Theorem: Narrow classical admission

Under the additional microlocal input:

1. The species-symmetrized polynomial $F_t$ is an element of the stated classical microcausal polynomial carrier.

2. The free-shell class of $t_{\mathrm{D}}$ has a microcausal representative, namely $t_{\mathrm{D}}$ itself.

3. Under the separately stated transport premise,

   $$\mathrm{WF}({\mathsf{W}}_t)\cap ({\tilde{V}}_{+}^3\cup {\tilde{V}}_{-}^3)=\varnothing ,{\mathsf{W}}_t=({\mathsf{E}}_0\otimes {\mathsf{E}}_0\otimes {\mathsf{E}}_m)t_{\mathrm{D}}.$$

These are classical kernel and free-shell conclusions. They are not time-ordering, graph normalization, or BRST/BV observability statements.

Proof. The closed future and past causal cones are pointed. A nonzero tuple wholly in either cone cannot sum to zero. Equation (eq:tcone) therefore excludes the forbidden cone products.

Embed the typed kernel in the direct-sum field bundle and apply the normalized species symmetrization. Permutations preserve compactness and the sum-zero covector condition. Evaluating the symmetrized kernel on the repeated multiplet argument gives the same polynomial $F_t$; no extra factorial is inserted. This proves item 1.

Define the free-shell null space in the matching compact-kernel space by pairings with all smooth solutions of $\square a=\square c=0$, $P_m\varphi =0$. The class $t_{\mathrm{D}}+{\mathcal{I}}_{\mathrm{sol}}$ contains $t_{\mathrm{D}}$. Its own cone exclusion proves item 2 directly. It is unnecessary to enlarge the frozen-kernel quantifier of XI-G's printed criterion.

For item 3, the transport premise preserves the component covectors in the fixed Minkowski frames. A surviving tuple therefore still sums to zero and is nonzero. Pointedness again excludes both cone products.