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Title: Positive geometry for nonplanar $N=4$ super-Yang-Mills loop integrands
Plain: The amplituhedron covers only the planar limit, where the number of colors is infinite. Finite numbers of colors require nonplanar diagrams, and no geometric shape for them is known.
Precise: Find a region in a space of kinematic data whose canonical form (the unique differential form with logarithmic singularities on the region's boundaries) equals the subleading-color (nonplanar) or the full color-dressed $L$-loop four-point integrand of $N=4$ super-Yang-Mills; momentum twistors are unavailable because there is no planar ordering. Nonplanar integrands with only logarithmic singularities and no poles at infinity have been constructed at low loop order.
Settled by: An explicit geometry whose canonical form reproduces the known two- and three-loop nonplanar integrands and predicts higher loops.
Kind: well-posed
Answer type: identification
References: (empty)
Literature status: open
Status note: No geometry is known as of 2026.
Posed since: (empty)