Eigen

That's also very field specific. In chemistry, I would say it's not likely that an award would bump you from not accepted to accepted at any of the top schools- there's usually more than enough funding. 
More funding strapped fields will be different. 
Its important to remember that often even with the GRFP, the department needs to pay quite a bit for you (tuition + fees + insurance) even when you're on tenure, and will need to find funds to pay for you when you aren't on tenure. 
I would also recommend making sure post-award acceptances come with something in writing indicating a garauntee that they will support you (to what level and how) when you're not on tenure. It's again field dependent, but remember that a GRFP funds a good bit less than half of the average PhD. 

This is totally implied, but just to be clear- it also only applies to schools who have signed the CGS resolution in the US- not all have.

Guys, have you... been on the internet before? 
 

Perhaps the woman with the same application as you was not accepted above her abilities, but rather you were rejected from a position you could have otherwise earned for the sense of entitlement, argumentative attitude, and lack of scientific understanding you've shown here.
 
If you have "observed a consistent pattern," could you please share your data? Or is it just another anecdote or two with incomplete information?

.....
Honestly, you don't seem to really want to have a discussion. You either call other people's arguments absurd, narrow the field of your arguments so they can't possibly be as much of an expert as you are.... And at the same time want other people to provide sources and then argue that that's an "absurdly high standard of evidence". 
On an academic forum, for a discussion among academics, proper citations and peer-reviewed research with data is pretty much the expected standard of evidence, not "well I've heard" or "I know people who say..."

.....
Honestly, you don't seem to really want to have a discussion. You either call other people's arguments absurd, narrow the field of your arguments so they can't possibly be as much of an expert as you are.... And at the same time want other people to provide sources and then argue that that's an "absurdly high standard of evidence". 
On an academic forum, for a discussion among academics, proper citations and peer-reviewed research with data is pretty much the expected standard of evidence, not "well I've heard" or "I know people who say..."

There are many many factors that influence acceptance to a school. The other persons letters could've been different, their SOP different, maybe they had better networking. Regardless, it is very rare to find someone with even a "similar" application. As stated prior, many say it doesn't exist, while I state it exists but doesn't play a major role. In your particular example, unless you knew exactly that their letters were the same, that their SOP was the same, and every other thing was the same, I don't think you can say your applications were "similar" and that gender was the sole cause of that persons success. 
I'd also say that the issue is not sensitive, but rather the way you worded it was plain wrong. What do people think about discrimination in STEM? The first statement should be is there gender discrimination in STEM, not what people think about it. Why does nobody discuss this? Well if it doesn't exist, not much to discuss then. It seems like an important issue. If it did have a major impact on applications and did exist, then yes it would be an important issue. The problem is 1) From general consensus this does not apply in most fields of STEM so 2) no point in discussing it and 3) Not an important issue because it doesn't exist. Thus, people rightfully so asked you to explain yourself and provide evidence. Also, I don't think anyone is strongly opposed on a moral level, but rather on a factual level (they disagree not because they think it's right to discriminate based off gender, but because they think that gender discrimination in grad applications does not exist) 

Lol at this absurdly high standard of evidence to have a discussion about something. Fine, it is my "conjecture"....

I was going to write a thoughtful long response, as someone who is active in advocating against gender discrimination in my field. I have a lot to say, both from personal experience and based on a large dataset I've collected along with a committee I'm active on showing bias in almost all aspects of a woman academic's life once she graduates from college. (Before you ask for the data, it's confidential and we're in the process of writing up a paper, so if you're *actually* curious, ask me about it in a few months.) I'm in a field where there are more women undergraduates and about as many graduate students as male students. But fewer women get onto shortlists for academic positions; in fact, even once on a short list, they are still less likely to get hired than a man on the same list; fewer women currently serve as faculty members; fewer women get chosen to present papers at conferences; fewer women have their papers published in peer-reviewed journals; fewer women get invited to contribute to handbook articles, which feature the top scholars in the field giving an overview of their main research topic(s); fewer women are invited speakers at conferences; fewer women get their work funded by government agencies. I could go on.
But this poster thinks that being a woman magically opens all doors for a candidate, from sample size N=1, and not even having a full picture of that particular one. To which all I can say is, Wow. 

I think also perhaps you should have posted a much more specific thread about math, rather than "sciences" in general, since you don't really seem to want to talk about the other fields and are focused on math being so very different. 

I can literally repeat questions from earlier exams on the final without seeing students pick up on it. 

I agree and I apologize if I implied that a mismatch of the gender distribution of graduate students and the general population means there is discrimination at the graduate admissions level.
(Aside: I do, however, think that each field of academia should have roughly equal distribution in demographics of academics as the general population. One reason is that people are often personally motivated to do research, often in areas of disease treatment etc. Another is that it is my opinion that the only point of doing any research is to benefit humanity and therefore, the people doing the research should represent humanity, not just a subset of humans who have the opportunity to do so. However, this is problem at a much larger level and I'd say it is beyond the scope of whether there is bias at the graduate admissions level or not.)
Back to the scope of this thread: The "control group" we must compare to is the pool of which the graduate applicants come from. So, if you accept the basic premise that men and women are equally brilliant**, then you would expect that the demographics of people accepted into grad school should match the demographics of people applying to grad school. If we do not see this, then there is something introducing bias at the admissions committee step. This is why I also brought up the fields medal example. The fraction of women in Math is much more than 1 in 56, yet the Fields Medal has only been awarded to a woman once in 56 times.
The bias could be many things, including unconscious ones due to the committee choices, conscious ones due to prejudiced committee members, and systematic ones that unfairly favour men over women that aren't directly in the control of the committee. An example of the last one is GRE scores. Findings from ETS show that men score higher than women. Again, if you accept the premise that men and women are equal, the only logical conclusion is that the test is biased to favour men. Incorporating these metrics into the evaluation means the committee will (knowingly or not) favour men.
Why does everyone think it is happening if no one can actually provide evidence for it. That doesn't sound like logical thinking to me and this also leads to observation/confirmation bias. I have had colleagues comment on the gender bias in conference sessions (i.e. "too many women were speaking") but when you actually count, it was representative of the field. When the norm is under-representation, equal-representation appears to be over-representation.
Also, I want to address this point too. I think you are setting up an unrealistic imaginary scenario and then trying to draw conclusions from it to apply to circumstances you appear to be observing in the math departments. For any pool of applications to grad school, it is very unlikely for two applicants to be otherwise identical.
However, there have been lots of actual studies done where reviewers/evaluators get two applications/proposals/resumes/etc. that are indeed identical, except for the name, and the result is that men are picked much more often than women. Here is an example. Same resume, different gender for a STEM job: https://blogs.scientificamerican.com/unofficial-prognosis/study-shows-gender-bias-in-science-is-real-heres-why-it-matters/

Yeah, just putting this out there... We don't censor and ban for pretty much anything. Obvious single post troll accounts being the notable exception, as those fall under "spam". Overt personal attacks and outing people are some of the only things, and those get a warning and are not usually censored. 
Given that one of the biggest complaints about moderation is that we don't censor enough, I find this comment quite surprising.

I like this one. Succinctly sums up what most of the posting here is all about. And warns new people what they're getting into. 

It fits quite well with the coffee theme of the board, and the all hours discussion. A slogan is a motto, not something that's supposed to explain a purpose.
That said, why does it need to clarify the purpose of the site? It's not like there are problems with getting new members, or with people not understanding the purpose of the site. 

It seems like you're starting with an assumption (the slogan isn't cutting it) and then working from there. 
Perhaps you'd like to share why you think the slogan isn't cutting it?

This is one of the main issues with any grant application: Reviewers take very little time reading it, and read a lot. Assume the person reading your application will take 5-10 minutes to read your entire file. Then that they will read a lot more that are really similar. 
You want your points to be as succinct and well outlined as possible. Underline things. Put them in bold. Make sure you talk about your outreach in the exact terms that the Broader Impacts document from NSF uses. 

This is one of the main issues with any grant application: Reviewers take very little time reading it, and read a lot. Assume the person reading your application will take 5-10 minutes to read your entire file. Then that they will read a lot more that are really similar. 
You want your points to be as succinct and well outlined as possible. Underline things. Put them in bold. Make sure you talk about your outreach in the exact terms that the Broader Impacts document from NSF uses. 

You should always have senior mentors, but I generally encourage my students to choose a junior faculty for a committee chair if they have the option. You can have *bad* mentors that are junior or senior, but when you're picking a committee chair or primary advisor, my experience biases towards the better experience with the junior.
Taking this point by point:
No junior faculty member at an R1 has no experience as a mentor. You don't get to that position if you don't.  Generally, getting tenure at most schools will involve successfully graduating students. For senior faculty, they don't really care if you finish or not- it's incredibly important that you finish successfully for a junior faculty member.  It's unlikely that a junior faculty member is going to move, but my statement qualified "new" assistant professors- the first likely move would be at tenure, by which you'll likely be either graduated or OK to stay on your own at the old institution.  New professors don't necessarily have smaller networks- and they have a lot more sway with the networks they do have, imo. Older faculty can, if they socialize well, keep extensive networks- but they can also fall into the rut of just associating with the same old group while new faculty are aggressively and broadly networking. No faculty member, old or new, is going to have a poor grasp of the broader field.  Similarly, you aren't going to get a faculty position at an R1 without a strong track record for (or potential for) publishing. Newer faculty are much more dependent on getting work out, which is why they're good to work with. Senior faculty can pick and choose what they want to work on, and can afford to take years perfecting a single work- new faculty can't.  Anyway, you seem to have an interesting view that in no way matches my experience with reality- you also seem quite arrogant in your assumptions of junior faculty, most of whom are exceptionally successful in their field or they wouldn't be there. There's a reason many senior faculty say they would not be competitive for the positions they're currently hiring among new faculty.
As said, overall fit with the mentor is the most important property, but taking out obvious red flags (interpersonal issues, major funding problems) and aligning research areas (both are fields you want to work with), I think the better bet is usually going to be the junior faculty member. It might be a bit of a higher risk/reward proposition, but down the road being one of the first graduates of a well-known faculty member will continue to serve you very well as you progress through your career, much more so than being one of many graduates they've had over the years. 
And the negatives of that position balance out by finding experienced senior faculty members to act as mentors- they can provide the insight and experience a young committee chair may lack.
Most of us give advice based on our personal experiences- we don't do multiple PhDs to be able to comment on parallel experiences with different areas. But in the programs I've been in, this has held true- as with the career trajectory of myself and my colleagues, so it's advice I continue to give to my students. 
Regardless, as mentioned, fit is the primary factor- choosing a senior person who's a worse fit for your research interests over a junior person who's a better fit (as is the case in the instances discussed in this thread) just because the person is senior isn't a great idea.

I like this one. Succinctly sums up what most of the posting here is all about. And warns new people what they're getting into. 

I like this one. Succinctly sums up what most of the posting here is all about. And warns new people what they're getting into. 

I like this one. Succinctly sums up what most of the posting here is all about. And warns new people what they're getting into. 

I like this one. Succinctly sums up what most of the posting here is all about. And warns new people what they're getting into. 

It fits quite well with the coffee theme of the board, and the all hours discussion. A slogan is a motto, not something that's supposed to explain a purpose.
That said, why does it need to clarify the purpose of the site? It's not like there are problems with getting new members, or with people not understanding the purpose of the site. 

I like this one. Succinctly sums up what most of the posting here is all about. And warns new people what they're getting into.