Scope, interfaces, and essential corrections

XI-D stopped before quantum composite admission and distinguished bare closure from closure of a complete observable XID. This paper takes the admission step by specifying Hadamard product data. It does not turn the three routes into a single BRST theorem.

Three distinctions prevent incorrect conclusions.

relations versus causal multiplication. An extension may retain every causal commutation relation of XI-B while using a different symbol product. The obstruction below is to extending the raw causal-Weyl contraction prescription to singular polynomial kernels with the stated continuity. It is not an obstruction to every algebra having those commutators.

elements versus old symbol formulas. An exact algebra embedding preserves old elements and products. It need not preserve their Weyl-symbol formulas after a change to normal-ordering coordinates. Agreement on smooth finite-rank symbols is stated with that ordering convention made explicit.

identities versus differentials. The expression $s_0{\mathcal{B}}_f+\frac{1}{2}\int fc\cdot \partial \varphi$ is a first-order coefficient calculation. The second term is the scalar co-transformation in that calculation. Naming it does not construct a differential on the on-shell algebra.

The extension is restricted to polynomials with fixed compact distributional kernels. We do not assert that a pointwise wavefront condition alone gives a closed algebra of every nonpolynomial smooth functional. This restriction also avoids the broader functional-analytic issue described in HRV.

The three endings are applied to each named route: Constructed, Proved obstructed, or Unresolved. Excluded calculations are not silently incorporated into a constructed route.