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Abstract

This article introduces a systematic framework to understand (not to derive
yet) the all-loop 4-particle amplituhedron in planar N=4 SYM, utilizing both
positivity and the Mondrian diagrammatics. Its key idea is the simplest one so
far: we can decouple one or more sets of loop variables (x,y,z,w) from the rest
by just setting these variables to either zero or infinity so that their
relevant positivity conditions are trivialized, then the all-loop consistency
requires that we get lower loop amplituhedra as "residues". These decoupling
relations connect higher loop DCI integrals with the lower ones, enabling us to
identify their coefficients starting from the 3-loop case. And surprisingly,
the delicate mechanism of this process is the simple Mondrian rule D=X+Y, which
forces those visually non-Mondrian DCI integrals to have the correct
coefficients such that the amplituhedron can exactly reduce to the lower loop
one. Examples cover all DCI integrals at L=3,4,5,6, especially, the subtle
6-loop coefficients +2 and 0 are neatly explained in this way.