What is wrong with Shiva Ayyadurai's statistical theory in this court filing? Dr Shiva Ayyadurai ran in the Republican primary for Massachusetts senator in 2020, losing this race to Kevin O'Connor. He then filed a lawsuit alleging irregularities in the vote count. The suit is currently pending.
His legal filing is embedded on this page https://vashiva.com/no-one-person-one-vote/ . In the document (on page 21, the previous pages can thankfully be skipped) he sort of makes an actual statistical argument, which I will try to summarize below.
I personally find his arguments extremely questionable, but if anyone with a better statistics background  would be willing to tease apart specifically what is wrong, I'd be fascinated to hear it.
Briefly, Ayyadurai provides a histogram counting the number of precincts in which he received N votes vs number of votes:

Then he counts the number of 'High-Low pairs' - by which he appears to mean points where the number of precincts he got N+1 votes is higher than the number of precincts in which he got N votes. He counts 9 instance of this for the first 22 bins (why 22 is not explained)
He then proceeds to argue that:

The  question  herein  is  how  likely  is  it  that  my  vote  count
in  Suffolk  County  could generate  9  or  more  “High-Low”  pairs
for  the  first  11  pairs  in  the  histogram,  assuming  the
reported  results  are  fair  and  unbiased.  To  answer  this
question,  I  have  modeled  vote counts  by precinct  and  candidate
using  a  binomial  distribution –of  298  precincts  x  2  candidates
for  596 total  distributions.

And concludes:

9 or more “High-Low” pairs should only appear in bins 1-22 of the histogram once in every 741 elections!

So my question is: what statistical fallacies have been committed in this analysis?
And related:

*

*What distribution should one expect this chart to follow? Surely it depends on the number of voters in each precinct

*Is there any legitimate use of this 'high-low pair' concept or is that just something he made up?

Thank you!
 A: Seems to be a lot of cherry-picking in this analysis.
Massachusetts has 14 counties (Wikipedia), of which he has picked just one, Suffolk County, for his analysis. He concludes that the election there, and in 10 other counties which used "predominantly electronic voting systems", was weighted against him. He justifies choosing Suffolk County by saying "Given Boston, the largest city with the largest voter turnout, is in Suffolk County, we began by analyzing Suffolk County." Wikipedia says there are two other counties with higher population (though it's possible they had lower total turnout). As far as I can tell, he never applies the same analysis to any other county.
He says that in Franklin County, which uses paper ballots, the reported data "closely track" with his "de-weighted" results from the 11 electronic voting counties. He doesn't analyze whether the results from the remaining counties (which presumably also don't use electronic voting), Dukes County and Nantucket County, also closely track with his de-weighted results.
The "high-low pair" concept looks like his own invention. I couldn't find a definition. Your definition seems plausible but it doesn't match up with what he says about the histogram associated with O'Connor, his opponent:

In Kevin O’Connor’s histogram and corresponding table chart, the incidence of “High-Low” pairs is much less pronounced

According to your definition, O'Connor's histogram actually has more high-low pairs in the first 22 buckets: I count 11 buckets whose bar is higher than the previous bucket. Hard to say more here without being sure of the definition.
He claims that his votes were reweighted by multiplying by 0.666 and rounding. Since this maps multiples of 3 to even numbers and everything else to odd numbers, he says it explains why among the first 22 buckets, the even buckets are smaller than the odd buckets. But it doesn't explain why the difference between even and odd vanishes for buckets above 22 (which make up more than half the x axis).
Overall, it's a very unconvincing argument.
