= Finite-torus crossing amplification
Take the increasing event that a finite square <torus graph> contains some long horizontal or vertical rectangle crossing. Translations and a quarter turn make this a <transitive increasing event> on its bonds. A positive critical crossing bound and the <Friedgut-Kalai sharp threshold theorem> make its supercritical probability tend to one. A bounded family of shorter, thicker rectangles intercepts every possible long crossing. The <square-root trick for positively associated events> transfers the high probability to each fixed intercepting rectangle. Using every possible translate would lose the useful bounded exponent.
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