Two Ext groups and a residue

The present constitutes the key lemma which was hitherto missing in the author's proof of, inter alia, the Green-Griffiths conjecture for surfaces with enough 2-jets, e.g. $13\mathrm{c}_1^2> 9\mathrm{c}_2$. As to why this is so is not the immediate concern of this article since it is a lemma within a whole. By way of what is essentially an appendix to the current paper, a guide to the whole is provided at http://www.mat.uniroma2.it/~mcquilla.