Near-Null Limits and Contact Distributions for Maxwell Composite Operators

We investigate the simultaneous near-null, longitudinal-continuum, and ultraviolet limits of four-point correlation functions of quadratic composite operators in Maxwell theory. Taking the external momenta to be collinear and spacelike, we weight each loop integral by a smooth test function of the routed longitudinal momentum. All three connected Wick contraction cycles are generated from a common Gaussian source functional and regulated by a cutoff on the total proper time. For the resulting integrals restricted to a fixed ultraviolet proper-time interval, we establish the uniform bound $\left|\mathcal I_{\delta,M}^{\epsilon}[\phi]-\mathcal I_{0,\infty}^{\epsilon}[\phi]\right|\le C\|\phi\|_{C_b^8}\left(\delta\epsilon^{-3}+M^{-2}\delta^{-3}\epsilon^{-1}\right)$, where $\delta$ denotes the dimensionless near-null tilt parameter and $M$ specifies the longitudinal lattice resolution. This estimate determines the joint scaling regimes under which a universal counterterm subtraction yields an identical renormalized finite part. We further show that the longitudinal logarithmic divergence of the alternating chiral cycle is governed by the measure $8\mathbf 1_{(0,1)}\mathrm d\xi+3\delta_0+3\delta_1$, whereas each of the other two cycles contributes $8\mathbf 1_{(0,2)}\mathrm d\xi$. These distributional coefficients emerge from the complete covariant numerators, including the instantaneous longitudinal terms. Finally, for the alternating cycle and any smooth compactly supported test function coinciding with $\xi^2$ in a neighborhood of $[0,1]$, the critical trajectory $\delta=c\epsilon^3$ with $M^2\delta^3\epsilon\to\infty$ leaves a finite discrepancy of $30c$, demonstrating the sharpness of the tilt scaling. The proof combines exact Gaussian moment reductions, whole-line quadrature error bounds, and an explicit evaluation of boundary jets.