\section{Introduction}

With their experimental study of temperature and composition dependences 
of transport properties of the disordered crystalline solid 
SrNbO$_{3 - x}$N$_x$ \cite{Oka.etal.2021}, Daichi Oka and co-workers 
aimed to contribute to a better understanding of electron localization. 
Interpreting their measured data according to the currently prevailing 
theories, they came across a gold nugget, but ignored it. That is, they 
obtained some data which clearly contradict these theories. Although 
those data are graphically presented in Ref.\ \onlinecite{Oka.etal.2021}, the 
shown serious inconsistency is not mentioned in its text with any word. 
-- This is all the more astonishing since the authors cite a review and 
a comment, their Refs.\ 30 and 32 (here Refs.\ 
\onlinecite{Moebius.2019,Moebius.1989}), to which discrepancies of this type 
are central. -- Those data are very interesting because they challenge 
the widely presumed continuity of the metal-insulator transition (MIT)
in disordered solids, and thus the applicability of the highly cited 
scaling theory of localization, Ref.\ \onlinecite{Abrahams.etal.1979}. In 
detail:

In identifying the MIT, Daichi Oka and co-workers rely on the 
theoretical ideas of Anderson localization 
\cite{Anderson.1958,Abrahams.etal.1979} and of the interference of 
electron-electron interaction and elastic impurity scattering of 
electrons \cite{Altshuler.Aronov.1979}. In particular, they do so when 
analyzing the temperature dependence of the conductivity, $\sigma(T)$, 
of the sample with $x = 0.96$, referred to as sample 0.96 in the 
following. According to the $T \rightarrow 0$ extrapolation of a 
$\sigma = a + b \, T^{\, 1 / 2}$ fit \cite{Altshuler.Aronov.1979}, this 
sample is classified as metallic in Ref.\ \onlinecite{Oka.etal.2021}; thus the 
authors conclude the MIT to happen within the $x$ interval (0.96,1.02). 
That data evaluation, however, is invalid for several reasons as the 
present reconsideration from three different perspectives shows.


\section{The logarithmic derivative of $\sigma(T)$}

Consider the logarithmic derivative 
$w(T) = \mbox{d} \ln \sigma / \mbox{d} \ln T$. If measured data really 
obey $\sigma = a + b \, T^{\, p}$ with positive $a$, $b$, and $p$, then 
the slope of $w(T)$ is positive -- as claimed in the text of Ref.\ 
\onlinecite{Oka.etal.2021} to be the case for sample 0.96 at low $T$ -- and 
$w(T) \rightarrow 0$ as $T \rightarrow 0$. The validity of this 
statement on $w(T)$ should be verifiable by looking at Fig.\ 1d of Ref.\ 
\onlinecite{Oka.etal.2021}. Unfortunately, however, evaluating the low-$T$ 
part of the $w(T)$ data for sample 0.96 therein is complicated by the 
wide width of the $w$ range presented. Nevertheless, a thorough 
inspection of this diagram, enlarged by means of the PDF viewer, yields 
information on sample 0.96 that differs qualitatively from the 
expectation regarding the ansatz $\sigma = a + b \, T^{\, p}$:
(i) With decreasing $T$, between about 8~K and the lowest $T$ value
considered, roughly 2~K, $w(T)$ no longer decreases as for
$25\ {\rm K} > T \gtrsim 11\ {\rm K}$, but even increases slightly; this 
contradicts the above mentioned claim of Ref.\ \onlinecite{Oka.etal.2021}.
(ii) In the whole $T$ range shown, $w(T) > 0$. 
Taking these two points together, one naively expects that $w(T)$ tends 
to some finite positive value $w_0$ as $T \rightarrow 0$, so that 
$\sigma(T)$ vanishes as $T^{\, w_0}$ in this limit -- or that $w(T)$ 
diverges, so that $\sigma(T)$ vanishes in some exponential manner --. 
Therefore, the electric transport in sample 0.96 is most likely 
activated instead of metallic.

To shed light on the contradiction between this conclusion and the 
interpretation in Ref.\ \onlinecite{Oka.etal.2021}, we digitized the
information on sample 0.96 given in Figs.\ 1b, 1c, and 1d of Ref.\ 
\onlinecite{Oka.etal.2021} by means of WinDIG 2.5. Furthermore, for 
comparison, data for the sample with $x = 1.02$, presented in Figs.\ 1c, 
1d and in Supplementary Fig.\ 5 of Ref.\ \onlinecite{Oka.etal.2021}, were read 
out. The latter sample is referred to as sample 1.02 from now on.

\begin{figure*} [t]
\includegraphics[width=0.90\linewidth]{fig_1_moebius}
\caption{Temperature dependence of the logarithmic derivative 
$w(T) = \mbox{d} \ln \sigma / \mbox{d} \ln T$ of the conductivity 
$\sigma(T)$ for the sample with $x = 0.96$, marked by black 
{\large $+$}. These data points were obtained from Fig.\ 1d of Ref.\ 
\onlinecite{Oka.etal.2021}. In the left panel, they are compared to the 
logarithmic derivative of the fit function shown in Fig.\ 1c of 
Ref.\ \onlinecite{Oka.etal.2021}, given as red line here, and, additionally, 
to the logarithmic derivative of the power law interpolation between 
$\sigma(4.7\ {\rm K})$ and $\sigma(10.7\ {\rm K})$, shown as dashed 
magenta line. The right panel contrasts these data points for the sample 
with $x = 0.96$ with $w(T)$ data points for the sample with $x = 1.02$,  
where different $w$ scales (with equal zero) are used. The latter data 
points, read out from Fig.\ 1d of Ref.\ \onlinecite{Oka.etal.2021} and 
Supplementary Fig.\ 5 of Ref.\ \onlinecite{Oka.etal.2021}, are marked by blue 
{\large $\times$} and orange ${\textsf Y}$, respectively.}
\label{fig1}
\end{figure*}

The left panel of Figure \ref{fig1} of our reconsideration makes the 
above described observations easily comprehensible. Therein, the data 
points of sample 0.96 given in Fig.\ 1d of Ref.\ \onlinecite{Oka.etal.2021} 
are redrawn, where a tenfold zoomed in $w$ scale is used. These data are 
contrasted with the analytically calculated logarithmic derivative of 
the $\sigma = a + b \, T^{\, 1 / 2}$ fit shown in Fig.\ 1c of Ref.\ 
\onlinecite{Oka.etal.2021}; for related details see below. Three 
characteristics of this set of redrawn data points catch the eye:
(1) When $T$ decreases from 25~K to about 10.5~K, $w(T)$ decreases too, 
though more and more slowly.
(2) Between about 10.5~K and about 4.7~K, $w(T)$ exhibits a peculiar 
feature, that is a pronounced hump and a subsequent swale. 
(3) Finally, below about 4.7~K, with further decreasing $T$, $w(T)$ 
slowly rises again; $w(T)$ increases by about 9~\% from 4.7~K to 2.0~K 
-- whereas the logarithmic derivative of the fit function (red curve) 
decreases by 31~\% over this interval --.

It is instructive to compare the behavior of $w(T)$ for sample 0.96 to 
that of $w(T)$ for sample 1.02, nominally SrNbO$_2$N. For the latter 
sample, two data sets are available from Ref.\ \onlinecite{Oka.etal.2021}, 
that is from Fig.\ 1d and from Suppl.\ Fig.\ 5, in the following denoted 
as data sets A and B, respectively. Surprisingly, they differ 
considerably from each other, as is obvious upon first look at the 
respective $T$ regions around 10~K in both diagrams. These $w(T)$ data 
sets are reproduced in the right panel of our Fig.\ \ref{fig1}; they are 
contrasted with the $w(T)$ data for sample 0.96 therein. 

According to this diagram, in case of data set A, $w(T)$ smoothly runs
through a minimum when $T$ decreases from 25~K to 2~K; the minimum is 
passed at about 15~K. In contrast, the graph of set B exhibits a 
pronounced hump and a subsequent swale in the intermediate $T$ region,
that is between 10.5~K and 4.7~K. Moreover, set B is clearly less dense 
than set A. Above 10.5~K and below 4.7~K, however, the graphs of sets A 
and B roughly agree with each other. -- The small shift between these 
graphs at low $T$ would have been further diminished by ascribing the 
difference quotients to the respective interval centers, instead of to 
the lower interval boundaries as done in Ref.\ \onlinecite{Oka.etal.2021}. 
This way, by utilizing the symmetric difference quotient \cite{numdiff}, 
the numerical error would have been considerably reduced; see also 
Appendix C of Ref.\ \onlinecite{Moebius.2019}. --  

To render the comparison of samples 0.96 and 1.02 as informative as 
possible, we use different $w$ scales for the individual samples in the 
right panel of Fig.\ \ref{fig1}. The left scale is valid for sample 
0.96; it agrees with the $w$ scale of the left panel of Fig.\ 
\ref{fig1}. The right scale applying to data for sample 1.02 has the 
same zero as the left one. Its unit is chosen such that, at 
$T = 10.7\ {\rm K}$, the data points for both the samples roughly fall 
together in this diagram.

Two striking similarities between the peculiar features of the $w(T)$ 
for sample 0.96 and for set B of sample 1.02 are clearly recognizable in
the right panel of Fig.\ \ref{fig1}:
(i) Both humps appear in the same $T$ region. 
(ii) Both humps have approximately the same shape and the same relative 
magnitude. 
These two findings, together with the absence of such a peculiar feature 
in the data set A for sample 1.02, suggest the following interpretation. 
Both the peculiar features are very likely artifacts. Presumably, they 
originate from similar errors of the related $T$ values, caused by any 
shortcoming of the measuring procedure. 

Below 25~K, data set A for sample 1.02 contains almost twice as much 
data points as data set B. Thus, it is likely that set A results from a 
remeasurement following an improved procedure. -- Surprisingly, however, 
the data for sample 1.02 published in Fig.\ 1c of Ref.\ 
\onlinecite{Oka.etal.2021} are not consistent with those in Fig.\ 1d of Ref.\ 
\onlinecite{Oka.etal.2021}; the former ones relate to the same $T$ values as 
the ones of data set B. -- Since Daichi Oka et al.\ apparently took 
values with slowly sliding temperature, a change of the cooling / 
heating rate may have been the origin of the peculiar features of the 
$w(T)$ for sample 0.96 and for set B of sample 1.02. 

The positive message from the above comparison is: the peculiar feature, 
that is hump plus swale in the intermediate $T$ range, does not question 
the validity of the $w(T)$ data points for $T \le 4.7\ {\rm K}$. With 
respect to sample 0.96, the trustworthiness of these low-$T$ data points 
is further supported by the logarithmic derivative of the power law 
interpolation between $\sigma(10.7\ {\rm K})$ and 
$\sigma(4.7\ {\rm K})$. This function provides a plausible estimate of 
$w(T)$ for the intermediate $T$ range, $w = 0.090$, see the left panel 
of Fig.\ \ref{fig1}. Therefore, it seems to be highly probable that the 
slow increase of $w(T)$ with decreasing $T$ below about 4.7~K is real, 
that it is not an artifact of temperature or resistivity measurements.

Apart from the mysterious feature hump plus swale, the graphical 
presentations of the $w(T)$ data sets of sample 0.96 and of sample 
1.02 in our Fig.\ \ref{fig1} have qualitatively similar shapes -- though 
they differ in the position of the minimum and in the associated $w$ 
value --. This resemblance provides additional support for the above 
reclassification of sample 0.96. 

In this context, we stress that, apart from hump and swale, the 
appearance of both the $w(T)$ data sets of the samples 0.96 and 1.02 is 
rather typical for disordered solids, even if the minimum value of 
$w(T)$ is as small as roughly 0.1. The reader may compare them with the 
$w(T)$ data sets for compensated crystalline Si:(P,B) in Fig.\ 1 of 
Ref.\ \onlinecite{Hirsch.etal.1989}, an unusually unbiased and constructive 
reply to a comment. Furthermore, he may compare with data for 
crystalline Si:As and CdSe:In in Figs.\ 4 and 13, respectively, of the 
review Ref.\ \onlinecite{Moebius.2019} as well as with the ones for the 
amorphous solids a-Si$_{1 - x}$Cr$_x$ and a-Si$_{1 - x}$Mn$_x$ in Figs.\ 
7 and 8, respectively, of the review Ref.\ \onlinecite{Moebius.Adkins.1999}.
Correspondingly, having the four qualitative scenarios analyzed in 
Section 5 of Ref.\ \onlinecite{Moebius.2019} in mind, under certain 
conditions, one may consider the $w(T)$ data of sample 0.96 as an 
indication of the existence of a finite minimum metallic conductivity.

Nevertheless, alternatively, it might be speculated that, for sample 
0.96, the negative slope of $w(T)$ below about 4.7~K could arise from 
only a part of the sample exhibiting metallic conduction, means from the 
sample being significantly inhomogeneous; compare Ref.\ 
\onlinecite{Hirsch.etal.1989}. 

However, due to a scaling law discovered more than three decades ago, it 
seems not unlikely that the observed negative slope of $w(T)$ for sample 
0.96 is a generic feature: for a-Si$_{1 - x}$Cr$_x$ films deposited by 
means of e-beam evaporation, the $T$ and $x$ dependences of $\sigma$ 
were observed to satisfy $\sigma(T,x) = \sigma_{\rm{sc}}(T / T_0(x))$ 
if 
$\sigma(T,x) < \sigma_0 = 270 \pm 50 \ {\rm \Omega}^{-1} {\rm cm}^{-1}$ 
and $100\ {\rm mK} < T < 50\ {\rm K}$ \cite{Moebius.etal.1985}. In 
words: under these conditions, $\sigma(T,x)$ is completely determined 
by the quotient of $T$ and an empirically introduced characteristic 
temperature $T_0(x)$. Since $\sigma_{\rm{sc}}(T / T_0) \rightarrow 0$ 
as $T / T_0 \rightarrow 0$, this behavior is related to activated 
conduction. It implies the existence of a finite minimum metallic 
conductivity, in contradiction to Ref.\ \onlinecite{Abrahams.etal.1979}. -- 
This scaling behavior, however, breaks down in consequence of annealing, 
that is in consequence of the formation of nanocrystallites, see 
Subsection 3.3 of Ref.\ \onlinecite{Moebius.etal.1985}. Nevertheless, the 
appearance of $\sigma(T,x)$ varies only gradually in this process, not 
abruptly. --  

When we have the possibility of such a scaling in mind, what do we have 
to look out for when studying other solids? For the reconsideration at 
hand particularly important: within the validity range of this scaling 
law, the slope of $w(T)$ is always negative. In various cases, the 
latter feature has motivated the study of $w(T)$ as cross-check of the 
interpretation of $\sigma(T)$ in terms of metallic conduction, see Ref.\ 
\onlinecite{Moebius.2019} and references therein.

For the reconsideration at hand similarly important: the validity range 
of the mentioned scaling law encompasses not only a conductivity region 
in which fast, exponential $T$ dependences are observed, but, 
furthermore, also a conductivity region where $\sigma(T)$ varies slowly, 
in some non-exponential manner with $T$. -- Concerning exponential $T$ 
dependences, such scaling relations were observed to hold in several 
other homogeneous systems too \cite{Moebius.1985}. -- 

The scaling referred to in the previous three paragraphs has a
fundamental consequence: {\sl the closer the composition of the sample 
to that at the MIT, the higher is the lowest experimentally accessible 
temperature}. This only at first glance absurd statement relates to the 
point that it is not the ratio of $T$ to the absolute unit Kelvin that 
is crucial here, but that it is the ratio of $T$ to the characteristic 
hopping temperature, $T_0(x)$: for $T \gg T_0(x)$, $\sigma(T/T_0(x))$ 
varies in some non-exponential manner, even if e.g.\ $T = 1\ {\rm K}$. 
Therefore, since the parameter $T_0(x)$ apparently converges to 0 as 
the MIT is approached from the insulator side, the following 
interpretation difficulty arises in any experiment: there is some finite 
$x$ range prior to the MIT in which only non-exponential $T$ dependences 
of $\sigma$ can be observed; see the schematic diagram Fig.\ 2 of Ref.\ 
\onlinecite{Moebius.Adkins.1999}. Without very thorough data analysis -- as 
the evaluation of $w(T)$ --, such $\sigma(T)$ data may easily be 
misinterpreted to indicate metallic conduction.


\section{Approximation by $\sigma(T) = a + b \, T^{\, 1 / 2}$}

Assume now, a reader is not aware of this principal difficulty and has 
no information on $w(T)$. Then Fig.\ 1c of Ref.\ \onlinecite{Oka.etal.2021}, 
in particular the seemingly rather good quality of the 
$\sigma = a + b \, T^{\, 1 / 2}$ fit to the data of sample 0.96, may 
mislead him: it could make him feel sure about this sample being clearly
metallic. So the following questions arise. Is the quality of the 
approximation shown in Fig.\ 1c of Ref.\ \onlinecite{Oka.etal.2021} really as 
convincing as it seems at first glance? How reliable are the parameter 
estimates of this approximation? Why does it work quite well over a 
rather wide $T$ range? 

\begin{figure*} [t]
\includegraphics[width=0.90\linewidth]{fig_2_moebius}
\caption{Temperature dependence of the conductivity, $\sigma(T)$, for 
the sample with $x = 0.96$. According to Refs.\ 
\onlinecite{Oka.etal.2021,Altshuler.Aronov.1979}, it should be linear in such 
a $\sigma$ versus $T^{\, 1 / 2}$ plot. Data points read out directly 
from Fig.\ 1c of Ref.\ \onlinecite{Oka.etal.2021} are marked by 
{\large $\times$}, whereas the points which were refined additionally 
incorporating Fig.\ 1d of Ref.\ \onlinecite{Oka.etal.2021} -- see text -- 
are given as {\large $+$}. Left panel: Comparison of three fits of 
$\sigma = a + b \, T^{\, 1 / 2}$ to the refined data points focusing
on different temperature ranges: the solid red, dashed green, and dashed 
dotted blue lines relate to the intervals 2 to 14~K, 7 to 19~K, and 
2 to 4~K, respectively. Right panel: presentation of the fit concerning
the $T$ interval 7 to 19~K in an extended $T$ range. For that,
additionally, resistivity data were read out from Fig.\ 1b of Ref.\ 
\onlinecite{Oka.etal.2021} by means of automatic pixel by pixel digitization.} 
\label{fig2}
\end{figure*}

Insight into these interpretation issues is gained by studying under 
which conditions significant deviations of the measured data points of 
sample 0.96 from the adjusted function $\sigma = a + b \, T^{\, 1 / 2}$ 
occur. For such an analysis, one needs as precise as possible 
$(T_i,\sigma_i)$ data. Thus, first, we checked the precision of the data 
read out from Fig.\ 1c of Ref.\ \onlinecite{Oka.etal.2021} by comparing the 
$w_i$ values calculated from the digitized $(T_i,\sigma_i)$ to the $w_i$ 
data read out from Fig.\ 1d of Ref.\ \onlinecite{Oka.etal.2021}. Then, we used 
the digitized $T_i$ and $w_i$ to refine the $\sigma_i$ values. This 
precision improvement bases on the digitized $w_i$ having a smaller 
discretization error than the differences between neighboring $\sigma_i$ 
as read out. Additionally, a tiny constant shift of all $w_i$ was 
allowed to minimize the sum of the squares of the small corrections of 
the $\sigma_i$ values in the refinement; the optimum constant shift 
corresponds to roughly half a pixel in Fig.\ 1d of Ref.\ 
\onlinecite{Oka.etal.2021}.

The results are presented in our Figure \ref{fig2}. Its left panel 
displays $\sigma$ vs.\ $T^{\, 1 / 2}$ with much better $\sigma$ 
resolution than Fig.\ 1c of Ref.\ \onlinecite{Oka.etal.2021}. Two data sets 
are shown therein: the set of $\sigma(T)$ points obtained directly by 
means of digitization of Fig.\ 1c of Ref.\ \onlinecite{Oka.etal.2021}, as well 
as the refined set obtained as described in the previous paragraph. Both 
these data sets deviate only slightly from each other. Completing, the 
right panel of Fig.\ \ref{fig2} presents an overview over a considerably 
wider $T$ range, from 0~K to 49~K, though with somewhat lower $\sigma$ 
resolution. It combines the refined set and data read out from Fig.\ 1b 
of Ref.\ \onlinecite{Oka.etal.2021}.

The left panel of Fig.\ \ref{fig2} compares three fits of the function
$\sigma = a + b \, T^{\, 1 / 2}$ to different subsets of the set of the 
refined digitized data: 
(1) The solid red line corresponds to the approximation shown in Fig.\ 
1c of Ref.\ \onlinecite{Oka.etal.2021}: the parameters $a$ and $b$ are 
adjusted to optimally reproduce the measuring points within the $T$ 
interval 2~K to 14~K. This fit yields 
$\sigma(0\ {\rm K}) = a = 75.7\ {\rm \Omega}^{-1} {\rm cm}^{-1}$, in 
very good agreement with the value read out from Fig.\ 1c of Ref.\ 
\onlinecite{Oka.etal.2021}, $75.8\ {\rm \Omega}^{-1} {\rm cm}^{-1}$. 
\linebreak
(2) The dashed green line refers to a fit to the data points within an 
intermediate $T$ region, that is $7\ \rm{K} < T < 19\ \rm{K}$. 
According to this approximation 
$\sigma(0\ {\rm K}) = 78.6\ {\rm \Omega}^{-1} {\rm cm}^{-1}$. 
(3) The dashed-dotted blue line presents a fit to the data points taken 
at the three lowest $T$ values, $2\ \rm{K} < T < 4\ \rm{K}$; it 
yields $\sigma(0\ {\rm K}) = 70.2\ {\rm \Omega}^{-1} {\rm cm}^{-1}$.

Three conclusions can be drawn from this comparison. First, substantial 
deviations from the $\sigma = a + b \, T^{\, 1 / 2}$ approximation are 
clearly recognizable. Second, these deviations have primarily a 
systematic, non-random character: apparently, the experimental 
$\sigma(T)$ crosses the solid red line only twice. Third, the 
extrapolation to $T = 0 \ \rm{K}$ depends considerably on which $T$ 
interval is taken into account in the fit: the lower the upper boundary 
of this $T$ interval, the smaller is the estimated value of 
$\sigma(0\ {\rm K})$.

In spite of the above demonstrated low-temperature extrapolation 
uncertainty, the fit of $\sigma = a + b \, T^{\, 1 / 2}$ to the measured 
$\sigma(T)$ data given in Fig.\ 1c of Ref.\ \onlinecite{Oka.etal.2021} seems 
to yield a quite reasonable approximation within a rather wide interval. 
The question is why. Inclusion of data of sample 0.96 from Fig.\ 1b of 
Ref.\ \onlinecite{Oka.etal.2021} into the comparison solves this puzzle: the 
right panel  of our Fig.\ \ref{fig2} completes the set of refined data 
of sample 0.96 with data from Fig.\ 1b of Ref.\ \onlinecite{Oka.etal.2021} for 
$15 \ \rm{K} \lesssim T < 49 \ \rm{K}$. Contrasting this composed 
graph with the $\sigma = a + b \, T^{\, 1 / 2}$ fit to all refined data 
points within the range $7\ \rm{K} < T < 19\ \rm{K}$, marked by the 
dashed green line, suggests the following explanation. 

\begin{figure*} [t]
\includegraphics[width=0.90\linewidth]{fig_3_moebius}
\caption{Temperature dependences of the deviations between measured 
data and their $\sigma = a + b \, T^{\, 1 / 2}$ approximation, relating 
to the interval $2\ \rm{K} < T < 14\ \rm{K}$, for the sample with 
$x = 0.96$. The left panel presents the absolute conductivity misfit, 
while the right panel shows the relative temperature misfit, see text.
} 
\label{fig3}
\end{figure*}

Overall, the experimental relation $\sigma(T^{\, 1 / 2})$ shown in 
the right panel of Fig.\ \ref{fig2} has an inverse S shape and, 
correspondingly, an inflection point within the considered $T$ range; 
this characteristic temperature amounts to (very) roughly 10~K. So the 
dashed green line represents a linear fit focusing on the surroundings 
of an inflection point. Therefore, the quadratic term of the Taylor 
expansion at the fit interval center is almost negligible so that the 
region of approximately linear behavior of $\sigma(T^{\, 1 / 2})$ should 
be comparably wide. Thus, simply for numerical reasons, it is not 
surprising that the dashed green line provides a quite reasonable 
description over the rather wide $T$ range from 4~K to 25~K. 

As cross-check of this interpretation, consider the fit focusing on the 
interval 2~K to 4~K -- dashed-dotted blue line in the left panel of 
Fig.\ \ref{fig2} --. It concerns a region considerably away from the 
inflection point. As expected, it seems to have a far smaller 
applicability range; in this case, the judgement bases on the estimation 
of the upper half width of the range.


\section{Sample as secondary thermometer}

Now, we have a closer look at the deviations of the $\sigma(T)$ data 
points of sample 0.96 from their $\sigma = a + b \, T^{\, 1 / 2}$ 
approximation. The left panel of our Figure \ref{fig3} shows the 
differences between the refined $\sigma_i$ data and the corresponding 
values $a + b \, T_i^{\, 1 / 2}$ of the fit for the region 2~K to 
14~K; this approximation focuses on the same $T$ interval as considered
in Ref.\ \onlinecite{Oka.etal.2021}. -- Note: the resolution of the vertical 
axis of the left panel of Fig.\ \ref{fig3} is by a factor 13 higher than 
that of the $\sigma$ axis of the left panel of Fig.\ \ref{fig2}. -- 

Loosely speaking, the left panel of Fig.\ \ref{fig3} presumes the $T$ 
values to be exact and shows how the deviation of the $\sigma$ value 
from its estimate depends on $T$. Complementing this presentation, the 
right panel of Fig.\ \ref{fig3} relates to the opposite perspective: the 
$\sigma$ values are presumed to be exact, and we ask for the deviation 
of the measured $T$ value from the $T$ value estimated on the 
basis of the same $\sigma = a + b \, T^{\, 1 / 2}$ approximation. In 
other words, in the right panel of Fig.\ \ref{fig3}, the sample is 
considered as secondary thermometer: to each value of $\sigma_i$, a 
secondary temperature value, $T^{\rm sec}_i$, is ascribed. The 
corresponding relative misfit, $(T_i - T^{\rm sec}_i) / T_i$, 
is depicted versus the actual temperature, $T_i$, in this diagram.

The left panel of Fig.\ \ref{fig3} confirms the presence of substantial 
systematic, non-random deviations between measured data and fit. 
Furthermore, it provides the following two interesting messages. The 
very systematic $T$ dependence of the deviation between measured data 
and fit present above 10~K testifies the rather high precision of the 
resistivity measurements by Daichi Oka and co-workers. Simultaneously, 
however, this $T$ dependence exhibits a strange feature between about 
5~K and about 10~K, marked by a red ellipse. That presumable 
inconsistency correlates with the peculiar feature hump plus swale 
illuminated in our analysis of the logarithmic derivative of $\sigma(T)$ 
above. These statements are confirmed by the right panel of Fig.\ 
\ref{fig3}.

Finally, we point to two quantitative conclusions which can be drawn 
from the right panel of Fig.\ \ref{fig3}. First, above 3~K, the relative 
misfit, $(T_i - T^{\rm sec}_i) / T_i$, varies between as much as -9~\% 
and +8~\%. Second, below 3~K, where the relative misfit is positive, it 
increases rapidly with decreasing $T$ and reaches a value of 24~\% at 
2~K! Of course, this sharp increase should be checked by additional 
measurements at intermediate $T$ values. Furthermore, an extension of 
the considered $T$ range down to 1.8~K -- presumably in reach of the
equipment used in Ref.\ \onlinecite{Oka.etal.2021} -- should be very helpful
in clarifying the situation.  

Both these findings are further strong arguments against the 
significance of the $T \rightarrow 0$ extrapolation of the 
$a + b \, T^{\, 1 / 2}$ fit to the $\sigma(T)$ data presented in Ref.\ 
\onlinecite{Oka.etal.2021}. 


\section{Conclusions}

We summarize: regarding the sample with $x = 0.96$, Daichi Oka and 
co-workers comprehensively reported on their quite careful, albeit not 
problem-free determinations of $\sigma(T)$ and $w(T)$ in Figs.\ 1b, 1c, 
and 1d of Ref.\ \onlinecite{Oka.etal.2021}. In doing so, unfortunately, they 
used inappropriate $\sigma$ and $w$ scales. Here, zooming into Figs.\ 1c 
and 1d of Ref.\ \onlinecite{Oka.etal.2021}, we have uncovered a fundamental 
contradiction between the authors' theoretical interpretation of these 
$\sigma(T)$ measurements and their $w(T)$ data, obtained in an 
ansatz-free manner: the authors consider this sample as metallic, but 
$w(T)$ does not decrease correspondingly with decreasing $T$ below 
4.7~K. In the present reconsideration of the respective part of Ref.\ 
\onlinecite{Oka.etal.2021}, that problem has been analyzed from three 
different perspectives.

It is astonishing that the contradiction highlighted here was not 
mentioned in the text of Ref.\ \onlinecite{Oka.etal.2021} with any word. It 
was not mentioned though, concerning the interpretation of $w(T)$ data, 
the authors cite Ref.\ \onlinecite{Moebius.2019}; that recent critical review 
discusses observations and consequences of similar behavior of $w(T)$ 
for various other disordered solids in great detail. It seems,
none of the authors of Ref.\ \onlinecite{Oka.etal.2021} noticed the lack of a 
hint to the above illuminated contradiction.

Thus an unusual question suggests itself: to which extent is the 
narrative that the scaling theory of localization 
\cite{Abrahams.etal.1979} qualitatively correctly describes the MIT of 
various three-dimensional disordered systems a result of confirmation 
bias? Serious doubts about this theory have been on the table at least 
since the surprising identification of a metallic phase in 
two-dimensional disordered systems 
\cite{Kravchenko.etal.1994,Kravchenko.Sarachik.2004} already almost 
three decades ago. They arise from the existence of such a phase being 
ruled out by the scaling theory of localization 
\cite{Abrahams.etal.1979}.

Hence, further thorough and unbiased experimental studies of the MIT in 
disordered solids are urgently needed. In this respect, it will 
certainly be highly valuable if Daichi Oka and co-workers repeat the 
$\sigma(T)$ measurement of their sample with $x = 0.96$ with increased 
precision -- provided this sample is sufficiently long-term stable --. 
In doing so, compared to Ref.\ \onlinecite{Oka.etal.2021}, a greater part of 
the data points should be taken close to the lower end of the $T$ region 
considered: constant quotient of subsequent $T$ values is more 
appropriate than constant difference. The precision and significance of 
such measurements will benefit from carefully thermalizing the setup of 
sample and thermometer before each resistivity measurement, instead of 
measuring with slowly sliding temperature. Including into the proposed 
study additional samples with slightly different composition will 
presumably be fruitful. 


\vspace{0.5cm}

\begin{center} {\bf Acknowledgments} \end{center}

Various critical remarks by Manuel Richter were very helpful in 
improving the manuscript.

\begin{center} {\bf Ethics declaration} \end{center}

The author declares no competing interests.

\begin{center} {\bf Data availability} \end{center}

All data obtained in this analysis are available from the author upon 
reasonable request.

\vspace{0.5cm}

