A theoretical model to predict the age of traditional balsamic vinegar

Traditional balsamic vinegar (TBV) is aged for a long time in a set of barrels arranged in decreasing scalar volume. New cooked must is added and aliquots of product are transferred from barrel to barrel every year; this procedure generates a blend of vinegars of different ages. These ages have been described by sequences of real numbers depending upon the number of year of the barrel set and volume of vinegar transferred. A theoretic study of these sequences has shown that there is a finite limit for the age of vinegar and this upper limit can be formulated through the values of both the volume of vinegar in the barrels and the transferred one. Namely, we have proved that the real ages, in any single barrel, are strictly increasing and with a finite limit as the number of years of the barrels set goes to infinity and we calculated these limits. From a practical point of view, the proposed mathematical model allowed us to define formulas able to calculate the maximum amount of TBV that each producer can sell as a limited product, in the hypothesis that the vinegar has been reached the minimal legal age required of 12 years. In addition, since the independent agency that officially states TBV authenticity has also the authority to inspect the producer's factory and therefore to know exactly how much TBV is withdrawn, it is reasonable to hypothesize that the public agency is able to found all data required to calculate the vinegar age by the proposed model.