Theorem: Symmetric cubic form and joint limit

For real compact $f$, the fully Wick-ordered kernel defines a symmetric continuous sesquilinear form, and

The limit is joint and independent of the bounded profiles with the stated support properties. No transfer-zero distribution is added.

Proof. The bound (eq:kernelbound) is integrable with weight ${R_p}^{-20}$. For each pair transfer use $q\in {\mathbb{R}}^3$ and six remaining variables $y$; the integral is bounded by

The ultraviolet error has $R_p\gtrsim \Lambda$; extract ${\Lambda }^{-1}$ and retain an integrable weight. The transfer exclusion contributes $C({\delta }^3+\delta )$. The response-edge strip has length at most $2\epsilon$; the pre-integration estimate of Lemma lem:kernel gives $C\epsilon$.

In (eq:sector), the two Schwartz bounds yield $⟨z_j{⟩}^{-60}$ for each spectator, and ${\prod }_{\mathrm{external}}⟨p_j{⟩}^{-30}$ dominates $C{R}_p^{-30}$. Spectators therefore contribute $I_{\ast }^{n-s}$. The combinatorial factor is at most $3(n+3{)}^{3/2}$. The weights in $\mathfrak{p}$ dominate these factors, proving both estimates after summing sectors.

For real $f$, complex conjugation reverses all mode signs. Combined with signed permutation symmetry, this interchanges the $(r,s)$ and $(s,r)$ blocks and proves symmetry. Weighted integral convergence selects the locally integrable kernel itself.