An exact local kernel criterion, not a complete graph kernel

The finite Taylor multiplication identity gives

where $\xi$ denotes the passive chart variables. It follows directly by applying the Leibniz rule to a test pairing.

On a product subchart around $q=0$, define

Then

The independence used here is that of the finitely many one-dimensional delta derivatives after the coefficients have been reduced to $q=0$. Testing against a compact function with one prescribed finite $q$ jet extracts each $B_i$.

Equation (eq:localcriterion) is an exact criterion for the transcribed local module. It still contains distributional relations in $r$. It does not claim that the few relations in Lemma lem:relations generate every such relation. It also does not determine cancellations between different placements or between different operational entries in a common reconstruction.

The complete six-entry presentation space is a different object. XIV-J calls it a finite direct sum of the internal coefficient presentation spaces, with summation into the common tensor-valued kernel. That statement does not enumerate every coefficient space and every cross-entry identity in this patch. The $K_{47}$ entry, in particular, is not identified with one instance of the ${\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)$ local family.