Dual‑volume interpretation of loop integrands (dual Amplituhedron)

Ascertain whether loop integrands in planar N=4 super Yang–Mills theory admit a dual‑volume interpretation as canonical forms of a dual Amplituhedron, and, if so, construct the corresponding dual geometry realizing this interpretation.

Background

At tree level, amplitudes can be interpreted as volumes of dual polytopes, and integrands are positive inside the corresponding positive geometry. At loop level, the existence and structure of a dual Amplituhedron are conjectural, and it is unknown whether integrands admit a dual‑volume description.

Clarifying this would bridge positive geometry with integrated observables at loop level and explain observed positivity structures beyond the integrand.