Abstract

We freeze the classical completion route for the displayed second-order coefficient, without changing the action, the external switches, or the inherited carrier. The tested correction class has parameter-local unpropagated components and at most one new retarded metric application. A successful correction must also supply an even representation of the compact vector-field algebra preserving the complete free-shell equation ideal.

These requirements have a decisive consequence. A causal, zero-past field variation tangent to the homogeneous free shell vanishes there by Cauchy uniqueness. Every shell-descending action in the frozen class therefore induces the trivial representation on the existing classical shell quotient.

We specialize the copied XII-F witness family so that its detecting parameter is an explicit commutator of two allowed compact parameters. The auxiliary pairing is nonzero on that commutator. Its necessary cocycle condition consequently fails for every representation in the frozen class, excluding both the slot equation and an integrated compensating functional in that class.

An admitted parameter-dependent bilinear cochain cancels the copied value algebraically. Its kernel is displayed, but it is not the variation of a compensating functional under the declared action. This distinction prevents algebraic cancellation from being released as an integrated completion.

The ending is Proved obstructed for the explicitly frozen class, not for arbitrary nonlocal actions outside it. The quantum two-line extension remains Unresolved. No quantum gauge-consistency defect or interacting differential is opened.