Balanced jets give unique local functional representatives

Polarize each homogeneous field-degree-$r$ jet polynomial into $r$ field slots. Commutativity of the scalar fields identifies it with a polynomial in the formal derivative variables $p_a$, invariant under slot permutations. The invertible linear change $p_a=q_a+P\mathrm{/}r$ identifies the full polynomial algebra with $\mathbb{C}[P]\otimes \mathbb{C}[q_1,\dots ,q_r]\mathrm{/}({\sum }_a{q}_a)$. Slot permutations fix $P$ and act only on the relative factor. Expanding in the independent monomials of $P$, invariance holds if and only if every relative coefficient is invariant. Thus the invariant space is exactly $\mathbb{C}[P]\otimes {\mathcal{B}}_r$. In each finite degree choose a basis by ordinary finite Gaussian elimination on the real coefficient polynomials; the monomial ordering gives compatible choices across degrees. The Leibniz rule identifies multiplication by $P_{\mu }$ with the total derivative ${\mathrm{\partial }}_{\mu }$ of the local field polynomial. This is a direct tensor decomposition, so has no hidden relations between total and balanced derivatives.

Expand a density using step 1.1 as a finite sum $p(\mathrm{\partial })B_b$. In its integral against $f$, move $p(\mathrm{\partial })$ onto the compact coefficient by integration by parts, obtaining $p(-\mathrm{\partial })f$ multiplying $B_b$. This gives existence of the displayed representation and preserves compact support. Degree zero gives the scalar $c=F(0)$ and is separated from the positive field degrees. To prove uniqueness, suppose a balanced expression represents zero. Separate its field degrees by replacing $\varphi$ by $t\varphi$, polarize, and evaluate its $r$ slots on the smooth exponential fields $e^{2\pi i{k}_a\cdot x}$; complex exponentials are permitted by complex multilinear extension from real configurations. The resulting identity is a finite sum of ${\widehat{f}}_{r,b}(-{\sum }_a{k}_a)$ times the balanced polynomial $b(2\pi i{k}_1,\dots ,2\pi i{k}_r)$. For fixed total momentum, vary freely the relative momenta. Independence of the basis polynomials on ${\sum }_a{q}_a=0$ makes every coefficient Fourier transform zero at that total momentum. Since that momentum is arbitrary, [F2] gives every $f_{r,b}=0$; also $c=0$.

The same uniqueness is local. In center and relative coordinates for the polarized variational kernel on the thin diagonal, balanced differential operators differentiate only the relative delta variables; their coefficients are the $f_{r,b}$ in the center variable, up to fixed nonzero coordinate constants. Test with a center test times a relative monomial times a cutoff equal to one near the relative origin. Starting at the highest relative order and descending isolates every independent relative delta-jet coefficient. Thus vanishing of the kernel over an open center set forces each $f_{r,b}$ to vanish there. If $F$ is unchanged by all variations supported in an open set $U$, its polarized kernels vanish where any one field slot lies in $U$; near their thin diagonal this is precisely the preceding situation. Conversely, if every coefficient vanishes on $U$, no variation supported in $U$ changes $F$. This proves equality with the functional support. Under the action defined in the Statement, a Lorentz transformation substitutes $p_a\mapsto {\mathrm{\Lambda }}^{\mathsf{T}}{p}_a$ and transforms each coefficient by $f(x)\mapsto f({\mathrm{\Lambda }}^{-1}(x-a))$; $\det \mathrm{\Lambda }=1$ preserves $d^{4}x$. Thus $P$ and all $q_a$ transform by the same linear map and the tensor decomposition is preserved. Translations act on coefficients by the same formula with $\mathrm{\Lambda }=I$, and, because the chosen basis is real, complex conjugation sends each coefficient to its complex conjugate. Hence the decomposition is equivariant as stated.

For a general polynomial label write its unique tensor decomposition as ${\sum }_{\beta ,b}{c}_{\beta ,b}{\mathrm{\partial }}^{\beta }{B}_b$, and define its coefficient by ${\sum }_{\beta ,b}{c}_{\beta ,b}{\mathrm{\partial }}_{x_i}^{\beta }K(\dots ,B_b,\dots )$, independently in every entry. Commutativity of coordinate derivatives and the direct decomposition in step 1.1 make this consistent, linear, and the unique extension satisfying the displayed derivative identity. Smearing and integrating by parts gives exactly the coefficient normal form of step 2.1, so equivalent functional presentations yield the same map. The construction respects the equivariance and support statements of step 3.1. Finally, dilation of a distributional derivative gives $({\mathrm{\partial }}^{\beta }T{)}_{\rho }={\rho }^{-\mathrm{\mid }\beta \mathrm{\mid }}{\mathrm{\partial }}^{\beta }{T}_{\rho }$, so the added derivative increases a scaling upper bound by at most $\mathrm{\mid }\beta \mathrm{\mid }$. All sums and derivative orders here are finite.