3. Sentence three: «However, this formal-mathematical approach has never found a meaningful grounding in jurisprudence, sociology, mathematics, or other applied sciences». Which approach? The approach of superimposing the Gauss curve on social processes? Since «superimposition of Gauss curve» exist only in the brains of these gore-mathematicians, we should look for justification there. But I'm not a psychotherapist.

4. The next paragraph contains a banal statement that none of the electoral statisticians argue with, moreover: we have repeatedly emphasized this fact:

«No national or international normative act correlates «mathematical anomalies» with legally significant actions in the process of establishing the valid will of the voters and determining the final results».

5. What follows is another passage that shows the complete ignorance of the authors:

«Modern science is not aware of any arguments justifying the hypothesis that a certain kind of sociological distribution should obey a normal distribution…. The idea of normal distribution in sociology is unfounded».

Any statistician, and especially a sociological statistician, will tell you that it is – a set of words from people who don't understand what they are saying. «A certain kind of sociological distribution» is what? Okay, let's guess that the authors meant to say «a type of distribution of a certain social indicator». Do the authors of the report know how a sociological sample differs from the general population? Do they know that a sample in principle does not give any continuous distribution, which is the Gauss distribution? And that the question is only about statistical, scientific testing of the hypothesis that the results of sampling do not contradict the hypothesis of normal distribution of an indicator in the general population? And that there are plenty of such sociological indicators (although there are others)? Judging by the set of words in the quote just given, they don't know any of this and don't want to know it.

6. Next a collection of words: «The only mathematical component of these approaches is the similarity of the rationale for using a normal distribution to describe the electoral dependence of the number of votes cast at polling stations on the turnout percentage». What is «mathematical component»? What is «electoral dependence» and how does it differ from functional, statistical and dependence of election commissions on administrations? What is «number of votes cast at polling stations»? All votes or for a certain candidate? And where is there any «semblance of justification» of the dependence of this incomprehensible quantity on the turnout? It is not surprising at all that not even understanding themselves, the authors cannot understand us either:  «…the principle of construction of this mathematical model remains unclear and de facto unexplained». In order to understand the authors of the report should have gotten a different education, studied better, and cared less about their careers.

7. Apparently realizing this, the authors use an authority - Professor of St. Petersburg State University, Doctor of Physical and Mathematical Sciences V.N. Marachevsky, previously, however, not noticed among electoral statisticians. Actually, Dr. F.-M.Sc. should not have written such a thing, but the report attributed a quote to him: «theoretical description of the voting process by means of a single Gaussian function is possible only in the hypothetical case if the probabilities of voting «for» in all cities and regions take the same value and the same number of voters comes to each polling station in the country.».

Gosh, it's about Gauss again. I don't know what Prof. Maraczewski wrote in the original, but the above statement in itself means NOTHING, except that the authors want to declare that the models we use are useless. This statement, as I hope the professor realizes, has nothing to do with scientific argumentation.

8. In the next paragraph there is a reference to N.E. Shalaev, who is well-known in the circles of electoral statisticians: «the distribution of figures within the framework of known methods is not applicable to the search for electoral anomalies». Let me ask you, what does the Gauss distribution have to do with it? Here we are talking about another method of research of electoral statistics – about the method of frequency of the last digits in electoral indicators. I have a question for the reader: why is this link inserted here? Because I really wanted to string words together or because the authors don't understand anything at all?"

9. Next paragraph: «In fact, some of the empirical distributions may have a form close to normal, but this may occur not due to regularity, but due to variability of distributions, based on the theory of large numbers». I wonder whether this was written by Borisov or Marachevsky? In this phrase, for the first time there is a more or less conscious use of terms, although it is not quite clear what the «theory of large numbers» is (probably, we are talking about the law of large numbers, which has a very indirect relation to «variability»). With this phrase, the authors of the report seem to allow the distributions of electoral indicators to be Gaussian, although they are never purely Gaussian and cannot be.

I will skip the next two paragraphs (they are as informative as the previous ones) and immediately quote a fragment of the last paragraph of this section:

10. «The factual substantiation of deviations of the voting results from the regional average, as well as the data of observers from polling stations presented in the report, testify to the lack of correlation between the conclusions of the chart compilers and the actual situation on the ground».

It is not quite clear whose graphs we are talking about: the graphs of the authors of the report or the graphs of electoral statisticians. It is clear only that the authors of the report apply the term «correlation», often used in statistics, in a purely kitchen sense.

 Next comes a page and a half of conclusions, which we will omit for now: we should look at the justifications.

11. Section 1 is called «Scientific papers on the application of normal distribution methods of voting results to their estimation». It is not possible to understand the title of the section the first time, but eventually it becomes roughly clear what the authors wanted to say. Although, of course, «normal distribution methods» immediately alarms us. Once again we are convinced that the authors of the report did NOT understand ANYTHING in what their opponents write. Because their opponents do not write and study normal distributions, but abnormal ones!

And then, as is often the case in humanities dissertations, there is a long list of surnames (not references!) of authors that the author was able to find in bibliographies on his subject. The surnames are indeed relevant to elections (mostly – to the present, not the present Russian elections), but not relevant. 

12. Finally, on the second page of the first section, we meet the main thing:

«Since recently, on the Internet (not in scientific publications) began to appear materials related allegedly to the mathematical evaluation of election results by comparing the distribution of votes depending on turnout with «normal distribution» («Gaussian distribution», «bell-shaped distribution») applied to identical environments and random processes».

Во-первых, заметим, что «якобы математическая оценка» выдает ненаучный настрой авторов доклада. Second, «bell-shaped» and normal distribution – are not the same. Thirdly, «identical» environments are what? Fourth, «normal distribution applied to random processes» is another empty set of words (hardly the authors of the report know what «random process» is one of the most complicated concepts of mathematics).

Probably, the authors of the report under «recent time» mean 2007, when Sergey Shpilkin's pioneering research  on falsifications at the elections of the State Duma deputies in 2007 appeared on the Internet. (It should be noted that back in 1993 there appeared the work of[2] A.A. Sobyanin and V.G. Sukhovolsky, in which a model for detecting falsifications was presented. The assumptions of this model are close to the assumptions in Shpilkin's model). But in Shpilkin's works it was not about Gaussian distribution, but about comparison of distributions of votes for different parties. Later on, a lot was said about Gaussian distributions, but the propagandists were too lazy to go into Shpilkin's model. A myth was invented that foreign agents in the guise of electoral statisticians claim that the distribution of electoral indicators should be Gaussian. This was a common propaganda lie, but it was convenient for the deployment of the offensive. We have already written that the offensive had already begun in 2008. Despite the fact that we have repeated many times that this is a lie and a myth, the authors of the new report bluntly criticize the same myth.

By the way, propagandists are similarly using and introducing into the public consciousness the lie and myth that the movement for the protection of electoral rights «Golos» is a foreign agent and receives money from abroad.

For the sake of justice, it must be said that the myth in its kitchen version is also widespread in society. This is well illustrated by the famous poster «We believe Gauss, not Churov». Some radical lovers of Gauss really often recall him when drawing histograms of the distribution of electoral indicators.

13. «According to the proponents of this approach, vote rigging should reveal itself in the form of electoral anomalies – statistical properties of data that deviate from the average values and do not fit within the framework of mathematical models of random number distribution».

Oh, my God, what a mess! Falsification of voting results MAY be found in electoral statistics, in particular, in the distributions of electoral indicators. This fact is obvious to any sensible person, a graduate of any, even law and humanities school. Imagine two large groups of precinct commissions: one has honest commissions and the other – falsifying ones. Will they have different voting results or the same? The answer «identical» can be heard only from those election apologists who do not consider common sense and very much want to imagine an exotic, extremely unlikely case (such examples are given in the report).

The influence of some types of falsifications was investigated by me in the above-mentioned paper with the help of computational experiments. By the way, it was published in a journal considered to be scientific.

That is, the first part of the quoted phrase testifies to the rightness of «supporters of this approach» (although the approach itself is stated incorrectly). But the second part is a complete muddle. Think about it: according to the opponents' opinion «falsifications are found in anomalies that do not fit into the framework of mathematical models of random number distribution». Some nonsense…I will report here that absolutely everything fits within the framework of mathematical models of random number distribution, and the authors have no idea not only what random numbers are, but also what mathematical models are. However, I find it hard to believe that Dr. Marachevsky, a doctor of physics and mathematics, and Zadorin, a graduate of the Physics Institute, have no idea. Here I should probably sympathize with their squeamishness.

14. We omit the banal swear words about political bias and give an example of a technique often used by propagandists. Instead of arguments, quotations are given which are in NO WAY disputed by the opponent and which in NO WAY refute the opponent. But there are a lot of such quotes, respected people are quoted, and these quotes are supposed to give the appearance of solidity. Here is one, as I am not a fan of the same garbage style as the authors of the report.

«… «analysis of specific results of observations, in particular, measurement errors, always leads to the same conclusion – in the overwhelming majority of cases real distributions differ significantly from normal», – noted in his works Soviet and Russian mathematician, economist statistician, sociologist and cybernetician A.I. Orlov». So? I also hold this opinion: most distributions of electoral indicators cannot withstand the hypothesis of normality of distribution with any decent level of confidence. I've tested it myself. But what does this have to do with the honorarium received for the report?

Further on, a whole page goes on quoting quite scientific conclusions of A.I.Orlov, published by him in a quite scientific journal. But arguments about the applicability of classical criteria for hypothesis testing (in particular, Student's criterion) have no relation to the results of the opponents criticized in the report. Borisov and others quote Orlov, again without understanding what he is writing about! And Orlov is absolutely right when he writes that «These rules should be used only to identify «suspicious» observations, the question of rejection of which should be decided on the basis of considerations of the relevant subject area, and not on the basis of formal-mathematical considerations». Which we keep talking about by bringing in «considerations of the relevant subject area» which are no less annoying to apologists for our elections.

15. Next come two pages of references to political scientists and sociologists discussing the Gauss distribution as applied to sociological research. They are very close to the arguments of the authors of the report and boil down to a non-trivial conclusion quoted in the report: «Normal distribution works with random variables». But let's not repeat ourselves.

16. And, here, next, on page 9, comes some more or less substantive objection:

«A questionable argument of the authors of the normal distribution is that the correlation between the level of turnout at elections and voting for pro-government candidates indicates ballot stuffing and «spun turnout». According to the supporters of the «electoral theory of Gauss», such a pattern is allegedly not observed with respect to other parties or candidates. Researchers of the Faculty of Political Science of Moscow State University note that such correlation is obvious if we consider conservative strata of the population, for example, rural residents or pensioners, who are distinguished by conformist behavior and are inclined to vote for the incumbent authorities. And at those polling stations where these strata prevail and demonstrate high turnout, the vote for the party of power will be higher as well».

Hardly «the authors of the normal distribution» talked about this correlation, but this idea is indeed the basis of both the Sobyanin-Sukhovolsky method and the Shpilkin method.More precisely, the assumption is formulated as follows: «the probability that a voter who participated in the election will vote for a particular candidate (candidate or party) does not depend on turnout (i.e., how many voters participated in the election).». This assumption is indeed controversial, and its failure entails a violation of the conclusions of both Sobyanin-Sukhovolsky and Shpilkin. This assumption is a model assumption, which can be challenged and investigated in order to assess the quality of the model. This is what electoral statisticians do, unlike their critics.

Researchers at the MSU Political Science Department are quite right when they claim that the assumption under discussion is far from reality. The deviation has long been noticed. That's why the vote distributions show curled «right tails». These are – consequences of failure to fulfill the above assumption in heterogeneous communities. In the Caucasus and in the Kemerovo region, people vote really differently than in Moscow, in villages they vote really differently than in cities, and in pre-trial detention centers and hospitals – differently than in freedom.

Deviations from the Gaussian distribution (more precisely – from those histograms that can be explained by the Gaussian distribution) can be explained by socio-cultural heterogeneities. The more homogeneous the society is, the better the assumptions of the Sobyanin-Sukhovolsky and Shpilkin models are fulfilled. And the analysis of electoral statistics in large regions – Moscow, St. Petersburg, Moscow region and some others proves it well. And the statement that the discussed model assumption is incorrect – is not a discovery and cannot be a refutation of the results of the model, at least in the case of a fairly homogeneous electorate.

Another proof of the superficiality of this objection is that the reasons for the violation of the discussed assumption are not investigated. Meanwhile, it is violated not only because of socio-cultural factors, but also because of the «man-made» composition of the voting population. Due to coercion of subordinate workers to vote, mass voting of «socially dependent» strata of the population. These phenomena have been repeatedly noticed in Russia not by statisticians, but by observers.

Deviations from normal distribution (once again: this means the impossibility to identify the distribution histogram as the result of Gaussian distribution of the general population) can indeed be explained by socio-cultural heterogeneity, as the report reminds us. But even more easily they can be explained by violation of electoral legislation, which the report does not remind us of.

17. «Seemingly odd patterns in voting data do not necessarily indicate fraud» – is rather mildly noted in a research paper posted on the website of the US University of Cambridge Digital Library». And rightly so! They only give an excuse for further research, primarily empirical. For example, to compare protocols of precinct commissions with their copies, to study video recordings, etc.

The authors of the report, persistently pursuing their goal, substitute this thesis of the Americans cited by them with another one: electoral statisticians knowingly misinterpret the results of the research based on political considerations. The same is the case with election commissions that do not register undesirable candidates on the basis of «presumptive» conclusions of handwriting experts that «the date is probably not written by the voter's hand». The target is added to the assumptions and the desired result is obtained.

18. Then the authors of the report forget about Gauss for a while and ramblingly – in two paragraphs, on 12 lines - talk about quite other methods of research of electoral statistics. At the same time, of course, no arguments against these methods are given, i.e., the authors continue to fill their «report» with a set of words.

The question is about the detection of turnout peaks at round turnout values of 50%, 55%, 60%, 70%, and so on. On this topic, serious mathematicians have made much more serious statements than those referenced (see, e.g., https://tov-y.livejournal.com/21499.html). They then jump to a method of researching the latest figures for electoral indicators (e.g., the number of voters who cast ballots).

Slipping over these topics, the report's authors immediately move on to a purely propagandistic maxim:

«As sociologist I.V. Zadorin notes, identifying the causes of deviation of voting results from the normal distribution is not the responsibility of electoral institutions and often remains a field of speculation of various stakeholders. Unfortunately, these speculations, which interpret unusual voting results at specific polling stations solely as a consequence of deliberate fraud, are usually part of political technologies used to delegitimize election results».

Part of the paragraph is in bold, so it is not quite clear what sociologist Zadorin wrote and what – lawyer Borisov wrote. And if with the fact that «identifying the causes of deviation of voting results from the normal distribution is not within the area of responsibility of electoral institutions », It is still somehow possible to agree, then in the next sentence, the authors of the report express what their report is made for. And «to strengthen» their main thesis, the authors give the following repetition of it after a paragraph:

«From a scientific point of view, the results of the study clearly demonstrate that at the current stage of development, the search for electoral anomalies has turned out to be in the interests of achieving the set political effect, focused on the least fruitful approach to the analysis of aggregate data on electoral behavior, actually built on the study of the frequency of figures».

So, what are we criticizing: Gauss, the frequency of numbers, or just a non-state point of view?"

19. Section 2 begins with the words «The authors of the study analyzed publicly released graphs of the normal distribution of voting as a function of turnout in the 2016-2020 federal campaigns».

What study? And why did the graphs turn out to be normal distribution graphs after all? It becomes even «clearer» (in the words of the authors of the report) it is clear that they do not understand what they are writing about.

Further: «The authors of the distributed graphs do not cite any specific facts of violation of the legislation, but at the same time they make categorical conclusions about «unreliability of the result» in the subjects of the Russian Federation, even trying to accuse the organizers of the elections in a criminal offense – «total falsification», «a lot of drawing», «everything», «shameless drawing», «total drawing» , «total dobros», «much and varied thrown in», «original cluster plundered», «complete end» etc. п.»

The first part of the phrase – just a lie. I, being the author of the distributed graphs, just write more about specific facts of violation of the law. Specific falsifications are confirmed by documenting violations on video recordings, in written complaints and witness testimony, but the fact that they are total is really confirmed by electoral statistics.

A few pages are devoted to quoting Shpilkin's conclusions and his graphs on «all-Russian voting». The most radical examples of Shpilkin's statements are selected.

And then on several pages there is again «exposure» of what the criticized authors did not assert. For example, in bold type: «The peak of 99% turnout at polling station No. 3793, which voted «against» the amendments, in accordance with the imposed concept of the authors of the graphs, differs from the normal distribution and is a falsification (throw-in)». The authors of the report write nonsense, attribute it to others and criticize those others! Is this out of hatred, self-interest or stupidity?

This nonsense continues right up to page 19. The authors of the report give examples of individual low turnout precincts, explain the phenomenon of high turnout in those precincts, and claim that their opponents are supposedly passing these anomalies off as evidence of deviations from a normal distribution. I have previously seen these arguments in the «sociological» work of I.V.Zadorin. It is hard for me to imagine that a graduate of a phystech does not understand that he is criticizing nonsense invented by himself, so we have to speak about the scientific impurity of this sociologist. It is confirmed by section 3 of the report, which consists of Zadorin's presentation.

20. Section 2.3 is titled «Correlation of observers' estimates with plots of normal mathematical distribution». Well, let it be such a strange title. In any case, the whole text of the report consists of pearls, here, for example:

«The conclusions of the authors of mathematical graphs about the level of «anomalousness» in a particular territory are close to a random distribution of data based on probability theory and, obviously, cannot be the basis for reasonable conclusions about the course of the electoral process in the region».

«Mathematical graphs», «conclusions are close to a random distribution of data», «distribution of data based on probability theory»….- sets of words strung together for the sake of once again repeating the authors' conviction: «it is obvious that there can be no…».

21. The results of the research conducted under the leadership of I.V.Zadorin are outlined in Section 3. It is claimed that it was «a sociological study on the results of detection of «anomalies» by mathematical methods (!), which did not confirm the hypothesis of finding «throw-ins» by means of mathematical analysis».

I became acquainted with the presentation of this study about a year ago at a sociological conference. Even then I told Zadorin that the study has NOTHING in common with the conclusions of electoral statisticians and illustrates only the bias of the head of the study.

The 10 pages of the report are a piece inserted there from Zadorin's study. Zadorin proves that the outcome of voting in some precincts can have an unusual character. What a profound and original result, however! I wrote about anomalies at small and isolated polling stations back in the early 2000s, for example, in the work «Course of Anomalous Electoral Statistics».

Organizing 18 sociological expeditions to different regions of the country (that is, spending a lot of money of the client) Zadorin studied 18 polling stations, mostly small and not quite ordinary, and concluded that their anomalous results are explained by specific socio-cultural features. But Zadorin was commissioned to work on refuting the results of the electoral statisticians. Zadorin did not present his work to the electoral statisticians, but to the sociologists and the client.

And he wrote in conclusion: «Thus, even the experimental survey of a very small number of polling stations, where «electoral anomalies» were encountered in the 2018 presidential election of the Russian Federation, confirmed the initial hypotheses of researchers about the diversity of causes of deviations of voting results from «normal ones». I note that here Zadorin uses the word «normal» in a purely everyday sense. But this was enough for the authors of the report to increase the volume of their work by another 10 pages.

22. Section 4 (pages 34-43) – perhaps the only value of this report. Examples of deviations from unimodal distributions in different elections in foreign countries are given. Some examples are obviously incorrect, as they represent graphs (interpolation of histograms) of the distribution of the number of precinct commissions when the number of commissions is too small. The authors of the report do not tell what is drawn on the graphs: what step is chosen on the abscissa axis and from what the percentages are calculated. This is hardly what one would expect after reading the previous pages.

But some of the graphs are interesting and do show deviations from a normal distribution. As well as many of the graphs that are given with explanations in the works of electoral statisticians. But, on the other hand, the authors of the report do not provide numerous graphs of pronounced unimodal distributions of voting in foreign countries.

23. Section 5 «Peculiarities of mathematical modeling of the process» looks grotesque. It illustrates not only A.I.Krylov's phrase «Trouble if the cobbler starts baking pies and the cobbler starts making boots» but also a case when the cobbler is very angry with his customers.

Besides the headings, everything else in this section is a copied picture with mathematical symbols from some work! The authors of the report couldn't get the formulas right, so they just copied someone else's text in its entirety. This part takes up 7 pages.

As for the mathematical content of the text, I refer the reader to section 11.3 of real mathematician Alexander Shen's work «Elections and Statistics: the Casus «United Russia» (2009–2020)» (https://www.mccme.ru/shen/public/report-xelatex.pdf). Shen, after reading this part of the report, wrote roughly what I wrote above and added: «Outwardly this text resembles a mathematical text, but even with the most sympathetic attitude it is not easy to make sense of it.

24. In the next sixth part, the authors of the report return again to poor Gauss and ask the question that they, and only they, care about: «Does «normal» distribution in elections exist?"». The question characterizes the level of the report, but Part 6 itself has little to do with the main purpose of the report – to criticize electoral statisticians.

The authors of the report first tell readers about the Gaussian distribution as best they can, then draw their conclusion: ««The works» of S. Shpilkin are based on initially false statements that do not correspond to the object of research (statistical data on social processes). The conclusions made by Shpilkin on the basis of the Gauss function have signs of bias and are anti-scientific».

I repeat for the persistent propagandists: the conclusions drawn by Shpilkin are not based on the GAUSS FUNCTION (God forgive me for quoting such a vague wording).

And let me ask: on what function are the conclusions of the report based?

25. Section 7, which is called «Manipulative Technologies», in my view, belongs to the genre of «propaganda fiction». In it, the authors of the report express their fantasies about what manipulations electoral statisticians use in order to defame Russian elections.

It would be possible to quote these ideas in their entirety, but space is a pity, since the authors of the report once again repeat the same thing. I will cite selected passages. The reader can evaluate them for himself.

«We take the required hypothesis (e.g., all elections that do not fall under a normal distribution – «abnormal») as proven to be true». Lies…

«The likelihood of violations is presented as the presence of doubt…». What's that? A set of words…

«This technology, which has nothing to do with scientific knowledge, is used to manipulate public consciousness and is used for political purposes to delegitimize the election result». Well, here at last is the innermost thought for which the words were strung.

26. Section 8 contains «conclusion and recommendations». Here, too, we meet with a not very conscious set of words strung together.

«The still insufficiently high level of trust in the Russian electoral system, caused by a number of objective and subjective factors, forms the demand for «independent assessment» of the voting results, which is problematic to implement even at the international level, given the high politicization of electoral procedures».

….

«Attempts to assess the results of voting by means of the laws of mathematics and physics have been used repeatedly in elections all over the world, but they have not received serious attention from election organizers, authorities or even human rights activists». «Laws of mathematics and physics»…One can feel the illiterate babble of human rights activists like Brod, Borisov, Grigoriev and other ideological fighters.

«The above laws are applicable to physical homogeneous random processes and independent quantities, and their application in public legal relations is not correct and not correct from either scientific, logical or practical points of view» (grammar of the authors is preserved). Which laws? Boyle-Marriott or the central limit theorem (in the minds of the authors of the report, this is probably a mathematical law)? What are «homogeneous random processes» and «non-independent quantities»?

«The absence of scientific and public discourse on such approaches will form an information vacuum, which will immediately be filled with pseudoscientific theories and dubious evidence, which against the background of fake news flow in the modern open society will form an unfavorable picture of democratic institutions of power reproduction, reduce trust in elections and delegitimize elected bodies».

Yes. Some people think that «the unfavorable picture regarding our democratic institutions» is created by the lack of real elections, but it turns out it is created by a lack of discourse. Couldn't the authors of the report discourse with electoral statistics somewhere? At least in RIA «Novosti» or on Channel One?

«When solving the complex problem of legitimizing electoral procedures, it is necessary to pay attention to the fact that the most suspicious cases of organizing voting are invariably characterized by a large number of various anomalous properties, including in the mathematical analysis of the distribution of voting results». Or maybe the authors of the report did not expect anyone to read it? «The most suspicious cases of voting organization are characterized by a large number of anomalous properties

«And while our society, which has not yet reached a sustainable democratic state, will trust near-scientific findings and assessments used by interested actors for their political ends, a number of measures should be implemented to better inform voters about the electoral process and remove mistrust in the procedures of voting and the establishment of its results».

Yes, something must be done with our society, which still has some shortcomings. It is very gullible and therefore prefers Shpilkin to Borisov. But it is necessary to implement a number of measures to remove distrust in the procedures for establishing the results of voting». It is possible, for example, to establish not only a captcha for obtaining these results, but to declare them a state secret for disclosure of which one can get 10 years. And for the use of the Gauss»function, we should also deprive them of the right to correspond.

The time has come to fight back against the mathematicians who are invading our social processes!