Poincare-Miranda theorem

If each component of a continuous map on a box has opposite weak signs on the corresponding pair of faces, the map has a zero.

Projecting a sign-corrected displacement map back to a square turns Brouwer's fixed point into a zero of the displacement, proving that paths joining opposite side pairs intersect.

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The Poincaré–Miranda theorem is a result in topology that relates to the existence of continuous choices of functions under certain conditions. It is often used in the context of multiple variables and can be seen as a generalization of the intermediate value theorem for higher-dimensional spaces.