Tao's area inequality via conormal intersections: Ordered rectangles of every aspect ratio, and cyclic quadrilaterals

We give a proof of Tao’s area-inequality conjecture (Conjecture 5.6 of [1]): if four simple closed polygonal curves in the cylinder (R/LZ) × R, homologous to the horizontal circle, have vanishing alternating action, then they jointly inscribe a possibly degenerate square in the prescribed order. The same argument gives ordered rectangles of every aspect ratio and, with suitable weights, similar copies of any cyclic quadrilateral. A symmetric linear shear of cotangent coordinates sends the configuration locus onto the conormal bundle of the diagonal of a torus, and the vanishing of the alternating action is exactly the cohomological condition under which a Hamiltonian image of the zero section must meet that conormal. The intersection is proved by restricting a generating function quadratic at infinity to the diagonal and applying finite-dimensional Morse theory. The argument uses the classical existence theorem for such generating functions and the isotopy classification of essential curves in an annulus.