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Sorry for the slow response. I've been looking into this, and I suspect you are correct. I believe the radiant temperature should be calculated as:

$T_r=\sqrt[4]{\frac{\sum_1^n{A_iF_i'T_i^4}}{\sum_1^n{A_iF_i'}}}$

I am trying to track down the original Carroll document to determine if it was incorrect there.