# Unwinding the Amplituhedron in Binary

## TL;DR

This paper introduces a combinatorial and topological binary characterization of the amplituhedron, linking geometric properties to sign flips in projected data, and deriving physical principles like locality and unitarity.

## Contribution

It provides a new binary and topological framework for understanding the amplituhedron, connecting geometry with combinatorial sign flip patterns and offering a fresh perspective on scattering amplitudes.

## Key findings

- Binary description of the amplituhedron via sign flips
- Derivation of locality and unitarity from binary codes
- Introduction of a dual amplituhedron concept

## Abstract

We present new, fundamentally combinatorial and topological characterizations of the amplituhedron. Upon projecting external data through the amplituhedron, the resulting configuration of points has a specified (and maximal) generalized 'winding number'. Equivalently, the amplituhedron can be fully described in binary: canonical projections of the geometry down to one dimension have a specified (and maximal) number of 'sign flips' of the projected data. The locality and unitarity of scattering amplitudes are easily derived as elementary consequences of this binary code. Minimal winding defines a natural 'dual' of the amplituhedron. This picture gives us an avatar of the amplituhedron purely in the configuration space of points in vector space (momentum-twistor space in the physics), a new interpretation of the canonical amplituhedron form, and a direct bosonic understanding of the scattering super-amplitude in planar N = 4 SYM as a differential form on the space of physical kinematical data.