In this manuscript, the authors report on <<Anomalous fading correction in luminescence dating – a mathematical reappraisal>>. In my opinion, this manuscript is very interesting, as it addresses age correction in relation to the anomalous fading effect. The complexity of the expressions is not strictly mathematical but rather based on extended algebra, which makes them both accessible and applicable. I am confident that future readers will enjoy the manuscript and benefit from its content. Therefore, I recommend its publication in its present form.

This project is an important step forward in developing closed form expressions for values and uncertainty propagation for common fading corrections. I imagine this represents a significant amount of work to parse these equations into their present forms. Additionally, I enjoyed the having the historical progression of fading expressions laid out clearly. My comments below are less focused on these expressions, and more related to the editorial tone of the first half of the manuscript, which I found distracting. I also disagree with the authors' assertion that Visocekas meant to describe tunneling as a power-law phenomenon. Overall though, this is manuscript should be of great interest to the luminescence dating community.

Line edits:

l.27-31: The tone is unnecessarily pejorative. There is no need to characterize the broader community as confused, ignorant, and progress-avoidant with respect to fading phenomena. Nor to discredit efforts to provide transparent fading correction packages, such as those in the R 'Luminescence' package, for example, which implement several published correction routines. 

l.38-46: The authors are shadowboxing a bit here. No one in the luminescence dating community is hampered by Wintle failing to develop a mathematical expression for anomalous fading in 1973. And Huntley and Lamothe's statement is true: their g-value correction, as formulated within Huntley and Lamothe (2001), is only appropriate for the linear portion of the dose response curve. In Lamothe et al. (2003), for example, they extend the framework to deal with samples nearing dose saturation, but reiterate that the expressions laid out in the appendix of Huntley and Lamothe (2001), apply only when luminescence intensity is proportional to dose. That is, the linear portion of the dose response curve. At higher doses, intensity is no longer linearly proportional to dose and therefore this correction would be inappropriate. This was echoed in Huntley and Lian (2006): "Huntley and Lamothe (2001) showed how to correct ages for tunneling for samples which are on the low-dose linear region of the dose response, but we do not yet know how to correct ages for older samples which are on the non-linear region of the dose response curve."

"which nevertheless became a legitimate excuse for when corrected ages fall short of their independent constraints." This seems to ignore the subsequent paper, Lamothe et al. (2003), which extended the g-value framework to the laboratory dose-response measurements, their so-called "dose-rate correction equation." On a related note, I am surprised to not see this Lamothe et al. (2003) correction engaged with more thoroughly in this study, as this was one of the approaches advocated by the original others when dealing with high-dose regions. And subsequent comparisons between nearest-neighbor models and g-value models have compared Kars et al. (2008) to Lamothe et al. (2003). For example, in Kars and Wallinga (2009).

l.73-77: The rescaling of kappa is reported in the R Luminescence package documentation for the calc_FadingCorr function, though I agree with the authors that this expression is difficult to find.

l.82-82: What about the semi-analytical solution described in Jain et al. (2012) (doi: 10.1088/0953-8984/24/38/385402)? 

l.85-94: Here also I find the tone here to be extraneous. Impugning past workers ("taken as unquestionable truths" "to the best of its authors' understanding") is neither helpful nor appropriate. 

l.110-117: I disagree with the authors' assertion that the "proliferation of these faux units in the literature ("% [loss] per decade")...leaves little doubt that "logarithmic decay"...was in fact a misnomer." Visocekas (1985)'s Fig. 2 plainly shows decay that is logarithmic, not power-law. 

Maybe one could argue that there is ambiguity in the phrase "% loss per decade." For example, the phrase could be interpreted to mean "% (of original quantity) loss per decade" or "% (of remaining quantity) loss per decade." The authors view the latter concept as more useful, which is fine and the merits of this view should be developed. But based on the description, mathematical expressions and figures of Visocekas (1985), he intended to describe decay as logarithmic and he meant--unambiguously, I think--the former concept. 

More critically, redefining kappa as the slope on a power-law plot, rather than the slope on a logarithmic (% remaining vs. log(t/tc)) plot, and then describing this expression as kappa "sensu Huntley and Lamothe, 2001" has the potential to deepen confusion.

l.127-129: "This 'feature' of the model bothered Visocekas (1985)..." I am not convinced that it did bother him. From his p. 524: "The case of large storage duration t which, from equation (3), would lead to negative values for the TL remaining, is the most easy to deal with...At a sufficient long time the holes [within which all nearby traps have been emptied] will start to intersect. When this occurs the centers are no longer independent and the (1/t) law does not apply. The fading rate approaches zero as the spheres fill the entire space and no trapped charge remains." 

Based on passages like this, I interpret his view to be that logarithmic decay is a physically meaningful description, but only relevant when recombination centers are independent. It is a fair criticism that this leaves the behavior at long storage times undefined.

l.132: "Eq. (5) is the physical model that Visocekas probably stived for" The physical model described in his papers was of logarithmic decay. If the authors prefer another model, they should argue its benefits. The implication that the authors succeeded where Visocekas failed is gratuitous.

l.140: There is a factor of 100 missing from the definition of g-value.

l.183: Please expand on the meaning of 1.8. Ex., from Jain et al. (2012).

l.205-209: "such a definition is broad enough" I do not follow this statement. Surely D_e / Ddot_lab and t_lab = 30 days are very different numbers in this context? Please clarify. 

Eqs 10, 11; 16, 17: These equations are very useful contributions!