Completion status

No new completion class is frozen in this manuscript. Hence:

• inside ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$, completion is Proved obstructed (Input statement input:XIIID); the cochain ${\mathcal{C}}_{\xi }=-{\mathfrak{B}}_{\xi }[D\gamma ,\varphi ]$ is admitted, cancels (eq:eval) algebraically on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$, and is not the variation of one fixed functional under the forced representation; citing it as the completion would fail this gate;

• a pure auxiliary or ghost rule changes $\mathcal{O}$ by zero (Input statement input:XIIF);

• constant compact metric translations commute and do not preserve ${\mathcal{N}}_{\mathrm{sol}}$; they do not supply the representation needed for a cocycle test (XIII-A);

• a coefficientwise quotient by vanishing on ${\mathcal{C}}_W$ is incompatible with the inherited brackets (Input statement input:XIIIA); the evaluation identity of Theorem thm:eval is not promoted to a quotient;

• nonlocal Cauchy-data actions, nonzero past data, and other integration interfaces remain undecided; none is frozen or tested here.

A2 ends: Constructed as an evaluation identity on ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$ with the pairing printed on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$; Proved obstructed for unqualified invariance on ${\mathcal{C}}_{\mathrm{sol}}$ and for completion inside ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$; the completion in any unfrozen class is outside the claim.