Writing a short paper pointing to a mistake in a 2012 journal paper that got significant citations, while discussion with the author not successful I've very convinced that an author made a serious error in the main theorem in one of her 2012 papers in Statistics and Computing, which is a reputed journal in machine learning. I contacted the author, and we did have a respectful conversation, in which she neither agreed nor disagreed with me, but didn't seem to know a mathematically fundamental step mistakenly used in the proof of the theorem. Now, since I'm convinced that this is indeed a serious error, what should be my next step?
Along this line, I followed this question, and ideally I wanted to collaborate with the author, who did very kindly give me half an hour to discuss in real time, but I understand that there's no hope for writing a corrected version of the paper with her, since she neither agreed not disagreed with me.
Now since I'm in the beginning phase of my research career, I'm thinking of leveraging my discovery into writing a paper that'd be along the line of &quot;Notes on the paper X&quot;, which'll mostly focus on discussing why the theorem  and the proof is wrong. I do of course want to acknowledge the discussion with the author. How can I make this possible?
See the thing is: since the main theorem of the paper is wrong, it makes the subsequent experiments wrong as well, so I can't really do much to improve the results into something to make a brand new paper. However, I do want my contribution to the community (that catching the mistakes and explaining them why it's wrong with potential counterexamples) noted. This is the reason I want to publish my finding. Also I'm wondering even if I write to the editor, they acknowledge that it's a mistake, will they necessarily publish my notes in the same journal? Intuitive it seems no because it's just a correction (albeit to the main theorem), but yes, as this should be the appropriate choice, as this is the very journal the paper was published in. How should I go about it? Thank you!
If you are sure of your work, there is little reason not to write the paper and submit it. Cite the other work properly, of course, and make your case clear. If the editor of a journal doesn't reject it out of hand it will be passed to a couple of referees who will give their judgement on the work.
But you don't jeopardize your career by doing this. Your paper isn't public until it is published. But asking the editor first is probably a  mistake, since you can't develop your argument without writing the paper.
And, it isn't impossible that in writing it up formally you will find an error in your own work. Make the case. Submit it for publication if it is strong enough.