these detecting fixed data,

Consequently every shell-descending representation in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}(f,g,W)$ fails the necessary cocycle condition for the cancellation target. No member satisfies all declared interfaces.

Proof. The nonzero shell value (eq:B-theta) proves that ${\mathfrak{B}}_{\theta }\notin {\mathcal{N}}_{\mathrm{sol}}$. By Theorem thm:triviality, every eligible action induces $\bar{\delta }=0$. Therefore

At $u_{\ast }$ its representative value is $8I>0$. Apply Proposition prop:criterion.