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Promised authorship was not given; am I supposed to get an automatic acknowledgement?
TLDR: I spent a long time helping with a paper from a group at another university. The first author, who was at the same institute as me, promised multiple times I would be one of the coauthors. But, the night before submission, he informed me that I won't be on the author list. There are about 20 coauthors in total.
Now the paper is on the second round of revisions (and on arXiv). I learned that I would not even be acknowledged in the "thanks" list. What is the best thing I can do here?
The field of this paper is between applied statistics and computer science. The authors are not listed alphabetically.

Full story:
I am quite new to publication. A friend from another group come to ask for help. He is postdoc leading a project. I started helping by providing comments on writing the paper, polishing the paper, and providing a statistical model. He said with the model, I could be the second or third author.
Later he told me that I might not be the second author because they decided to not use my model, but I will still be on the author list. I understand this.
I trusted him so I don't have a signed document or any record to prove that he promised this. I was still providing comments. We discussed this specific project more than twice a week. Each chat lasted more than one hour. I asked him once to include my name, but he said the paper will not have any names on it until it is good enough for publication.
Right at the night before the submission, he called me saying that he cannot add my name because his supervisor doesn't want me on the list. I don't personally know his advisor, who has a dominant personality. He is the only first author.
His rationales of excluding my name were:

I did not write anything, and by the rules of the publication, I have to contribute in every major aspect of the paper.

He thinks: only first authorship will help my career. Even if he gives me the authorship, it won't benefit my career.

One of the supervisor's students who gave comments was also excluded by the supervisor.

He is new to that group, so he wants to build a better relation with his supervisor.

The supervisor said almost all authors must be in the group at the first place. (By group, I don't know if it is the research group or this project group. There is at least one author who is completely unaffiliated with the supervisor's group and never attends any group talks.)

To calm me down, he further promised me three things that night:

He can bring me into other projects he is leading in the future. This has never happened and now I don't think further collaboration with him will help me.

This current paper is unlikely to be published at a top journal but his next project is very likely to be published at a top journal.

He can later introduce me to his advisor for further clarification or collaboration, but this never happened.

Later, when I asked him for a written agreement regarding a future collaboration on another project, he hesitated, saying that he cannot be sure about the contribution and authorship until the work is finished. Since then, he's been occasionally ignoring my calls.
I disagree with rationale #1. He intentionally prevented me from writing down anything on the paper. He only took my feedback. Moreover, no one contributed on every aspect of the paper except the first author. The publication culture of this field is that everyone is highly specialized and only works in a  narrowly defined area.
He also admits that I spent more time than the second author on this paper. To my knowledge, most of the coauthors did not write anything on the paper.
Well, I was convinced by him that night. His point 1 is probably generally ethical. What do you think?
However, I thought I could at least be acknowledged, but now it seems like my name will not appear anywhere in the paper. I would like to be at least thanked in the paper, without having everyone think that I am trying hard to free ride. What can I do?
(Using "he" is not intended to imply that the first author is a male. The story should be read as gender neutral.)

The night before submission, he sent me a screenshot of the text exchange with his supervisor. The conversation goes like this:

He: I think we should probably add HighGPA because she provided a lot
of feedbacks and helped with the statistical methods.
Supervisor: My student Jane also provides feedbacks and comments, yet
I don't put her in the list. All authors should be in the group and
the weekly group talk from the beginning.
He: I respect you as the final decision maker. I actually promised
HighGPA an author slot a few months ago, and since then, she've been
providing helpful feedbacks and spending not less time than anyone
else in the group. Can you do this as a one time courtesy and I
promise that this will not happen in future?
Supervisor: Maybe next time. We already have 20 authors. Now add Joe
Doe because he helped with initial data collection.

A timeline is added for better organization of the events.
Before Jan 2020: A posdoc shortly shared the same advisor with me before. I worked for one project he was leading. In that project he trusted me: he let me edit and rewrite the paper, add my own contents, and then he blindly approved them.
Jan 2020 He joined a new group.
Feb 2020: Posdoc presented the idea to his new group. At the same day, he invited me to help with the project by providing remarks and helps in applied statistics. I start to help with the project by providing him feedbacks every week. I did not put my advisor or my previous advisor in the loop because they are less relevant to his subfield; this is the first mistake I made.
March 2020: He invited me to his PI's group meeting, with the condition that I, just like other members, have to read and present a paper once a while. I find this requirement ridiculous as I have my own things to do so I did not go. (This could be the second mistake I made. Back to then I underestimated the value of networking and presenting. Now I won't forgo any chance to preach my paper and idea.)
April 2020: I was asked to prepared a statistical method. He promised that with the method I can be second or third author.
May 2020: My statistical method was rejected in a group meeting. I am not sure if he brought me up for the method. He said to me that I cannot be second author but there will be no problem for authorship. He first asked if I can improve and revise the paper just like what I did before for the last project. I agreed but later he hesitated to let me directly write or comment on the paper. I respected him.
Late June 2020: He finished an early draft and I asked him to include my name. He said "no worry, the names will be added before submission." I am still providing feedbacks twice a week.
Sep 2020: The paper is about to be finished and he contacts me less for feedbacks. He contacted me asking if I can revise the final version. I agreed but he found reasons to delay sending me the up-to-date version.
Oct 2020: Finally he send me a semi-final version for my final feedbacks. He politely and indirectly ask me to not to directly write on the paper or comment on the paper. I still trust him and respect his request. I provided oral feedbacks instead. Though I find it weird, as he wrote 95% of the paper and based on his previous style, he will be more than happy if I am willing to write on it.
Late Oct 2020: The night before submission, he told me that his supervisor refused to include me as an author. He send me the screenshot. Note that he handles the submission process of the paper and his advisor hardly ever writes down a sentence.
The first author explicitly complained to me that the second author contributed very little, yet the supervisor "forced" the posdoc to put the person up as the second author. He chose not to fight the supervisor because he is new to the group and he wants to build networks with team members.
Early 2021 The paper is on arxiv without my name being thanked.
I agree with all of you that acknowledgement carries no weight in academic career. However, I am human so to be thanked or not to be thanked makes a huge difference.

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While we cannot conclusively determine whether or not you would have deserved co-authorship, for the reasons others mentioned (number of authors on the paper, the time you invested into this project, etc.), it seems likely that you should have been one - or at the very least acknowledged. Now, let me share with you my hunch why that never happened, and how you can maybe learn to protect yourself from a similar situation in the future.
The postdoc's PI never even knew about you. Everything you write about the situation boils down to keeping you in the background, probably the postdoc claiming what you contribute for himself:

"We don't add names before we're ready to publish."
"I'd like you on the paper, but my adviser thinks you shouldn't be on." It's not
me - it's them!
And he keeps you compliant and quiet by promising you future projects that never
happen.

Maybe my personal Hercule Poirot went overboard here, but this smells rotten. You will only know if you finally directly contact your postdoc's adviser. While contacting a senior PI might be culturally discouraged, the path through their underlings preferred (not where I am from, but in some parts of East Asia, say), I'd encourage you to always cover your bases in the future.

I actually agree with your Hercule Poirot. There are other evidences. Right after submission, he starts to ignore my calls, and ignore his own promises at the same time. So you are right that he promised the future project only to keep me quiet rather than truly help me.

@HighGPA: Well, it's still a sad world where, if it's true, this kind of stuff happens in academia. I don't know if I would push for anything regarding this specific case - it might turn into a he said-she said with no clear benefit for you. But do make sure to be more careful in the future. Best of luck!

@HighGPA A variation: 1. Postdoc promises authorship, without consulting PI. 2. PI is upset that the postdoc has overstepped his authority. 3. Postdoc should have come clean and apologized to you, etc.; instead, he tries to cover up his mistake, compounding the trouble he has caused. 4. Things fall apart as he continues to avoid blame.

@gnometorule Hi gnometorule, even with the added conversation, I still agree with you. The posdoc has his own funding and is less dependent on his advisor. He wrote almost every part of paper, and the idea was his. Earlier or later, he can surely add me in if he really want to. Again sorry that some details were missing from my first post as the whole process took eleven months and I cannot recall every detail right away.

Your “friend” is no friend to you at all, and behaved unethically in promising you coauthorship and making other promises that he has not kept, and that it’s not clear he had any intention of keeping. The other answers analyze at length whether you being a coauthor makes sense or not, but that seems beside the point to me. You were promised to be made a coauthor, and such a promise must not be made if there isn’t a good reason to make it, and must be kept if it were made and it is even remotely reasonable to keep it (which it clearly is).
As for your question of what you should do, the main thing you should do is learn from this experience and in the future not put yourself in a position where it is so easy to exploit your labor and talents. The reliance on verbal promises, lack of written documentation, and the fact that only one person knew about your contributions, have all put you in a vulnerable position.
Aside from that, you can try to argue that you deserve an acknowledgement, and you can try to make other people in the “friend”’s group aware of his nasty behavior. There are some arguments to support doing both of these things. It’s also possible that none of them will be productive and that trying to do such things will only lead you to a waste of time, frustration, and disillusionment with the academic research and publishing process. So another possibility is just to let it go, learn from the experience and resolve to avoid such situations in the future. Which one of those paths to take is your personal decision. In any case, good luck!

I trusted him, but now, as you said, the "friend" is no friend at all. As you said, I will definitely learn from this experience, and having written agreements before working on any projects. In  fact, there might be a downside of doing this. I've told every potential collaborators that I need a written agreement before joining the project. However, this seems to scare them off as they often said: we don't know the contribution yet so we cannot be sure about the authorship.

I should've let his advisor to know me personally and who is my advisor. That is also a mistake I made.

OP did what is recommended at the beginning of a collaboration, rather than an implicit agreement: they agreed explicitly ahead of time how authorship will work. Nonetheless, they were cheated out of authorship or at least acknowledgement. The collaborator's group acted despicably (assuming the PI was fully kept in the picture). Not sure how OP might have protected themselves against this. I do not think that written agreements is something I have ever seen, but clearly OP is seen as vulnerable and should be more cautious in seeking out collaborators. Avoid large groups, that's what I can say.

@CaptainEmacs I’m not referring to formal written agreements, but it’s good to have some written record of things said in conversation, and more than one person in the know including trusted parties. For example, following a phone conversation OP could send an email: “Hi [name], I enjoyed discussing [project] with you today. I’m really excited at the idea of becoming a coauthor on this paper, which you said would definitely happen based on my contributions so far. Let me know how else I can help. Copying [my advisor] to keep them in the loop.” ...

... If the other person does not reply to the email to deny or correct what you said, you have established a pretty convincing record of the promise. (And if they do deny it, well, I guess at that moment you have a clear picture of what kind of person you’re dealing with...)

@DanRomik Yeah, your formulation would work. Although it is pretty clear that it establishes a subliminal level of formality that everybody involved understands is a type of 'contract enforcement' and a ever so slight signal of distrust. I never did that, I must say, preferring to work with people I can blindly trust, but perhaps in nasty communities it's unavoidable.

@CaptainEmacs distrust? I have no idea what you’re referring to. The email is just about OP expressing her excitement at being offered coauthorship in a cool project/paper. Anything else other people read into it is simply their imagination working overtime...

@DanRomik Actually, putting your Advisor in CC on that is pretty much that. I do not think people are so naive as to not understand what that CC is for. Of course, you can claim plausible deniability, but still, that trick of enforcing agreements is as old as the invention of carbon copy paper or duplicate stone tablets, whatever came first. Again, I do not think your idea is bad, but it makes things quite formal.

I understand the point of your answer, but I tend to agree with the objection made by @CaptainEmacs: I'm not sure if "distrust" is the right notion to capture it, but if I got an email stating "I’m really excited at the idea of becoming a coauthor on this paper, which you said would definitely happen based on my contributions so far" the point that I would perceive as most outstanding is not the OP's excitement, but rather the OP's very determined way of ensuring their co-authorship. That need not necessarily be a bad thing, but as CaptainEmacs says: it keeps things at a more formal level.

@JochenGlueck it’s never a bad thing to be assertive and signal to those around you that you will stand up for yourself. This is a common misconception in academia (that I have seen expressed here far too often): people focus too much on being “nice” and on not behaving in a way that will be perceived by others as unpleasant or formal. Meanwhile they miss the fact (that I have personally witnessed many times) that if you show that you have a backbone and cannot be easily pushed around, even if that means being a bit unpleasant, the people around you will respect you more, not less.

@CaptainEmacs thank you for agreeing that my idea is not bad :-)

@CaptainEmacs Are you going to write an authorship-confirmation email after a phone talk? If so, are you going to cc the advisor? By saying "but it makes things quite formal", do you think that formal is a non-optimal thing here?

@HighGPA Dan believes this is the prudent behaviour. If you operate in a Machiavellian scientific community, he's right and it's probably the best thing for you to do. I personally do not like this, and choose to work with people I can trust blindly.  I haven't been fooled a lot, and limited myself to work with people I like to work with without exploitative hidden agendas. After all, if one is in academia, one should also enjoy what one does and with whom; else one can as well go into cutthroat businesses where one can earn proper money.

@HighGPA to clarify, the email idea was just one suggestion for how to create a little bit of protection for yourself. I’m not necessarily advocating always sending such an email or using that specific language. It depends on the details of the situation, personalities involved etc. In some situations other approaches will be better. The main message is to be aware of the problems that can arise and act to prevent them in a tactful way that won’t damage the relationships with your collaborators. Also, as CaptainEmacs said, choosing your collaborators wisely can also save a lot of trouble.

Very strictly speaking, the following is rather a discussion of the circumstances than a plain answer to the question "What is the best thing I should do here?" Anyway, I think the following points should be pointed out in some detail.
Should the OP be a co-author of the paper?
Many (not all) people here seem to agree that, given the information the OP provides, the OP should actually be a co-author of the paper. Still, I think it is worthwhile to list the following points in detail:

Giving extensive comments on a paper might or might not, in general, entitle you to be a co-author of a paper. This might depend on the cumstoms of the field and on both the relevance and the quantity of your comments.
(Buffy makes, in their answer, a point that contributing comments and suggestions does not make you a co-author; I can follow this argument to a certain extent, but I would also be careful with it. For instance, in pure mathematics, it is indeed very uncommon to give authorship to somebody who merely made comments and suggestions. But then again, authors are listed alphabetically - i.e., not ranked by their contributions - in pure mathematics, and typically papers in pure mathematics don't have so many authors. If it is possible and common to rank authors by their contribution, things might change. As I said above, the details might depend both on the quanitity and the relevance of the suggestions provided, and on the specific field.)
Remark: In an earlier version of this answer I said that I agree that the OP should be a co-author. Buffy's answer made me re-think this claim and phrase it a bit more carefully. But still, I'm under the impression that things did not go well here be any reasonable standards. Since the first author has already promised the OP authorship, it seems at least odd that the reasons that they are given now for not being listed as a co-author, contain several
non sequiturs (please see below).

The rationale that the OP "ha[s] to contribute in every major aspect of the paper" in order to be a co-author is, obviously, ridiculous. Is anyone seriously supposed to believe that each of the 20 co-authors of the paper contributed to every major aspect of the paper?

I'd like to specifically point out the following aspect that has already been briefly mentioned in Captain Emacs' answer: the mere fact that the OP's statistical model - which was initially seen as a major contribution by the first author - did not make it into the final version of the paper, is no indication of whether this model did or did not contribute to the findings in the paper.
It is a major point in scientific research that, at the beginning, we do not know how things will work out in the end (that's why it's called research, after all). Our only way to scientific insight is trial and error, and it happens very often that what we tried first does not work out well, but guides us the way to a better solution.
For instance, many of my papers have gone through many stages of "development" and it happens frequently that what seemed to be a key idea at the beginning got, at some point, completely removed from the paper. In general though, this does not necessarily make this idea an irrelevant contribution to the development of the paper!
Whether this is the case in the OP's concrete situation is, of course, not possible to determine from the information provided.

Why would they do this?
Several suggestions of why this has happened have already been suggested. I would like to add that there is actually quite a large variety of further potential reasons, and it might be extremely difficult - even for the first author and their supervisor - to figure out in retrospective which of them applied. Here is a - completely non-comprehensive - list of examples of potential reasons:

It might be their honest opinion that the OP's contribution was significantly less than the contributions of each person listed as co-author.

The aforementioned point might be only the supervisor's opinion. The first author might disagree, but might not have the courage to bring it up, given the supervisor's dominant personality.

On a related note, somekind of misunderstanding might have occurred between the first author and their supervisor. Consider, for instance, the following hypothetical dialogue:
First author: "I think OP should be a co-author of the paper, too."
Supervisor: "What's their contribution?"
First author: "They contributed many very useful remarks and comments on the paper."
Supervisor: "That's not a sufficient reason to be a co-author; my student xyz has also made several remarks, but they're not listed as a co-author, either."
First author: "Ok."
I've witnessed many "discussions" between people with "dominant personality" and their supervisees/employees that followed such a pattern (although on other topics). There are various problems involved: (i) A "dominant personality" is a character trait that is quite strongly correlated with forming strong opinions, and with forming them quickly. This can easily lead to a sentence such as in the fourth line of the above dialogue, where the supervisor expresses a certain opinion despite that they still have insufficient information to make this conclusion. (ii) People with "dominant personalities" are - almost by definition - difficult to argue with, and even more so if you are in some essential way dependend on them. This might result in the first author just giving in instead of pointing out that there can be various degrees of contribution by means of giving comments and making remarks.

Maybe the first author was indeed playing you false and never had the intention to add you as a co-author.

They (or only the supervisor) might follow a "policy" that people from outside their lab need to contribute more than people from their lab in order to get co-authorship. (Needless to say that such a policy would be ridiculous, but people tend to do a lot of ridiculous things).

For some weird reason they might even think that it is better for the OP not be co-author of the paper. (They might, for instance, argue that it could hurt the OP in the future to be a co-author of a paper when the OP is, in their opinion, not able to give good arguments of why are a co-author.)

Then again, it might be real malice, or somekind of discrimination, be it conscious or unconscious. (Maybe one them doesn't like people with blue hair, or people who are white on their right side and black on their left instead of the otherway round, or some other kind of nonsense that human prejudice is able to come up with.)

Another option is simple ignorance on their side. Maybe they think it's "not a big thing", and thus didn't bother to really reflect on what they were doing.

Of course, combinations of several such reasons could have occurred, too. My main point is that there are various possibilities which differ considerably with respect to the intensions and (non-)ethics involved, and it is, in general, extremely difficult to figure out which reason applies to which extent.
So my advice is to be extremely careful about making any assumptions (and even more so about making any accusations, even if only in private).
Relevance of the situation to the OP's career
It has been argued that having one more paper where the OP is one of 20 co-authors might not necessarily do a large benefit to their future career.
Still I think that there are several potential benefits which should be pointed out:

Even "not a large benefit" would be different from "zero benefit". In particular with regard to the very competetive academic job market we should not forget about this.

It is my understanding that the OP is a mathematician. So while being one of many co-authors of a paper in a field between applied statistics and computer science might not count as a huge scientific contribution, it can serve the OP another purpose which should not be underestimated:
It would give the OP the opportunity to argue (for instance, in front of a hiring committee) that (i) they are able to work in interdisciplinary projects, that (ii) their work/expertise is appreciated by people from outside their own field, and that (iii) they did not only work with people from their own lab.
In some hiring processes these points might make a significant difference.

One point which is, in my opinion, too often neglected is that there tends to be a considerable difference between a person's opinion and their actions. I've heard numerous scientists claiming that "qualitiy is more important than quanitity", that "counting publications is not a good indicator of how capable a researcher is", and that "Researcher X has a lots of papers, but most of them have a different first author, so we shouldn't conclude from this that X is really good."
However, whenever it comes to concrete decisions (in particular, to hiring), all those "irrevelant" or "non-conclusive" indicators suddenly play a very prominent role in the decision. Sometimes this can even be observed within a single sentence, when somebody says e.g. "Well, I have to say I really don't believe in citation counts, but if we compare candidates A and B it is really remarkable that candidate A's work got fourth as many citations."
So assuming that a paper with 20 co-authors would most likely not benefit the OP in the future is, to put it bluntly, a bit naive in my opinion. It might even happen that someone explicitly notes that this paper shouldn't count too much but, unconsciouly, still takes the higher number of total papers as an indicator of the OP's qualities as a researcher. (Of course, if we only restrict the discussion to this single paper, the effect won't be very high - but such things add up during a career.)

Another point is citations. Even if the OP would be just one of many co-authors, every citation for that paper would occur is an additional citation for the OP in a bibliometric database. In my experience, people don't make much of an effort to differentiate where all the citations of a researcher come from - they just look at a few numbers such as, for instance, the total number of citation, and maybe the top-cited articles and/or something like the h-index.

So to sum up, I'm somewhat reluctant to conclude that this publication would not have benefitted the OP much.
What to do?
Now we come to the actual question, and my answer will most likely be disappointing for the OP:
I don't think that much can be done about this.
Pressing people to make you a co-author could, in general, be a dangerous thing to do. If the reason why the OP was excluded is on the rather harmless end of the spectrum (for instance, a misunderstanding between the first author and their supervisor), a simple email with a kind request for clarification right after the decision to exclude the OP might have had some chance of success.
However, the paper is now in the second revision, and at this stage the (other) authors might even be a bit more reluctant to add another author. If they did so, it might be expected (at least in some fields) that the corresponding author at least briefly outlines (for instance, in a response letter that accompanies a revision) why the new author was included. "We first excluded OP as a co-author without good reason, but now we admit that we were wrong" is probably not a good thing to write there, so the corresponding author would need come up with a more decent formulation. Of course, this would be absolutely possible without too much work, but it still is one additional obstacle (at least psychologically) that the other authors would have to pass in order to grant the OP co-authorship.
Further remarks.

That the OP is not even mentioned in the Acknowledgements is a different point. This seems to be completely weird - giving credit in the form of a simple "thank you" at the end of the paper is essentially for free for the authors, and as a rule I would always thank everybody who helped me to improve a paper by providing me with a non-trivial amount of non-trivial comments.

Despite all that has been said above, it is important to note that such kind of "collaborations" still have the potential to benefit the OP in the future, even if they turned out like this.
For instance, depending on the personality of the first author, they might now be under the impression that they owe the OP (or they might not be under this impression - as mentioned above, this is difficult to say.)
The first author might also mention the OP's expertise to other people working in different groups (not necessarily the first author's group), which could increase the probability of future collaborations (not with the group responsible for the paper under discussion - the OP will probably not be particularly keen on working with them again - but with different and, hopefully, more obliging collaborators).
The point I'm getting to is that what has happened is reason enough not to work with this group again, but I would advise against adding any further damage to the relation with this group (should they ever suggest to the OP again to work with them on a paper - OP can just kindly decline, referring to lack of time or the like).

That's a well thought-out answer, but I don't agree with this point: "adding co-authors at this stage will typically cause the corresponding author more trouble" -- Revisions can lead to substantial efforts that might involve additional authors. So it's not given that the editor will ask for clarification. If they do, there are graceful ways to reply, such as "Mrs. X developed a statistical model that greatly helped to generate insights relevant for the revised version" (which would not be untrue, even if the insights were also relevant for the first version).

I quite agree with most of the advice. However, I am not sure about the "burning bridges". The bridge has been burnt by the collaborators. OP should, if they continue to communicate with the group at all, of course, be polite, but frankly there is nothing OP can want to have to do with that group of people. OP should really not fall for any future suggestion of collaboration from them. It will just be burnt time.

@lighthousekeeper: You're right of course. I didn't choose the wording well; what I meant is: adding a "new" author after the initial submission should, in my experience, typically be accompanied by a brief explanation in the "response letter", even without the editor explicitly asking for it. It would certainly be possible to write a good explanation for adding the OP as an author. It is, though, something which the authors would need to do "in addition" to the revision - which could reduce the odds that the other authors are willing to do this a bit further. I'll edit my answer accordingly.

@CaptainEmacs: I agree with you on that. Probably the idiom "burning bridges" is a bit too strong for what I intended to say. Indeed, I'd suggest to avoid working with these people again. But in addition, I'd also suggest to avoid any further damage to the OP's relationship to them that goes beyond what has already happened and beyond not working together in the future. (I'll change the wording in my answer accordingly.)

Hi Jochen Glueck. Much appreciated for your thoroughly thought-out answer! You made a  very good point on "citation count", which is definitely important; I neglected this. Furthermore, a paper with 20 people and a big group will have a lot of citations, as long as each author self-cites at least once. Following your advice, now I would not push for authorship. But do you think it is ok to push for acknowledgement, and, if the paper is rejected and significant revised, push for authorship?

Jochen you are so smart. The hypothetical supervisor-first author is almost the same as the real conversation. The real conversion goes like: HE: I think I should add HighGPA because of she provides many helpful feedbacks; Advisor: one of my student also provides feedbacks but I exclude her; HE (politely): A few months ago I promised an authorship to HighGPA, who later spent more time than everyone except me; Advisor: maybe next time as we already have about 20 authors. Advisor added: Now please add Joe Doe to the list because he help collect data....... I will add this in the main question.

"The point I'm getting to is that what has happened is reason enough not to work with this group again, but I would advise against adding any further damage to the relation with this group". Upvoted.

Basically, you spent XYZ hours on this work depending on your colleagues promise that you would be co-author. This is time you would have spent on your own research otherwise. Pacta sunt servanda and you did your part.
In addition, I agree with @JochenGlueck that if the paper already has 20 authors, there is no good reason why you shouldn't be also one of the authors.
You could try to get an acknowledgement, but, frankly, that's not worth much.
Probably, I would express a mail with a low-key message of disappointment, listing the most important of the above arguments, plus the arguments that you feel you considerably contributed to the results (that your work led to a dead end is not your fault and may have triggered their current line of insight) and you disagree with their judgement that you do not deserve co-authorship.
Papers, especially with many authors, are so complex today that if one really strictly applied the rule that authors must be authors in every major aspect of a paper, there would be authorless papers, simply because nobody has full overview over every aspect.
I would not demand anything, just let them know that you felt let down. With this, you should cut your losses and not work with these people again, declining politely if they ever had the cheek to ask you again in the future (yes, people are that gormless).

Hi Emacs, many thanks for your input! My friends and advisor were kind of shocked, too, because adding one person into a team of 20 should not be a big deal. However, I think I should not let my advisor to get involved as he knows nothing about the paper and the team. Shall I email the first author, or first author + advisor, or the whole team?

@HighGPA Hard to know. Did you ever communicate with the supervisor? If you think that it will help, you could write to your "colleague" and their supervisor. If they would have the power to damage your career, it might be prudent to limit it to your colleague. In any case, stay low-key.

@HighGPA And you are completely right to keep your advisor out. You learn to fight your own battles, especially as they were not involved originally, good decision.

His supervisor is a Big Name with dominant personality. However, I don't think the Big Name can harm my career as we are at different field.

I agree with many points made in this answer, but I'd be a bit careful about the suggestion to "list[...] the most important of the above arguments". If not phrased very carefully (and very briefly), such a list of arguments might contradict the - very important - constraint of keeping the email low-key. (Lists of arguments are often perceived as, well, argumentative.) A second thing to keep in mind is that the removal of the OP from the list of authors apparently occurred some time ago (the paper is in the second revision now), so writing such an email now might come across a bit odd.

@JochenGlueck I kind of agree with you, Jochen. Probably, if the paper is rejected or it needs further revision, I can start to ask for inclusion of my name in a future version. I forgot to mention that the paper is now also published on Arxiv, without my name being acknowledged.

@JochenGlueck I didn't notice that it is already quite far into the process, thanks for noticing. If so, probably dropping the whole matter is the best course of action. OP is not going to gain anything anyway then.

Upvoted. This is the nice answer I was lazy to write while commenting under OP question.

Sorry, but you may not like this answer.
Spending time and effort helping someone on a paper, even giving advice on presentation, doesn't make you an author. Had they used your statistical model, assuming that you created it and it is of publishable quality, then you would, of course, be an author. But if they decided not to use it, then you have no real claim to authorship of this paper. If the model is a standard one that you merely suggested that they use, it probably doesn't make you an author if they use it.
However, if you helped them understand the problem better, perhaps that being the reason that they didn't use your model, then they really should acknowledge your contribution to their thinking even if it isn't directly represented in the paper.
I'd suggest writing to the authors, maybe to all of them, asking for an acknowledgement of the form "Thanks to HighGPA for contributing ideas to early versions of the paper". That much is owed, I think.
If they object, then I'd let it go and focus on your own work, unless you want your advisor to get involved. But your advisor, speaking informally with theirs, can make things happen.

And writing to all of them is probably more effective than writing to only the lead author(s), as you then become harder to ignore.

Uh, writing to all authors might be perceived very negative imho.

I truly appreciate your answer, Buffy. Regarding your question, I did help him the first author understand and solve problem better, but it seems like that he is the only one who I directly spoke with and the only one who clearly knows my contribution, for now.

Without knowing the field you're working in, it's difficult to say who should be author.
In some fields, co-authorship is given for any number of reasons, such as to acknowledge the PI who originally collected the data, even if they weren't directly involved in the paper.
In other fields, authors are only the people who both worked on the paper and are asserting that it's true.  They would instead directly acknowledge other people's help and/or cite something to indirectly acknowledge the work upon which the paper is based.
I would think that you should get some recognition for the contribution that you described, but I don't know that it would qualify you for authorship without knowing your field.  And I also don't know if you actually agree with their findings, and thus maybe it's better that you were not and author, but simply acknowledged.
I'd also like to mention what I've taken to calling a 'null acknowledgement'.  That is, you used someone else's work, but then went another direction.  This is especially important in data citation, as there are a number of observational fields where someone might see something of interest, look at data from an instrument to get context and determine if it's a known type of event before doing a deeper analysis.  Although you might not cite that instrument or its data, you still used it.  And not acknowledging that you used it might affect their continued funding.
Similarly, if someone helped you determine that an approach to analysis was not correct, they still helped you get to your solution, and deserve some recognition.

Much appreciated for your answer and your suggestion! I agree with their finding. The field of this paper is between applied statistics and computer science.

I guess I should've been more specific ... I can only comment on my own field, where it tends to be "MainAuthor, Supervisor/PI, AlphabeticalListofEveryoneElse".  As a reviewer, I once saw a paper that I'm pretty sure the co-authors never read (and told the editor, as it neglected to compare their project to similar ones that their co-authors had done.  The editor refused to believe me until I pointed out that there were only 5 references, and none of them were to papers by the co-authors that should've been cited).  I think I got black-listed as a reviewer for that.

Hi Joe, you've done a good job reviewing. Why you got black-listed? Probably you deserve to review for a better journal.

I'm IT support in an obscure field of physics.  I was only reviewing papers about new tools / software in the field so it was at most one per year.  They might've thought I was racist as I had first pointed out the number of grammatical errors, and primary author was a non-native English speaker, and at least three co-authors were American or British.  And I said that the whole paper needed to be re-written (citing the style of a paper by one of their co-authors)

If the supervisor of the postdoc and your own are not in the loop yet, they should be.  At the very least, they should be able to provide perspective.
My feeling is that if someone has worked on a project, even if their particular solution didn't work out, they should be considered for authorship.
I don't know what field you are in, but the claim that papers don't count if you are an author but not the first one is dubious.

Many thanks for your input! The field is between applied statistics and computer science.

While some others disagree, I think you have a strong case for co-authorship of the paper, particularly since you did work on the project in consideration of a promise to be included as an author.  The role you played sounds sufficient to me to warrant co-authorship, particularly on a paper that already has so many authors (who presumably each did a small part).  Irrespective of this, there is a legitimate issue of research ethics in making a promise to you that you would be a co-author and then reneging on this after you did your work.
In terms of what to do now, if you wish to pursue the matter I recommend you seek a resolution from an independent adjudicator at your university.  For example, you could ask for the matter to be adjudicated by an independent faculty member in the university's research committee/research office.  There should be some kind of faculty member at the university who has expertise on the ethics of research authorship, who can adjudicate an internal dispute of this kind and provide a resolution sanctioned by the university.  Indeed, that kind of adjudication would generally fall directly within the purview of academic staff in the university research office.
To get the ball rolling, I recommend that you write to the corresponding author of the paper, note your disagreement on the matter of authorship, and inform him that you would like your authorship/non-authorship to be adjudicated by an independent academic in the university research office.  Write to the university research office and ask for an appointment to explain the matter and to seek an internal adjudication.  This might also be a matter where the university ethics office might want to get involved, in view of the promise of authorship and later withdrawal of that promise.
Regardless of whether you are eligible for authorship, the university is likely to take a dim view of the fact that you were promised co-authorship and then this was withdrawn after you had made your contribution to the project.  There may be an ethics breach involved here, due to misrepresentation of your right to authorship, and you have a good case for some kind of resolution in your favour.  Certainly I would think that an adjudicator would be sympathetic to your position, even if they feel that your contribution falls short of what is required for co-authorship.

Hi Ben, much appreciated for your input! I am truly sorry that I forgot to make it clear that they are from a different university. The posdoc recently moved from my institute to the new university. Does the same procedure applies? Shall I write an email to a faculty member on the ethics team of that university? You mentioned that the adjudicator will be sympathetic to my position. This really encourages me and I appreciate this. However, I am still pessimistic because I did not write any part of the paper -- I only provided verbal feedbacks and remarks, which are hard to track.

I would think that an adjudication at their university would be fine.  So yes, you could instead write to the research office at their university and seek an adjudication.  Procedures will differ by university, but initial contact with the research office should let you know if they are happy to perform an adjudication of this kind and what the relevant procedure is (if one even exists).  As to the view of the adjudicator, I can only speculate, though you also did some modelling for the project (which was ultimately not used).

I think you should definitely discuss the situation with your own advisor before starting off any formal procedure, especially one that drags in another institution!

Authorship can be a contentious issue. In this case, I think it is actually impossible for any of us, including the OP, to know whether it should have been granted. While the OP helped the main author via discussions and suggestions, the final version of the paper may look very different and be the product of many other discussions, suggestions and contributions by other people. In all the projects I have worked on, I have had conversations with people and heard suggestions. They have offered this freely without making co-authorship demands, nor have I offered co-authorship because people close to me in academic positions have told me that would not be appropriate. To me, work that has co-authors involves more collaboration. Devising the idea together, being part of the team that evaluates and discusses approaches.
The postdoc in question may not have ever been in the position to offer authorship. They are working on a project defined by their supervisor and funded by a project their supervisor submitted. The main ideas for resolving the problem may have come from the supervisor. I am a postdoc, but I would not offer coauthorship of anything without discussing it with my supervisor. It may be, in fact, that the LW's statistical model was excluded by the supervisor because it was part of a collaboration that the supervisor was not comfortable with and that was outside of the scope of the funded project.
I think an acknowledgment would be nice, but it is not something to be demanded and probably does not make a difference for the OP except a warm feeling. Lessons for the future are more that not everything is about authorship and the scope of most projects predetermines who can and cannot make the author list. This answer is offered since most people have said that the OP should be a co-author, and it is possible that the scope of the work was much greater than the OP's contribution, and there was simply not a good argument for the OP's inclusion on the list. The OP may also have a dominant personality that makes it difficult for the postdoc to voice that perspective.

Thanks for your answer. “evaluates and discusses approaches" was exactly what I did in the first few months. The idea of the project was come-up by the posdoc.

Perhaps I do have a mild dominant personality, I am not sure. By "mild" I mean I could have forced the posdoc to adapt my method anyways; this is because the supervisor did not handle the data or write the paper, so if the posdoc simply tell the supervisor that this method works the best among alternatives (which was true), then the supervisor would have no option but to accept it. I did not do this and I did not even think about doing this back to 2020, so I doubt that I had a dominant personality.

I don't mean this as critically as it came across. I just mean that there is usually another side to the story. It is hard for me to understand how a project the two of you did all the work on morphed into something with 20 other collaborators. That makes me think there must be more to the story. Something was not transparent here, but it is hard for me to read this without imagining a project much changed from the first version.

This is only a partial answer, specifically discussing the point of whether the OP should be a coauthor or not.
While I would agree with the general premise of other answers discussing this point (e.g. @Buffy and @JochenGlueck), I would like to take a different point of view. In my opinion, the details of the contribution of the OP and whether their statistical model made it into the paper or not do not matter much, since the promise alone is (or should be) sufficient to gauge the magnitude of the OP's contribution.
The reason for my conclusion is that I think the metioned answers do not sufficiently acknowledge the nature of a promise to become a co-author. Undoubtedly, the decision whether one should be a co-author of a scientific paper is based on whether one contributed or not, as @Buffy and others write. However, at the point where one is promised authorship by the first author, the relevant parties have already determined that your contribution is indeed valuable. A contribution to the scientific progress of a project cannot be nullified by further progress afterwards.
Of course, in the real world, the first author may have hastily promised authorship. Even if that is the case, however, the hasty promise in itself is malfeasance. As the OP states, they invested a lot of time into the work. They may not have done that if they had not been promised that the resulting contribution will warrant authorship.
So in my view, the situation is a clear case of the OP's work not being valued appropriately by its co-authors, one way or another. The OP was either tricked by the first author or their authorship was wrongly denied.
With regards to recommendations for action, I do not have anything to add to the other good answers.

Hi Wolpertinger thanks for your answer! I am quite interested in what you said: the nature of a promise to coauthor makes the contribution sufficient. So generally, who can give this type of promise? I am guessing that, if all the first author(s) agree on adding someone, then the promise is given? Does the last corresponding author ethically have the power to give or deny promises? In my case, it is clear that the corresponding author believes that she has the full power to determine the co-authorships.

The first author explicitly complained to me that the second author contributed very little, yet the supervisor "forced" the posdoc to put the person up as the second author.

Hi @HighGPA ! I would maybe not say that it "makes" the contribution significant, but the first author figured your contribution was so significant that they promised you authorship. A corresponding author should not have the power to "decide" authorship. It is the contribution that counts and a first author postdoc should be able to gauge what your contribution actually a was. A promise, in my view, is just the stamp on it.

The main point of my answer is to say: You did nothing wrong here; you were treated badly; your contribution was valuable. I'm sorry this happened to you and I hope it will not drag you down.

With regards to "who can decide authorship": Nobody should be able to decide on their own. Practically, the judgement is often differed to PIs. But again, with a first author postdoc (someone who should really be able to judge your contribution) going so far as to make you a promise of authorship... to me that is a clear case of either the postdoc lying to you directly about the significance of your contribution or a deserved authorship being stolen from you. No matter which it is, it is not ok, which is where I (mildly) disagree with some other answers.

Hi Wolpertinger I definitely agree with you, including your comments on "stamping" the contribution and I do agree that the first author, who wrote 95% of the paper, should be able to decide whose contribution is more significant. However, I am not sure if you agree on this: the sad reality in many groups is that the supervisor has the power over the first author in determining author orders. I am probably too pessimistic but I think there is little we could do about this sad reality...

@HighGPA Yes, I fully agree. I hope that this will not discourage you from doing great work and contributing to science though! My answer was mainly intended to not make you doubt your contribution and search for a mistake that you did yourself.


text:
Concurrent workers independently published part of my results. How to convince the referees that I got the results independently?
In my primary research project (not coauthoring with advisor), I got two results, say result A and result B, back in 2020. I wanted to submit it in 2020, but my advisor says that I need a further result, say result C, to be considered for a top journal.
I agreed with him as I find the (hypothetical) result C way more exciting. Now I am half-way through getting result C.
A competing group tried to get result A (with mistakes) and submitted a paper in 2020. The referee rejected their paper because they made a fatal mistake. They corrected the mistake and submitted the corrected version of result A again. They got their paper accepted at a top journal in March 2021.
I don't know how to speak with my advisor about this without offending him. My advisor says it is a good thing as it proves that I was at the right direction and a super active subfield. I think he is just trying to sooth me by saying nice words.
Now I am submitting the results A and B by myself. I do have timestamps on public cloud services to prove that I did get the correct version of result A before them. However, the timestamp is just on a poorly written draft rather than a fully written paper, because my advisor asked me to work on C.
It is my first time dealing with this issue. Shall I also submit the timestamps to prove that I had the result A?
I am thinking about putting result A into appendix and emphasizing result B in my paper. Am I correct? Is it best to submit the paper to the same journal?
I will cite their paper of course. I want to say that I am a concurrent worker work in parallel. I am afraid that the everyone think I am the follower, who falsely and unethically present myself as a concurrent worker.
What shall I say to make clear in my paper what happened?

PS:
I was not worrying before March because I thought:

The competing group made mistakes so they are not competent competitors

Result A alone is not enough for top publications. Their paper still contains some technical mistakes and exaggerated claims.

However, facts proved that I underestimated the competing group. The hard lesson learnt.
The field is applied statistics. This is a primarily theoretical work that studies the properties of two widely-adopted statistical methods.
Related: Pre-print service like arXiv but with private option, so I can correct my mistakes without publicly advertising my mistakes?

https://academia.stackexchange.com/questions/88219/how-do-i-address-my-research-publication-being-scooped

@AnonymousPhysicist Many thanks for sharing the link. I actually bookmarked the post. But it seems to focus on a non-independent collaborator scooping the other?

I think not publishing your approach is a disservice to science. What if a couple years down the line someone discovers that the other group made an error in their proof that slipped through peer-review? Then your proof might still hold. Just make clear in your paper what happened.

@Joooeey I fully agree with your points, Joooeey. May I ask how could I make clear what happened in my paper? What should I say in the paper?

I haven't been in Academia very long. Ask your advisor. They know better how to deal with the situation. Just make sure to explain how you got result A and to mention that the other group got result A as well while you were finalizing your manuscript.

You've been scooped, I'd say it's common. You could write: Further to publication of our preliminary results [1], X et al. showed ... Or: In parallel with this work, X et al. showed ... (You could cite your technical report elsewhere in the latter instance.) You can then explain how your work improves on theirs, especially as your results are way more exciting.
Your approach to result A may be different to X et al., which may allow you to present the result in the main body. Also, your result A may be slightly different to their result A (i.e., you may have slightly different results). Speak to your advisor about how to approach this.

You are right the our approaches are different. And my result A is slightly different from their A. Many thanks for your help! Are you sure that I can say that their work is "in parallel"? I think I might be attracting bad opinions on me by saying this, as their work is published first.

@HighGPA You conducted the work in parallel, you'd merely be stating a fact. Albeit, there's a time window for how long you can use that. Maybe a year is okay, imo

That is the best encouraging and motivating line for me in 2021! I don't know how to express my thanks to you!

I never published the "technical report". It is only a poorly written draft without introduction and literature review. Is it unethical to cite it as a 2020 work?

That is the best encouraging and motivating line for me in 2021: You're welcome! I don't know how to express my thanks to you: Go publish an awesome paper!

I never published the "technical report": I misunderstood your cloud services remark, my apologises. It is only a poorly written draft without introduction and literature review. Is it unethical to cite it as a 2020 work?: It is ethical, but it's rather pedantic. Your readers are unlikely to be interested. Focus on selling the best parts of your work, don't work so hard on selling the part you've been partially scooped on

I am assuming this is not experimental work.

How to convince the referees that I got the results independently?

Don't.  How you got the results is irrelevant to the fact that they have already been published.

It is my first time dealing with this issue. Shall I also submit the timestamps to
prove that I had the result A?

No.

I am thinking about putting result A into appendix and emphasizing result B in my paper. Am I correct? What is best thing to do here?

That is reasonable.  Cite the other paper.

Of course I must cite the other paper. Much appreciated for your suggestions, Anonymous Physicist. Shall I add in cover letter that I have been working in this project for a long time?

Good answers here already, but I feel some points are still missing:
How you will behave depends on your research area, but I've never seen someone trying to revert a scoop situation and succeeding. Try to establish that your work came first is irrelevant in my opinion. You had the result and didn't publish it, and frankly, it doesn't make a lot of sense academically. Some areas are more competitive, but I feel researchers often forget that replicate experiments (or, in your case, run them simultaneously) are as important as reaching new findings. And you can always add or expand on former findings based on slightly different methodology.
Two researchers (or teams) working on the same topic will eventually get results for the same questions. That doesn't mean your work is wasted. At the very minimum, you have your own independent results to corroborate (or contradict) the already published material. This will only strengthen your publication with possible result C as you can present your findings: "results A and B, published in XYZ, are confirmed in my independent experiment, and we further explored our research, reaching the astonishing result C." Simultaneously, you are in a position to evaluate the competing group's work in your future publication (that makes more sense in some areas than others).
I understand it would be nicer if the first findings (results A and B) were in your publication, but it seems the ship has sailed.
Finally, as stated in other answers, speak to your advisor about the situation. It depends on your relationship with your advisor, but stating the facts and discussing the next steps with no finger-pointing should not offending him.

Thanks for your answer, Meneses. It will offend him because now he still think that I should do more work and complete the (hypothetical) result C. I think his previous requirement was one of the reasons why I lost this priority, though I have not directly bring this up to him.

@HighGPA That's why I wrote "It shouldn't" ;), relationships are complex. Right now it seems (I don't have the full picture here) that without result C it will be hard to publish on high-level journals. Unfortunately, replicating experiments/results are not an academic priority unless you prove the former results wrong.

To expand on earlier answers, the following is offered.  One of the hardest things to do is to not worry about the primacy of one's research.  That other researchers independently determined your answer A should not detract from your own independent work in the determination of answer A, or consequential answer B, or the culmination in your determining C.  This can be easily dispensed with in publication of your own work within the explanation of the development of C (for instance) by stating that independent to your development of answer A, researchers X and XW have subsequently published this result. This does not detract from your own independent work.  Nevertheless, the statement is factual and establishes your own work as original independent thinking on the problem, without overtly calling attention to who was first. Don't let establishing the primacy of your results distract from the importance and correctness of your work.
Many researchers are faced with the realization that other people are interested in the same research area and are capable in making the same determinations that one is able to make, or that one has already made.  The total state of independent research is obviously unknown.  Only what is published is known if not lost in obscurity.  Consequently, people readily glean valuable insights from published research abstracts or grant applications.  Beware, there is also research espionage where people glean information from conferences or prepublication sites, hidden ftp sites or through unauthorized electronic access, or even thru guest visitations to research offices and laboratories.  Hidden information can be determined by casual observation and making mental notes; this happens all the time. Tacit information is sometimes inadvertently disclosed, and that can open the whole book on what one is doing or thinking.
So lets presume that one's thinking on a specific research problem is largely, or entirely, known only to one's self and closest colleagues in confidence, being in one's notebooks and hidden from view, not generally or openly discussed except in closed, confidential meetings in the direction of the work.  Very few, if any, other people have reached one's state of thinking and development regarding this particular research problem.  Deep thinking on particular technical aspects of this mathematical or statistical problem is not likely being done by many people independently, and simultaneously.  And even if others are working in the same arena, they may not have achieved the same state or advanced position of one's own thinking.  Tempered with self doubt, one may become overly concerned that the path to get to the current state of one's research was relatively easy and that others have achieved this and may be far ahead in their own advanced developments, or at least not far behind.  Yet realistically this may actually not be so.  Your work has been developed by your own applied critical reasoning to attain its current state.  You have seen the result in care you have exercised, and the consequences in the work of others fraught with error who have not used that same care.  As suggestions for thought: Persevere and publish your own complete work when its impeccability is certain. Be dedicated but not hurried or competitively driven by others.  Don't be overly concerned with what has passed elsewhere in print, especially if of no ultimate consequence to the direction of your own work.

Very detailed answer! Great advices and directions that I will follow. You made a point that my development process of A, B, C cannot be replicated by others. This is truly encouraging.

I want to stress just a single point:

Now I am submitting the results A and B by myself.

I would not do this. Your advisor is most probably part of the journey to reach A and B. Publishing the results without his name is very likely misconduct.
This might harm the relationship and trust between your advisor and you!
He told you, that A and B is not enough for a top journal. I guess he is right. He thinks you can also get C. So focus on achieving C. There is no point in regretting not having published A and now reclaiming you were first.
The price waiting for you is getting C published first. Without any scoop situation, without getting in conflict with your advisor.
If you insist, you can still put your proof for A und B into the paper containing C. It might not fit the paper and a reviewer might ask you to remove it. Difficult to say without knowing the whole story.

Much appreciated for your answer! I will definitely ask for his permission before submitting anything.

"Publishing the results without his name is very likely misconduct."  Not very likely.  Possibly.

If you look up most groundbreaking discoveries you'll find that many if not all* end up in this situation: two completely independent researchers or research groups come up with pretty much the same result (be it a prediction, measurement, discovery...) around the same time**. This is exactly the reason many Nobel Prizes are actually shared among multiple people, e.g. the Brout-Englert-Higgs boson.
If you're aware of the other publication, read it, digest it, and integrate its presence into your own work. As you say, they may have glanced over certain aspects. Highlight the differences (and reasons for these differences), and expose the similarities. Don't just think the others made "mistakes". Maybe it is you that made the "mistake", or both of you just approached it differently (e.g. disregard a specific class of influences, or chose to neglect certain parts of an interaction because you deemed them "small" for some reason, be it mathematical or just based on "a feeling"/experience/...).
*This may be a small exaggeration from my part, but I do find this simultaneity occurs more than you'd first expect.
**Time here is relative; while experimental progression is often fast (and simultaneously reported discoveries will lie close to each other in absolute time), more theoretical or more general discoveries can be spread over decades, only later to be "united" as really being the same thing all along.

I agree with you. This is very encouraging to me. I will not loss my hope. Thanks again.

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text:
Is it okay to reference something well agreed by only one source, [e.g. 1 source]?
I am wondering if it is okay write something like:
"Many scholars hold the view that this is like that" and then give one example reference (e.g. Gans et.al. 1965) ?
The reason I am wondering because it is generally accepted and no one seems to disagree on that. Is it better (1) not to reference at all, (2) reference the most original paper and skip e.g., or (3) mention few?

Here are a few ways you can keep the citation with just that one resource. I'd rewrite to avoid the "many scholars hold the view" part (which you would need to defend by citing multiple papers). Instead, you can emphasize the idea itself.
If you're setting it up to disagree with it:

It has long been held that X is like Y (see, e.g., a comprehensive argument in Gans et al. 1965).

(Though if you're getting ready to argue against it, you will want to go into further detail and cite several different sources.)
If you're just trying to get that fact in there as background you're building on, you can say:

It can be useful to use Y as a model for X (see, for instance, a treatment of this question in the context of Z in Gans et al. 1965).
X can be thought of in the same way as Y (e.g. Gans et al. 1965).
X resembles Y (e.g. Gans et al. 1965).

All of these formulations shy away from asserting X is exactly like Y, or that X is a type of Y, but they get the point across and give people further reading.
Given all the time in the world to track down every bit of nearly-common knowledge, you would find and cite the original scholarly reference for the idea, AND (if needed) recommend a later paper that would help the reader because it is easier to read or is written in a context similar to the context of your own paper or it uses modern notation. Readers who are aware of the original source might be amused/disturbed to find only a "lesser" reference there without explanation.

I have used the following construction:

Many scholars hold the view that x is like y (Smith et al. 2000; Brown et al. 2001; and many others as cited in Stone et al. 2005).

Usually I try to reuse citations. If I already cited a paper which also states something more obvious, I cite the paper a second time instead of adding a new entry to my references just for a trivial statement.

If something is considered common knowledge in the field (for instance, Newton’s second law in a physics paper), then no reference is required. If it’s been developed multiple independent times, though, I’d use more than one reference to establish it.

However, Newton's second law in a paper from some other field: do include a reference. For example a standard physics textbook.  One reference is enough.  However: also include the page number in your reference, do not merely cite the whole book.

@GEdgar I can't imagine citing a textbook for Newton's second law in any non-school context...

If you are certain that several scholars hold the view, then presumably you know where they said that, and you should cite several sources. If you are not certain, you should look for sources.
Be careful to avoid propagating untrue information and wrong citations. For example, Friedman & Koller (2003), a machine learning paper about learning Bayesian network structure, says

in small domains with a substantial amount of data, it has been shown that the highest scoring model is orders of magnitude more likely than any other (Heckerman, Meek, & Cooper, 1997).

But Heckerman, Meek, & Cooper (1997) only showed the claim to be true in one specific case. Friedman & Koller's claim might be true, but their implication that Heckerman, Meek, & Cooper (1997) showed it to be true is not.
References
Friedman N, Koller D (2003), Being Bayesian about network structure: a Bayesian approach to structure discovery in Bayesian networks. Machine Learning, 50, 95–125
Heckerman D, Meek C, Cooper G (1997), A Bayesian approach to causal discovery. Technical report MSR-TR-97-05, Microsoft Research

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Should I add a non-requested work to my major revision of a scientific journal paper?
I am wondering if I should follow the advice of my co-advisor or a relative of mine who is also a researcher and used to write a scientific paper in the same field (but currently with less experience then my co-advisor).
I will make it a bit more clear:

I am a Ph.D. student in CS, specifically in networking, I wrote a scientific paper that I have sent to an Elsevier journal.
Last month, the journal replied to my submitted paper and requested a major revision; so I know that I have a bunch of rework to do.
One of three reviewers was complaining about the contribution and evaluations/simulations that I have done, while the other two didn’t complain in the same manner.
Following to this and in order to make it clearer for the complaining reviewer, my co-advisor advised me to add a small simulation/evaluation to my paper to clarify the contribution and justify why we considered various scenarios in terms of scale.
In the meanwhile, I asked a relative for help, he said that it is not a good approach to add this simulation/evaluation since no reviewer asked for it, and could be harmful to my work/my chances to being published.

While I can agree with my relative about adding a non-requested work, my intuition pushes me to follow my co-advisor, since this is a little work that can clarify, justify more the need for considering various scenarios (which I have done and the reviewer didn’t understand why I did it).

Ok, to say it in a correct way, it is my co-advisor not my advisor anyway he acts like if it is one, since my advisor does nearly nothing. I corrects the post

My parent is nearly in the same-field, he is a networking guy

FYI, I suspect that the word you're looking for is "relative" rather than "parent"; I think you're being misled by the similarity with French parent. (And if he really is your parent, then the usual word for a specifically male parent is actually "father".)

@ruakh you are absolutly right, I Will correct it and yes it is a relative

I would want to answer this based on my recent published collaborative work on a research paper.
We were in a similar situation where one reviewer was not convinced with experimental evaluation. However, I along with my collaborators came up with an idea to include a small simulation [#] to justify our procedure for kind of experiments that we had conducted with the hardware.
In fact, the new simulation study was included to the paper as a separate sub-section under "Results and Discussion" section. We submitted the revised article, and all the reviewers were pretty convinced with the new results and discussion section.
So, if you want to add something new, make sure that it really is closely related to your work. It should not diversify your present story that is conveyed by your manuscript.
[#] This simulation has to be related to the work you are doing. For example, if you are doing something on security protocols in network, then you could add a simulation showing the energy efficiency.

I am working on routing protocols, I tested protocols performances and small and large scenarios, but what my co-author advise is to re-do a simulation under a small scenario, while changing the position of my Acces Point (AP). So I guess that it is still very relevant to my work ;). Anyway thank you for your valuable answer

@HanniBaL90 I m sure your co-guide is asking the right thing to do in this case. Good luck!

As somebody in a different area of computer science, I find it hard to see how including a little more true, relevant information could possibly be harmful. The reviewers think your paper is worth publishing with revisions. The worst case I can see is that, when you submit the revised version with this extra unsolicited material, the referees ask you to remove it again.
You can always mention your parent's opinion to your co-authors and see what they think.

What you said above is totally true, but I thought that reviewers may doubt of my work if I decide to add something in my ow. Anyway, I don't want to mention the mind-conflict that I have to my co-authors. Thanks ;)

Your issue seems to be a social rather than a scientific one...
You commented this

Anyway, I don't want to mention the mind-conflict that I have to my
  co-authors

Why not? What else do you have your co-authors and advisors for if you cannot use their experience?
In another comment you mention something about an advisor not really being an advisor.
Furthermore, the fact that the conflicting input came from your parent should not be of any relevance. In an academic discussion your parent is a fellow scientist and their opinion should be weighed as such.
So, my unsollicited advice would be to get your act together. Do you feel a loyalty conflict between your parent and your advisor? Get over it. Is your advisor not helping you? Deal with the issue.
You need to feel comfortable asking for help/input/advice to those who want to help you. And you need to feel comfortable with not implementing the suggestions you get if you have solid reasons.

By no means you are restricted by what the reviewers said.
Often I don't have enough time to do everything I want to do in a paper, end up sending it "incomplete", and then add it after it gets accepted. Including in places where there is only one round of review (some conferences).
As long as you don't drastically change the contents of the paper, it's fine. It's a journal anyway, the reviewers will go over it again.
Personally, I feel the opposite of your relative, I like when the authors don't take the comments literally, but rather understand what is motivating the comments and go an extra mile to make the paper better.

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Would several years of appointment as a lecturer hurt you when you decide to go for a tenure-track position later on?
Academic jobs are always on a state of high-demand, more people seeking positions than there are the latter. I am an international worker in the US searching for a university position. Currently in my situation, lecturer positions seem more reachable than tenure-track because of various factors such as the number of publications, start date (lecturer seems to have more variable starting date than assistant professor, is this correct?), my visa period (which will expire in 4 months, so preferably can find an academia position since the H1B for these jobs is not capped). But ultimately, I still desire a research-focus job. Assuming I get an offer to teach as a lecturer and after 2-5 years decide to switch to assistant professor whether in the same institution or somewhere else, would my application be undermined in some way by the fact that I spent a couple of years not focusing on research? Also, how manageable is it actually to do a lecturer while doing your own research, e.g. collaborating with your former postdoc labs or getting a research grant?

What do you mean by "lecturer"? It has vastly different meanings in different places. Were you an adjunct in US, or a formal lecturer in UK, for example?

@Buffy I think it's clear from context (US, non-tenure track, non-research focused) that OP is talking about a US-style "lecturer" position which is a non-tenure track (often explicitly temporary) university teaching role.

@BryanKrause, can we hear from the OP, please? I've never been anywhere in the US that uses Lecturer as a title though I assume they exist.

@Buffy, Sorry, the question is US-centric. In the US, there are two kinds of faculty positions, the research-focused and teaching-focused roles. There are further several different types of teaching positions, one of which the university lecturer.

@Buffy, I am the OP.

Then please explain the nature of the lectureship. A temporary position or renewable contracts?

I am not sure if it is uniform across the country but I have only heard that it is on contract, but I maybe wrong.

there are definitely permanent "lecturer" spots in the US, if less common, which indeed have a reduced research requirement.  I guess some departments find this helpful for staffing, for example in mathematics where they are required to serve students from across the institution.  (such as in my masters institution , like in this Statistics faculty listing:  https://stat.as.uky.edu/people)

Also, how manageable is it actually to do a lecturer while doing your own research, e.g. collaborating with your former postdoc labs or getting a research grant?

This is the question that is intimately tied to your title question.
Situations vary from position to position, but the overall impression I get is that lecturers are overwhelmingly expected to contribute their full time to instruction, and although part-time lecturer positions may be available the salary is typically too low to survive on at an expected standard of living for someone with a college education without a partner making a normal salary.
That means there is effectively no time to dedicate to research and it is not possible to be competitive with people who do have dedicated research time.
It's not so much that your application will be "undermined", but that if you currently lack the research productivity to be offered a tenure track job, you should not expect that spending far less time on research will change that for the better.
It's possible you would be more competitive for tenure-track jobs with a teaching emphasis at institutions with less focus on research, but that's probably true now, too.

Thanks for your answer. How do I identify that the role in a job ad is a tenure-track that focuses on teaching rather than research? Does it have a special name? Have just heard about it from you.

@nougako You could actually ask that as a separate question. As far as I know it's never been asked here before, and if you are wondering then maybe others would wonder too.

@nougako Mostly by the type of institution; these would be typically be at small liberal arts institutions, teaching-focused satellite schools of state college/university systems, etc. Schools that aren't known for their research output.

@nougako Oh, I did just find the following, which might help: https://academia.stackexchange.com/questions/195379/is-a-non-tenure-track-lecturer-position-in-math-a-suitable-stepping-stone-to-a-t

@DavidWhite The link is definitely relevant, however, the OP hold a grant by the time they were looking for the job. That's a huge boost which is not applicable in my case sadly.

@BryanKrause Would that be something that can also be determined from the university's R number classification (R1, R2, R3)?

@nougako They'd mainly be the ones that aren't an "R" and don't have graduate (or at least not doctoral) programs.

"No time for research" - It's perhaps worth pointing out that there are two issues here. First is the obvious one: (hours in week) - (time needed for teaching) is likely to be a small number (and is not guaranteed to be positive). The second is that whatever spare time you do have will be heavily fragmented. Depending on your personality, and the nature of your research, you may find it hard to make progress when you have to keep stopping and starting.

I largely agree with Bryan's answer, but because he mentions not being sure if things would be different at a more teaching-oriented place, I wanted to chime in. I'm a professor at a liberal arts college, that values teaching more than research, and who has been on many hiring committees. For us, there's no expectation of winning a grant.
Unfortunately, I do think candidates who have been in non-tenure track "instructor" roles for too long are at a disadvantage, even if they have managed to do enough research to be "above the bar" on research. I wish it weren't this way, but I think there's some natural human bias that enters the minds of search committee members, causing them to ask themselves why this person stayed in "instructor" jobs instead of securing a tenure track job earlier. There's also bias in favor of shiny new PhDs, because people can forecast in their mind all the amazing things this person could do (vs someone who has been working for enough years that you know exactly what they produce each year). At my university, we now undergo training to try to resist biases like this, and part of that training involves reading research papers illustrating that the bias is real and has real consequences in searches, historically.
For this reason, when we hire an instructor (usually on a 3-year contract), I advise them to push hard to get enough research done to be a viable candidate at a tenure track job, and then to apply for such jobs as early as possible. I think 2-3 years as an instructor can be viewed as analogous to a postdoc, or a chance to learn the ropes of a faculty job. If someone has been an instructor for 5 or more years, I think their chances of getting a tenure track job anywhere decrease a lot.
Part of the reason, in addition to the bias I mentioned, is that many teaching-oriented places prefer to hire folks who are fairly fresh out of their PhDs, and then mentor those folks in the job, teaching them how to effectively teach our specific students. If someone has been an instructor for years elsewhere, they maybe less malleable. Of course, it also means they might bring a radically different and better approach, but it seems the general thinking is still more that it's a negative.
Lastly, in my experience, it seems that searches usually do not result in hiring an internal candidate (i.e., someone who has been working as an instructor in the department already). I think this is because of the bias I mentioned above regarding forecasting the potential of a candidate you just met vs. the tried-and-true person you've been working alongside for years. Again, I think this is lamentable, and I've often wanted to hire our instructors into tenure track jobs, if we had the chance. I think the best advice is to not get too comfortable in a non-tenure track job, and try to stay fresh and exciting (in both research and teaching) so you can outcompete younger people when on the job market.
Let me try to address a few specific questions the OP asked.

Lecturer seems to have more variable starting date than assistant professor, is this correct?

No, that's not correct. In the USA, for both, the starting date is tied to when classes start, e.g., in August or January for universities on the semester system. The vast majority start in August.

ultimately, I still desire a research-focus job...would my application be undermined in some way by the fact that I spent a couple of years not focusing on research?

Yes, absolutely. You'll be competing with folks who had a much lower teaching load, plus explicit research mentorship, and they were in more "prestigious" jobs. You can't afford to not focus on research if you want a research job.

Also, how manageable is it actually to do a lecturer while doing your own research, e.g. collaborating with your former postdoc labs or getting a research grant?

This varies widely. Most lecturers do not do much research. That could be self-selection (i.e., because they chose to take this job), or lack of time, or lack of mentorship, or a combination of factors. However, our lecturers are able to do good research if they want to. They just need to make it a priority, and a bit of mentorship/encouragement helps a lot. I think it's very hard to win a research grant as a lecturer, at least in math. Those tend to go to people doing a postdoc (with little or no teaching, like the NSF postdoc) or to professors who are using the money to fund grad students and postdocs.

Thanks a lot David for sharing your experience of the matter. I also don't plan to remain a lecturer for too long, 2-3 years seems ideal indeed. But now with the answers I am getting, I am reconsidering if I would really be willing to take the lecturer path to TT.

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Review recommendation when you can't review all of the paper
I'm reviewing a paper on a theoretical topic in the physical sciences. My opinion is that one part of the paper is valuable and deserves to be published, but another part needs attention from a reviewer with expertise that I don't have. (It wasn't obvious when accepting the review that this would be the case, otherwise I would have declined to review it.) I feel that that part of the paper should be reviewed by someone with the relevant expertise before publication.
I said as much in my initial review. However, the paper has now come back to me along with the other reviewers' reports, and none of them commented on the part that I wasn't able to review.
This puts me in a funny position, because if I recommend acceptance there is a possibility that it will be published without anyone checking the relevant section, but if I recommend reject then I'm recommending rejection purely on the basis that I personally didn't have the expertise to review part of the paper, which doesn't seem fair. I could explicitly state that it shouldn't be published without another review, but that seems like I would be overstepping my responsibility as a reviewer.
What is the right thing to do here? Should I contact the editor, or am I just over-thinking this?

Why do you feel like you can't just tell the editor what you said in the first paragraph? Your input as a referee is not limited to either recommending acceptance or recommending rejection.

@AdamPřenosil I did tell the editor that, in my initial review. However, it has now come back to me along with comments from other reviewers, which don't address the section in question. So the question now is whether I should insist that another reviewer be found, since the editor seems to have felt it wasn't needed.

"... there is a possibility that it will be published without anyone checking the relevant section" - this is none of your concern, it is the editor's job. You may (and should) reiterate your point to the editor, but you can not assume responsibility for what is an editorial decision. Your job here is to review what you can and clearly communicate the limitations you have faced to the editor.

Do you take into account, at least as a possibility, that the other reviewers read and understood the part in question but didn't comment because they thought it's OK? How sure are you that that part wasn't in fact competently checked?

Recommend to the editor that the paper needs further review of the parts you can't grok and that it needs to be someone with expertise.
An ultimate resolution, though not up to you, is that the paper be separated into several if any of it is to be published.
Just be honest with the editor about the state of the paper and the review process as you see it. It isn't your sole responsibility.

Should this sort of thing be written in the confidential comments to the editor, or in the main review?

It is for the editor who has additional work to do on this paper. Warn them. It isn't information for the author. If you know of someone with the required expertise you could recommend them to the editor.


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How to do an open-ended all-matching search in Web of Science
I would like to find out the total number of publications in the Web of Science.
(Actually, I'm interested in certain subcategories over time, but not restricted by word content at all. The categories and years can be accessed after doing the primary search).
What is the most general possible matching string / syntax?
My best idea so far is using
all=(the)

in the advanced search.  This is obviously not perfect. I tried to combine something like x OR NOT x but those get rejected.
There is a related question here:
Global number of publications over time
to which one could give more up to date data based on this question.

Ask your librarian whether there's a better way to do this? (Ideally, you would like to run a database query that only counts and doesn't go to the trouble of retrieving all of the bibliographic information!)

I wouldn't recommend using search to dump all of someone's database. This can easily be considered rude or abusive behavior.
Maybe not as serious as the example I'm about to give, but as a thought experiment I'm thinking about how printing at schools I attended pre-university was free of charge. I'd expect no trouble to photocopy a few pages for an essay I was writing (that's the intended use), but I'd expect to be in big trouble if I printed out whole books of every source I included in my paper due to the ink and paper costs (copyright issues aside).
You could contact the administrators to try to obtain this information another way, but please don't abuse search functions on websites or try to bypass limits placed on search.

This is not useful. It is premised on false suppositions about my intent and about what is possible, and is irrelevant to the question. Dumping the search result  is neither possible (as far as I know) nor asked for. I am simply trying to count by category, which is part of that for which my University is paying great money to Web of Science.

@CPBL - your university also pays for wonderful research librarians who know how to get the most amazing information quickly and easily…

@CPBL I'm not assuming anything of your intentions except that you want to perform a search that returns something from all rows of the (very large) database you are searching. There are going to be much easier ways to do this than with the functions made available to you for other purposes.


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How to cite in-text same source multiple times (chicago style)?
is there a more economical way of citing something like this:

In her essay titled “Electronic Literature: What is it?”, N. Katherine Hayles (2007) gives an account of the state of electronic literature. Hayles (2007) maintains [...]

Can I avoid the second in text citation? When do I need to put it back (like different paragraph and so on)?

What does the style guide of your target journal say?

@EnergyNumbers no target journal, I'm writing my my dissertation.

OK, so what does the dissertation style guide for your department / university say? What did you supervisors say? What have other recent theses from your department done?

I may be missing something here, but can't the second "Hayles(2007)" be replaced by "She"? The context (namely her work) should still be clear.

This question sounds like one that should be on English SE rather than Academia SE.

@DCTLib Can you please convert your comment into an answer so that I can vote for it?

There is no necessity to repeat a citation each and every time you refer to the paper. Whenever it is clear from the context to which work you refer to, there is nothing wrong with using words such as "they", "he", or "she" to refer to the author(s) of other works.
For example, you could write:

In her essay titled “Electronic Literature: What is it?”, N. Katherine Hayles (2007) gives an account of the state of electronic literature. She maintains [...]