Changing a distribution does not prove a change of class

The supplied XI-G excerpt defines a free-shell class of a compact cubic kernel $u$ on $M^3$, with two symmetric-tensor slots and one scalar slot XIG. Let $\mathcal{K}$ denote the vector space of compact kernels with this slot structure and tensor-slot symmetry. Write

Here $P_m$ is the scalar free operator used in that excerpt. Compact support allows the displayed pairing with smooth solutions.

The class is

The reported XI-G result is

for the frozen data and hypotheses of its obstruction theorem XIG. The symbol ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ denotes the predecessor's microcausal polynomial target space; no new admission theorem for that space is assumed.

deduction: the exact transfer condition. If a future prepared observable kernel $u_D\in \mathcal{K}$ satisfies

then it belongs to the same class and inherits the class obstruction:

. The set ${\mathcal{I}}_{\mathrm{sol}}$ is a vector subspace by linearity of the pairings. If $u_D-u\in {\mathcal{I}}_{\mathrm{sol}}$, then

Equation (eq:XIGobstruction) then gives (eq:sameclasstransfer).

A nonzero difference $u_D-u$ does not exclude (eq:sameclasscondition). Thus the sentence “the distributions are different, so XI-G does not govern the new distribution” is not justified without a free-shell comparison.

There is a further distinction: $J_D$ is a source label, whereas $u$ in (eq:XIGobstruction) is a compact observable kernel on $M^3$. The supplied material does not define the map from $J_D$ to a corresponding kernel $u_D$. It is therefore not legitimate to identify these objects merely because both are described as distributions.

The corrected conclusion is:

XI-G remains valid for its frozen class. Its applicability to a kernel induced by $J_D$ has not been established or excluded. A same-class proof would transfer the obstruction. A different-class proof would remove that particular transfer, but would not prove admission of the new class.

Even if a kernel induced by ${\sigma }_{\epsilon }{J}_3$ were admitted to some declared class, that would not establish admission of the kernel induced by ${\sigma }_{\epsilon }{J}_3+D_{\sigma }$. The correction and the total kernel must be analyzed.