Proposition: Zero-sum reduction does not assign a joint section

Cotangent addition sends every tuple in (eq:diagnostic) to the zero covector of $T_w^{\ast }W$.

Accordingly, reducing the diagnostic tuple by its copied linear dependence does not produce a point of ${\Lambda }_{\mathrm{char}}$. The diagnostic alone cannot define an ordinary principal-symbol section of the grouped coefficient.

Proof. The sum is

By definition, the zero covector is removed from ${\Lambda }_{\mathrm{char}}$. An ordinary conormal or Lagrangian principal symbol is a section over its punctured cotangent relation, not a value assigned at this zero covector.

The calculation specifies a cancellation of input covectors. It specifies neither membership of an output distribution in a conormal class nor an amplitude on an output relation.