Theorem: Complete polynomial smoothing estimate

The bound (eq:mainbound) holds for every compact $L⋐U$.

Proof. A compact distribution of order at most eight in dimension eight satisfies

Apply its finite-order estimate to the translated differentiated profile.

The complete internal kernel has order at most four. Four derivatives of a kinetic or trace-kinetic vertex test require at most five crossed derivatives and at most five joint-test derivatives. The mass test requires no more. The resulting conservative power is ${\delta }^{-8-8-5}={\delta }^{-21}$. The smooth observations, the fixed root, and finite sums enter only the compact-local constant. Restore $|\lambda {|}^3{\hslash }^2$.

Proof. The action, source, density, root, and crossed formulas were proved or explicitly prescribed before substitution. Theorem thm:substitution identifies the insertion kernel; Theorems thm:seed and thm:ledger give its complete positive-scale action. Theorem thm:combined proves the exact regrouping, and Theorem thm:moderate proves the bound.