Both internal placements of K 5

For an amplitude of degree at most four, split its coefficient polynomial by degree:

Terms with no degree-four contribution have $A_{[4]}=0$.

The splitting (eq:split) is exact. On normalized annular tests,

is uniformly bounded, and

in the finite-seminorm distribution topology of Theorem thm:dyadic, uniformly on compact parameter sets.

Every lower-degree term of the finite differential formula is retained in $A_{[\le 3]}$. The regular opposite-sheet term $b_s$ in (eq:pole) remains part of the full multiplier in both pieces.

Proof. Each degree-$d$ term scales as $L^{d-2}$. For $d\le 3$, division by $L^2$ gives at most $L^{-1}$. There are finitely many terms, so the uniform estimates can be summed. No term of $M_{\mathrm{R}}$ was dropped in defining the generators.

This is not a smoothing remainder theorem for the quantum coefficient. It is a proved, complete lower-degree estimate for the named internal module.

reconstruction

The map (eq:reconstruction) is a continuous linear map from the coefficient module, with its compact-set smooth topology, to ${\mathcal{D}}^{\prime }(V)$.

It is also the iterated normal inverse Fourier transform of the generators (eq:generators), followed by the specified coefficient multiplication.

On compact subsets of the original $W$ chart, its estimates hold for arbitrary joint tests in $(a,b,z,y)$. No restriction to a fixed singular residual value is used. A fixed derivative in the original variables raises the finite polynomial degree by at most its differential order.

Proof. The tensor product ${\delta }^{(j)}(q)\otimes {{\partial }_r}^{\beta }R(r)$ is defined. Its pullback to the adapted chart merely adds passive variables. Multiplication by a smooth coefficient is defined.

For a compactly supported joint test, the pairing is first a finite $q$ derivative evaluated at $q=0$, then a pairing with the fixed distribution ${{\partial }_r}^{\beta }R$, and integration over the remaining compact tangential variables. The distributional finite-order estimate for $R$ on the projected compact set bounds this by finitely many derivatives of the coefficient and test. This proves continuity.

Separate Fourier inversion gives

This proves the reconstruction identity without evaluating an oscillatory amplitude at a singular point.