The paper discusses possible phase factors for the S-matrix of planar $\mathcal{N} = 4$ gauge theory, leading to modifications at four-loop order compared to an earlier proposal. These modifications result in a four-loop breakdown of perturbative BMN-scaling but preserve Kotikov-Lipatov transcendentality in the universal scaling function for large-spin twist operators. A particularly natural choice, unique up to a constant, modifies the overall contribution of terms containing odd zeta functions in the scaling function based on a trivial phase. The authors present evidence that this choice is non-perturbatively related to a recently conjectured crossing-symmetric phase factor for perturbative string theory on $AdS_5 \times S^5$, once the constant is fixed to a particular value. This proposal, if true, might resolve long-standing discrepancies between gauge and string theory in the context of the AdS/CFT correspondence. The paper also explores the analytic continuation of the dressing phase from string theory and derives a closed form for the constants and a concise integral expression for the summed dressing kernel. The results suggest that the correct crossing-invariant choice of constants is twice the analytic continuation of the conjectured strong-coupling dressing phase constants. The authors conclude by discussing the implications of their proposal for the AdS/CFT correspondence and potential tests of their proposal through higher-loop calculations and comparisons with string theory results.The paper discusses possible phase factors for the S-matrix of planar $\mathcal{N} = 4$ gauge theory, leading to modifications at four-loop order compared to an earlier proposal. These modifications result in a four-loop breakdown of perturbative BMN-scaling but preserve Kotikov-Lipatov transcendentality in the universal scaling function for large-spin twist operators. A particularly natural choice, unique up to a constant, modifies the overall contribution of terms containing odd zeta functions in the scaling function based on a trivial phase. The authors present evidence that this choice is non-perturbatively related to a recently conjectured crossing-symmetric phase factor for perturbative string theory on $AdS_5 \times S^5$, once the constant is fixed to a particular value. This proposal, if true, might resolve long-standing discrepancies between gauge and string theory in the context of the AdS/CFT correspondence. The paper also explores the analytic continuation of the dressing phase from string theory and derives a closed form for the constants and a concise integral expression for the summed dressing kernel. The results suggest that the correct crossing-invariant choice of constants is twice the analytic continuation of the conjectured strong-coupling dressing phase constants. The authors conclude by discussing the implications of their proposal for the AdS/CFT correspondence and potential tests of their proposal through higher-loop calculations and comparisons with string theory results.