Generators, coefficients, and the reconstruction rule

The integer $m$ in this definition is the amplitude-filtration parameter. It is not the scalar mass appearing in (eq:vertex). The source uses the same letter in different contexts; the distinction is maintained here.

The coefficient condition is $C^{\infty }(V)$, with $w=(u,\zeta ,q,r,y)$. It permits dependence on $q$ and $r$. It does not require such dependence for every coefficient, but it does not exclude it. There is no later restriction in the supplied definition to coefficients independent of these coordinates.

Thus the proposed free-family interpretation over chart functions cannot be retained. Physical coefficient multiplication takes place after inverse transformation. It can annihilate or relate normal distributions even though the corresponding frequency expressions have distinct polynomial degrees.