Scalar graph coefficients have finite upper scaling bounds

Differentiation adds at most its order to a scaling upper bound: $⟨({\mathrm{\partial }}^{\alpha }T{)}_{\rho },\phi ⟩=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\rho }^{-\mathrm{\mid }\alpha \mathrm{\mid }}⟨T_{\rho },{\mathrm{\partial }}^{\alpha }\phi ⟩$. A surjective linear pullback does not increase that bound: in coordinates $(y,z)$ with $y=Lx$, $L^{\ast }T=T(y)\otimes 1(z)$, and integration of a test in $z$ cancels the extra dilation Jacobian. The same observation shows that a translation-invariant kernel, written as a relative distribution tensored with a constant center coordinate, has the same full-coordinate and relative scaling bounds. Its wavefront covectors satisfy ${\sum }_i{\xi }_i=0$: with a product localization in center and relative coordinates the center Fourier transform decreases rapidly, and the convolution localization argument of [F3] gives the statement for arbitrary localizations. Thus every cross edge ${\mathrm{\partial }}^{{\alpha }_e}W(x_i,x_j)$ has upper scaling bound $2+\mathrm{\mid }{\alpha }_e\mathrm{\mid }$, while every lower-order block kernel has covectors whose sum within its block is zero.

Fix a nontrivial partition $I\mid J$ of the vertices. The extended subgraph kernels $t_I$ and $t_J$ are distributions on their entire separate coordinate blocks, including their own total diagonals. Lift the two kernels to the full vertex-coordinate space by tensoring with the constant distribution in the other block. Step 1.1 preserves their finite nonnegative upper scaling bounds $b_I,b_J$. Their covector tuples occupy disjoint blocks, so their product is their tensor product and [F4] bounds its degree by $b_I+b_J$. A one-vertex block contributes the constant kernel and bound zero. Multiply this tensor product by all cross-edge kernels oriented as ${\mathrm{\partial }}^{{\alpha }_e}W(x_i,x_j)$ with $i\in I,j\in J$. At a full-diagonal point every nonzero wavefront contribution from a cross edge places a future null covector at $i$ and its negative at $j$. If the sum of all factor covectors were zero, summing its entries over $I$ would give a sum of future null covectors equal to zero, because the internal block covectors sum to zero by step 1.1. The future causal cone is pointed, so every cross-edge covector must vanish. The two remaining block tuples occupy disjoint coordinate blocks, and hence must separately vanish. This proves joint no-cancellation at the full diagonal. It also proves it at other points, since the wavefront condition on $W$ has the same orientation everywhere.

By [F3] and step 2.1 the causal block expression $B_I=t_I\otimes t_J{\prod }_{e=(i,j),i\in I,j\in J}{\mathrm{\partial }}^{{\alpha }_e}W(x_i,x_j)$ is a distribution even at the full diagonal. Apply [F4] there and step 1.1 to obtain the finite upper bound ${\mathrm{sd}}_0(B_I)\le b_I+b_J+{\sum }_{e\text{ crossing }I\mid J}(2+\mathrm{\mid }{\alpha }_e\mathrm{\mid })\mathrm{.}$ All factors are already defined at the center where [F4] is used. This argument requires no product of coincident $H_F$ factors and no directional property of an arbitrary unextended punctured-stratum product. Translation invariance makes the full-coordinate estimate exactly the relative-coordinate estimate.

The causal split sets $C_I=\{x_i\mathrm{\notin }{J}^{-}(x_j):i\in I,j\mathrm{\notin }I\}$ cover the complement of the total diagonal. Indeed, choose a timelike linear time function taking unequal values on two distinct entries of a configuration, then split the finitely many values at a level attained by none of them. These open sets are invariant under translations and positive relative dilations. Their intersections with the relative unit sphere form a finite open cover of that compact sphere. Using [F6] in its coordinate patches, choose a finite smooth partition $f_I$ there, subordinate to the cover, and extend it homogeneously by $f_I(\rho z)=f_I(z)$ to punctured relative space. On $C_I$, [F1] and lower-order causal factorization identify the graph coefficient with $B_I$. The lower-order expressions agree on overlaps: refining two causal splits into their intersections expresses both as the same block product, and the two incomparable blocks are spacelike, where $W=W^{\mathsf{T}}$. Hence the graph distribution is $t_{\mathrm{\Gamma }}={\sum }_I{f}_I{B}_I$ off zero. For a fixed punctured test $\phi$, each $f_I\phi$ is a fixed smooth compactly supported punctured test; homogeneity gives $(f_I{B}_I{)}_{\rho }(\phi )=(B_I{)}_{\rho }(f_I\phi )$. The finite sum therefore has the maximum of the finite bounds in step 3.1.

For a primitive graph, the definitions above mean every proper induced vertex subgraph lies below its divergence threshold. Induct on vertex number with the engineering upper bound $b_I={\sum }_{e\text{ internal to }I}(2+\mathrm{\mid }{\alpha }_e\mathrm{\mid })$ for a block $I$, and likewise for $J$. For independent balanced subgraph coefficients choose same-degree extensions by [F5]; their total-derivative descendants and any allowed local normalizations retain the contract's engineering upper bounds. Thus the lower-order block kernels, including any permitted finite delta-derivative terms, have these nonnegative upper bounds, starting from the zero bound at one vertex. Step 3.1 then counts every edge exactly once and step 4.1 gives ${\mathrm{sd}}_0(t_{\mathrm{\Gamma }})\le 2\mathrm{\mid }E(\mathrm{\Gamma })\mathrm{\mid }+{\sum }_e\mathrm{\mid }{\alpha }_e\mathrm{\mid }$. If a proper vertex subgraph has a chosen local counterterm, [F5] makes it a finite sum of finite-order delta derivatives. Each has finite relative scaling degree equal to its relative dimension plus its derivative order. Its translation-invariant lift still has block momentum sum zero by step 1.1, so steps 2.1–4.1 apply with a finite nonnegative upper bound for that chosen block. There are only finitely many vertex subsets and finitely many counterterm jets at this fixed graph order. Taking their finite maximum proves the stated general finite upper bound.