Graviton scattering in the null surface formulation. Part III: Fourth-order Bondi shear and the tree-level amplitude

We complete a trilogy on quantum graviton scattering in the null surface formulation (NSF) of general relativity by computing the fourth-order Bondi shear $\sigma^+_4$ and establishing three results of general scope. The perturbative $S$-matrix of the NSF is UV-finite at every loop order. This follows from the kernel scaling $K^{(n)}\sim\omega_{\mathrm{ext}}/q^{n-2}$, which we derive by induction on the recursive null-cone scattering equation; the $L$-loop integrand then scales as $dq/q^{4L}$, which is convergent for all $L\geq 1$ without regularization. We show that a simple loop-counting formula, $L=(n_1+n_2-6)/2$, classifies the topologically distinct contributions to $2\to 2$ graviton scattering by the perturbative orders $n_1$, $n_2$ of the out-operators. Tree level ($L=0$) is exhausted by $\mathcal{M}^{(22)}$, $\mathcal{M}^{(33)}$, and $\mathcal{M}^{(24)}$, which together reproduce the Weinberg--DeWitt amplitude $\mathcal{M}_{\mathrm{tree}}=-\kappa^2s^3/(4tu)$. The complete 1-loop amplitude requires, in addition to $\sigma^+_4$, the fields $\sigma^+_5$ and $\sigma^+_6$. At order $n\geq 4$ the standard Jordan--Pauli argument, which equates the advanced and retarded null-cone contributions, must be extended. The advanced cone receives additional contributions from $\delta\sigma^+_j$ ($j<n$), the nontrivial scattering corrections determined at previous orders. We formulate this as a generalized Jordan--Pauli relation that provides a systematic, order-by-order procedure for computing $\sigma^+_n$ from the free incoming datum $\sigma^-$. The computation of $\sigma^+_4$ uses three retarded-cone pairs, the conformal factor $\delta\Omega^-_4$, and -- for the first time -- advanced-cone corrections from pairs $(1,3)$ and $(2,2)$ built from the known $\sigma^+_2$ and $\sigma^+_3$