Quantum graviton scattering with definite helicities in the null surface formulation. III: Fourth-order recursion and ultraviolet finiteness
We extend the perturbative null-surface formulation (NSF) scattering map to fourth order and derive an all-order recursion for the quantum cut. After the antipodal matching, both cone sources are evaluated on the same retarded solution determined by the free incoming radiative data. The perturbative NSF equations therefore determine every coefficient $Z_n$ from that data, without introducing new independent asymptotic information. The partial cut $Z_{[N]}=\sum_{j=1}^{N}\ep^j Z_j$ defines the cumulative operator $U_{ω,[N]}=\exp[-\iiωZ_{[N]}]$. An exact factor recursion for this operator gives a generating formula for $δa_{n,λ}^{\mathrm{out}}$ in terms of the order-$n$ cone source and lower-order operators. The scalar flux term $Σ$, which begins quadratically, is included on the cut side of the matching equation; its free quadratic part cancels between future and past infinity and it never introduces a new order-$n$ radiative operator. For smooth smeared radiative data, every finite-order cut is well defined and self-adjoint, so $U_{ω,[N]}$ and its recursive factors are unitary and bounded. The frequency powers in the perturbative coefficients are thus part of the expansion of a bounded unitary operator, rather than separate ultraviolet enhancements. At fourth order we formally determine $δa_{4,λ}^{\mathrm{out}}$ and identify the mixed one-loop sector $\mathcal M_{24}=\mathcal M^{(24)}+\mathcal M^{(42)}$. A general radial power-counting proposition proves ultraviolet finiteness at arbitrary perturbative order for the flat-cone two-vertex sectors. In particular, $\mathcal M_{24}$ and the previously obtained $\mathcal M_{33}$ both scale as $\int^\infty \dd K/K^4$ in the uniform radial ultraviolet region.