We consider the universal behavior of half-BPS correlators in \( \mathcal{N} \) = 4 super-Yang-Mills in the cusp limit where two consecutive separations \( {x}_{12}^2 \) , \( {x}_{23}^2 \) become lightlike. Through the Lagrangian insertion procedure, the Sudakov double-logarithmic divergence of the n-point correlator is related to the (n + 1)-point correlator where the inserted Lagrangian “pinches” to the soft-collinear region of the cusp. We formulate this constraint as a new graphical rule for the f-graphs of the four-point correlator, which turns out to be the most constraining rule known so far. By exploiting this single graphical rule, we bootstrap the planar integrand of the four-point correlator up to ten loops (n = 14) and fix all 22024902 but one coefficient at eleven loops (n = 15); the remaining coefficient is then fixed using the triangle rule. We verify the “Catalan conjecture” for the coefficients of the family of f-graphs known as “anti-prisms” where the coefficient of the twelve-loop (n = 16) anti-prism is found to be −42 by a local analysis of the bootstrap equations. We also comment on the implication of our graphical rule for the non-planar contributions.
