This paper discusses the phase factors for the S-matrix of planar N=4 gauge theory, focusing on modifications at four-loop order. While these modifications break perturbative BMN-scaling, the universal scaling function for large-spin twist operators may still preserve the Kotikov-Lipatov transcendentality principle. A natural choice of phase factor modifies the contribution of odd zeta functions in the scaling function. The authors argue that this choice is non-perturbatively related to a crossing-symmetric phase factor for perturbative string theory on AdS₅×S⁵. This proposal could resolve long-standing discrepancies between gauge and string theory in the AdS/CFT correspondence. The paper explores the integral equations for the scaling function, showing that the dressing phase affects the universal scaling function. The scaling function is derived from the Bethe ansatz and is shown to obey the Kotikov-Lipatov transcendentality principle. The authors analyze the effects of the dressing phase on the scaling function, finding that it can cancel odd-zeta contributions or synchronize their signs with even-zeta contributions. They propose a specific form for the dressing phase constants that preserves transcendentality and agrees with string theory results. The paper also investigates the analytic structure of the scaling functions and their convergence properties. The results suggest a singularity at λ = -π² with exponents ±1/2. The authors compare their results with string theory calculations and find good agreement, supporting the validity of their proposal. They also test their proposal by computing the scaling function at four loops and find agreement with existing results. The paper concludes that the proposed dressing phase is non-trivial and preserves the transcendentality principle, providing a consistent description of the AdS/CFT correspondence.This paper discusses the phase factors for the S-matrix of planar N=4 gauge theory, focusing on modifications at four-loop order. While these modifications break perturbative BMN-scaling, the universal scaling function for large-spin twist operators may still preserve the Kotikov-Lipatov transcendentality principle. A natural choice of phase factor modifies the contribution of odd zeta functions in the scaling function. The authors argue that this choice is non-perturbatively related to a crossing-symmetric phase factor for perturbative string theory on AdS₅×S⁵. This proposal could resolve long-standing discrepancies between gauge and string theory in the AdS/CFT correspondence. The paper explores the integral equations for the scaling function, showing that the dressing phase affects the universal scaling function. The scaling function is derived from the Bethe ansatz and is shown to obey the Kotikov-Lipatov transcendentality principle. The authors analyze the effects of the dressing phase on the scaling function, finding that it can cancel odd-zeta contributions or synchronize their signs with even-zeta contributions. They propose a specific form for the dressing phase constants that preserves transcendentality and agrees with string theory results. The paper also investigates the analytic structure of the scaling functions and their convergence properties. The results suggest a singularity at λ = -π² with exponents ±1/2. The authors compare their results with string theory calculations and find good agreement, supporting the validity of their proposal. They also test their proposal by computing the scaling function at four loops and find agreement with existing results. The paper concludes that the proposed dressing phase is non-trivial and preserves the transcendentality principle, providing a consistent description of the AdS/CFT correspondence.