Generally one responds to a working paper with a meta analysis, not the other way around.
I was trying to parse what this meant and it didn't make sense to me for a while. But I get it now. That maths mumbo-jumbo you posted is the "meta analysis" which trumps what I posted the (working paper). A meta-analysis needs to be grounded in the actual data however. So in other words, your saying your hypothetical / theoretical symbol manipulation exercise trumps the actual data we're discussing? That's pretty silly, since it's the actual data that we are disagreeing on.
I posted evidence from papers that shows that if you take the Bureau of Labor's own data and separate women with no kids from women with kids, then the gender gap completely goes away for childless women: the gender gap is too small to measure if you only consider people without children. That's pretty good evidence that almost all the measurable gender gap comes from the interplay of family vs work, rather than a direct penalty for being a woman. And it's that "direct penalty" that the Paycheck Fairness Act is entirely focused on. In other words, there's an existing theory as to the main cause of the gender pay gap, but there's very little actual data to support that theory, and a detailed look at the numbers actually disputes that employer discrimination is even a measurable component of the overall "78 cents in the dollar" gender gap.
Another example is that I said that if you look at hourly wages then the wage gap narrows to 86%. You dismissed that as straight up bullshit but failed to engage with the actual BLS data when I presented it. That's the equivalent of going "i'm right! you're wrong! not listening!" with your fingers in your ears. In fact, it should be plain common sense that the rate will be more equal if you exclude confounding factors (differences in weeks worked per year, differences in hour worked per week).
Well, you turn around and say at one point "your missing out all the hidden discrimination variables". Well, how do you know the hidden variables all fall on the "sexist" side? The non-hidden variables we can look at don't really support such a bias, so the existence of a significant trend in that direction in the hidden variables is entirely hypothetical. Maybe those variables skew the other way. For instance, single women with no kids outearn single men with no kids by a full 8%, so it's entirely possible for some hidden variables we haven't yet looked for to be in favor of women some of the time (in that case, driven by the higher college graduation rates for women).