The puzzle of the lowland Livingston Y-DNA results is, not in our view so much the question of origins, as it is that these results seem to be all over the place, even when genealogy indicates that seemingly mismatching men are from the same (Callendar) line. It is at least arguable that, in this case, the number of mismatches is too great to be attributable to false paternity. 

One objective of this project is to use Big-Y test results for potential Callendar Livingston descendants to identify the origin of the eponymous founder, Leving. Was he a Gael with ties to the MacLeas, an Anglo-Saxon, a Hungarian nobleman, a Flemish entrepreneur, a Norse Viking from Normandy? A more significant objective is to gain greater knowledge of the mathematics governing the transmission of non-coding DNA by studying the Livingston Y-DNA data. The broadest goal of this Project is to collect data that pertains to an ancient patrilineal line so that, by comparing the Y-DNA profiles of the probable and possible members of this line, we may better understand the mathematics governing Y-DNA mutation rates. In particular, we are interested in data that is anomalous in the light of the assumption that mutation rates are uniformly uniform, are the same in the direction of the past as in the direction of the future or "past-future symmetric".

It's a radical idea, but we suspect that this widespread assumption is false, and we are investigating the hypothesis that Y-DNA mutation rates can in some conditions exhibit a strong "past-future asymmetry" in the sense that they tend to be faster in the direction of the past, and so that there is a loss of randomness or volatility in the direction of the future. We also dispute the related assumption that Y-DNA mutations always or primarily involve gaining rather than losing an SNP, i.e., that they are always forward-mutations. We hypothesize that—although they pass unnoticed in the small window of the present—back-mutations are increasingly common in the direction of the past. These phenomena are familiar in mathematics (the distribution of the primes) and in physics (the arrows of time) but as yet are not believed to play a role in genetics. 

The asymmetry hypothesis predicts the existence of apparent mismatches between men who according to genealogical tradition are relatively closely related. Coupled with the hypothesis about back-mutations, it predicts the existence of apparent matches between men who are not closely related. One possible example of an illusory mismatch is that between Y-DNA obtained from the presumed skeleton of Richard III and presumed contemporary patrilineal descendants of Richard's great grandfather Edward III. Another is the mismatch between the Y-DNA obtained from a handkerchief supposedly used to collect blood from Louis XVI on the occasion of his execution and presumed contemporary members of Louis's patrilineal line...