High order U-spin breaking: A precise amplitude relation in D0 decays

U-spin breaking corrections up to third order are studied in $D^0$ decays to pairs of a charged pion or kaon. We show that first and third order corrections in $\sqrt{|A(D^0\to K^+K^-)A(D^0\to \pi^+\pi^-)|}/\sqrt{|A(D^0\to K^+\pi^-)A(D^0 \to \pi^+K^-)|}=1$ vanish, while second order corrections cancel each other experimentally at a one percent level. We compare this ratio with three previously studied ratios of amplitudes involving these same decays, for which expansions up to second order are obtained. Real parts of two distinct first order U-spin breaking parameters of different physical origins are calculated using measured values of $|A(D^0\to K^+\pi^-)|/|A(D^0 \to \pi^+K^-)|$ and $|A(D^0\to K^+K^-)|/|A(D^0\to \pi^+\pi^-)|$.