>The next step is to understand why the QBO period requires the Draconic lunar month (27.2122 days) and not the Tropical lunar month (27.3215 days).
Hint: symmetry

Sorry I dont see this. I took the tropical month because thats (if I havent misunderstood some explanations) the
lunar period, as perceived from earth.

Concerning the above behaviour. Since the draconic month is rather close to the tropical month you'll also get a similar behaviour, just that the "hickups" are now at different places. The interesting point would be to see how long one stays in that pattern (were are the hickups), but today I couldnt do the computations even until 150 as in the above comment but only to 120 before running into a General::timeout: Timeout reached.
$Aborted
. May be I should say at this place: the computation time at mathics was sponsored by Angus Griffith, a young australian mathematician.

Here you see the same for the draconic month:
Clear[modf,aomod,iolist];
modof[n_]:=N[(n*27.21222/365.241891-Floor[n*27.21222/365.241891])*365.241891];
aomod={};
iolist={};
For[i=1,i<120,i=i+1,If[modof[i]<365.241891/12,aomod=Append[aomod,N[(i*27.21222/365.241891-Floor[i*27.21222/365.241891])*365.241891]],iolist=Append[iolist,i]]];

aomod
{27.21222,15.7291890000000012,4.24615800000000037,19.9753470000000016,8.49231600000000077,24.221505000000002,12.7384740000000011,1.25544299999999999,28.4676630000000024,16.9846320000000015}



and the months are:

1,14,27,41,54,68,81,94,(95),108,

>There is actually a statistically exact value for this!


If I understand correctly the blue moon is in a "hickupmonth" like for the tropical month the 108 and for the draconic the 95. What is a statistically exact value?