[yummyfajitas]

But what I meant was that I would expect the numerator to grow faster than the denominator, at least some of the time, as both inequality and regressivity increase.

Not possible. Define P = progressivity index = tax rate for top 10% / tax rate for bottom 90%. Define R = income of the top 10%, B = income of the bottom 90%, and inequality I = R/B. Tax paid by the rich = RP x tax rate for bottom 90%.

% of taxes paid by rich = RP / (RP+B) = IP/(IP+1). (Tax rate for bottom 90% factors out)

(% tax paid by rich) / (% income earned by rich) = [IP/(IP+1)] [(R+B)/R] = P(I+1)/(IP+1)

Simple calculus shows this quantity always increases with increasing P and decreases with increasing I. So a higher ratio implies either a) a more progressive tax system or b) less inequality.