Color-kinematics dual form of the five-loop $N=4$ super-Yang-Mills amplitude
In plain words
The squaring trick needs the gauge-theory answer arranged so that its momentum-dependent parts obey the same algebra as its color (charge) parts. At five loops nobody has found such an arrangement for four-gluon scattering in the most symmetric gauge theory.
Precise statement
Determine whether the five-loop four-point integrand of $N=4$ super-Yang-Mills admits a representation on cubic graphs with local numerators $n_i$ satisfying every kinematic Jacobi relation $n_i + n_j + n_k = 0$ that mirrors a color Jacobi identity (a BCJ representation). An answer is an explicit representation verified on all unitarity cuts, or a proof that none exists with polynomial numerators of the allowed power counting.
What would settle it
An explicit BCJ representation checked on all cuts, or an exhaustive ansatz no-go proof.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
The 2018 five-loop $N=8$ supergravity integrand was built with the generalized double copy, which does not require a BCJ form; no BCJ form of the five-loop four-point amplitude is reported as of 2026.
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