A proof of Merca's strong truncation conjecture for the quintuple product identity

Merca conjectured coefficientwise sign inequalities for two truncations of the quintuple product identity after removing three factors from the normalizing infinite product. These inequalities were previously known asymptotically for each fixed choice of parameters. We prove both inequalities for all admissible parameters and every coefficient. The proof uses partial theta transformations and a simultaneous pairing of adjacent terms in the resulting tails. Finite subset expansions then reduce the required signs to polynomial identities and a positivity lemma for a rational function in three variables.