In reality TOA DOWN emissivity = 0 because there can not be direct thermalisation of absorbed IR. (joannenova.com.au)
And in that case, the entropy all has to be produced at the point of thermalisation, which means it has to be a lot colder there. (rabett.blogspot.ru)
Thus going back to 1846 and John Tyndall, anyone with statistical thermodynamics» knowledge knows that thermalisation of IR from a higher temperature source can not occur in the gas phase at local thermodynamic equilibrium. (judithcurry.com)