# Extended Comment on Nature 586, 373 (2020) by E. Snider et al.

D. van der Marel  Department of Quantum Matter Physics, University of Geneva,
24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland  J. E. Hirsch  Department of
Physics, University of California, San Diego, La Jolla, CA 92093-0319, USA

###### Abstract

Recently the discovery of room-temperature superconductivity was announced for
a carbonaceous sulfur hydride (CSH) under high pressure Snider  _et al._
(2020). The evidence for superconductivity was based on resistance and
magnetic susceptibility measurements. In the figures showing the
susceptibility it was stated that “the background signal, determined from a
non-superconducting CSH sample at 108 GPa, has been subtracted from the data”.
From a thorough data analysis we show that the data are incompatible with the
notion that the susceptibility data are obtained from the “measured voltage”
using a background correction. On the other hand the data are compatible with
the reverse procedure, namely the “measured voltage” is obtained by adding a
“background signal” containing noise to what was reported as the background-
corrected susceptibility. For all 6 of the reported pressures our analysis
leads to the conclusion that: (i) the reported background-corrected
susceptibility data are pathological, (ii) they were not obtained by the
method described in this paper nor by any one of the alternative 3 methods
that were subsequently provided by the authors and (iii) the “measured
voltage” data are not raw data.

##  I Introduction

In Ref. Snider  _et al._ , 2020 it is reported that a material termed
carbonaceous sulfur hydride (hereafter called CSH) is a room temperature
superconductor. Data for resistance versus temperature and ac susceptibility
versus temperature at six different pressures show drops suggesting
superconducting transitions. Recently two of the authors of Ref. Snider  _et
al._ , 2020 have posted the numerical values of the data for the ac
susceptibility curves presented in Ref. Snider  _et al._ , 2020, as well as
the underlying raw data, on arXiv Dias and Salamat (Dec. 28,2021). The raw
data and the data presented in Ref. Snider  _et al._ , 2020 are called
“measured voltage” and “superconducting signal” respectively in Ref. Dias and
Salamat, Dec. 28,2021, for which we will use the notation
$\chi^{\prime}_{mv}(T)$ and $\chi^{\prime}_{sc}(T)$ . In Fig. 1
$\chi^{\prime}_{mv}(T)$ and $\chi^{\prime}_{sc}(T)$ are displayed for all 6
pressures. In general $\chi^{\prime}_{mv}(T)$ exhibits a prominent step-like
feature that is attributed in Ref. Snider  _et al._ , 2020; Dias and Salamat,
Dec. 28,2021 to the superconducting phase transition.

Initial analysis of these data was reported in Refs. Hirsch, 2022 and Hirsch
and van der Marel, 2022a, where we pointed out that the fact that the data
show much less noise than the raw data and the background signal is
incompatible with the statement in Ref. Snider  _et al._ , 2020 that the
background signal was obtained in an independent measurement. Furthermore, we
called attention to the fact that the data for pressure 160 GPa show highly
anomalous regularities, and showed that they result from the superposition of
a quantized component and a smooth curve Hirsch and van der Marel (2022a).

In this paper we give an analysis of the ac susceptibility data for all
pressures. We begin with $\chi^{\prime}_{sc}(T)$ and $\chi^{\prime}_{mv}(T)$
at $p=160$ GPa. Next the $\chi^{\prime}_{sc}(T)$ and $\chi^{\prime}_{mv}(T)$
for all other pressures are analyzed. Our analysis leads to the conclusion
that the statements in Refs Snider  _et al._ , 2020, Dias and Salamat, Dec.
28,2021 and Dias and Salamat, 2022 that the reported “Measured voltage” was a
measured voltage is not compatible with the facts.

Figure 1: “Measured voltage” (green) and “superconducting signal” (dark blue)
reproduced from the tables in Refs. Dias and Salamat (Dec. 28,2021); Hirsch
and van der Marel (2022b) for the six different pressures reported in Ref.
Snider  _et al._ , 2020. Figure 2: a and b. Enlarged scale of the
susceptibility data (“superconducting signal”) for CSH at pressure 160 GPa,
from the numerical data of Table 5 of Ref. Dias and Salamat, Dec. 28,2021. c
and d, The data of panels a and b after unwrapping with integer multiples of
$\Delta_{q}=0.16555$ nV. The different colors refer to disconnected segments
of panels a and b. e. The difference between neighboring points of panel a and
b divided by $\Delta_{q}=0.16555$ nV. f. Same as panel e but now using the
unwrapped data of panels c and d. The Excel file with the quantities shown in
this figure can be downloaded from Ref. Hirsch and van der Marel, 2022b.

Figure 3: Top panels. Left from top to bottom: Unwrapped component of the
susceptibility and it’s first, second, and third derivatives. Right (middle)
from top to bottom: 15-node cubic spline with natural boundary conditions
generated with an older (newer) version of commercial plotting software and
it’s first, second, and third derivatives. The nodes are indicated as red
crosses in the top panels. Bottom panel. Enlarged view of $d^{2}P/dT^{2}$,
showing the fourteen linear segments for the second derivative. The inset
shows the center part enlarged further. The Excel file with the quantities
shown in this figure can be downloaded from Ref. Hirsch and van der Marel,
2022b.  Figure 4: a and b, Quantized component of susceptibility data
(“superconducting signal”) for CSH at pressure 160 GPa. c, The difference
between neighboring points of panel a,b divided by 0.16555 nV. Figure 5: a,
“measured voltage” reported in Ref. Dias and Salamat, Dec. 28,2021 for 160 GPa
(black points), compared with quantized component $q(T)$ of
$\chi^{\prime}_{sc}$ (red points). b, Background signal inferred from
$\chi^{\prime}_{bg}=\chi^{\prime}_{mv}-\chi^{\prime}_{sc}$ in Ref. Dias and
Salamat, Dec. 28,2021 (black points), compared with background signal $-P(T)$
inferred from unwrapping of the susceptibility data (red points). $-P(T)$ and
$q(T)$ have been shifted vertically for ease of comparison.

##  II Analysis of the 160 GPa data

Fig. 2a,b shows the susceptibility corresponding to pressure 160 GPa shown in
Extended Data Figure 7d of Ref. Snider  _et al._ , 2020. The numerical values
are given in the second column of Table 5 of Ref. Dias and Salamat, Dec.
28,2021 (labeled “superconducting signal”). A superconducting transition
appears to take place around $T=170~{}K$. Because of the steep rise at 170 K
the regions above and below 170 K need to be displayed in separate panels. A
similar zoom of the 160 GPa curve was previously shown in Fig. 9 of Ref.
Hirsch, 2022. One of the striking features is a series of discontinuous steps.
These steps are directly visible to the eye in the temperature ranges where
$\chi^{\prime}_{sc}(T)$ has a weak temperature dependence. However, they are
also present in the range where $\chi^{\prime}_{sc}(T)$ rises steeply as a
function of temperature, as can be seen by calculating the difference between
neighboring points

$\Delta\chi^{\prime}_{sc}(j)=\chi^{\prime}_{sc}(T_{j})-\chi^{\prime}_{sc}(T_{j-1}).$
(1)

This quantity, shown in Fig. 2e, exhibits an intriguing “aliasing” effect in
the “shadow curves” displaced vertically by integer multiples of $0.1655$
(below this will be refined to $0.16555\pm 0.00005$ by examining the noise of
the 3d derivative). To make this crisp, the vertical axis of Fig. 2e
corresponds to $\Delta\chi^{\prime}_{sc}(j)/0.16555$. Clearly this is a set of
curves vertically offset by an integer $n=-1,0,1,2,3$ and $4$. The most
systematic offsets in sign and size occur between 169.6 K and 170.1 K.

By shifting continuous segments of the curves by an amount $0.16555n$, with
$n$ integers that can be read off from Fig. 2e, it is a simple and
straightforward task to apply a commonly used procedure for removing
mod-$2\pi$ discontinuities of the phase of a periodic signal, termed
“unwrapping” of the vertical offsets. From here on we will reserve the
notation $P(T)$ for the resulting “unwrapped curve”. The result for the two
separate ranges above and below 170 K is displayed in Fig. 2c,d. Comparing
panels c,d to a,b it is possible to verify that the resulting curves are
extremely smooth and completely free of discontinuities. The steep rise at 170
K is absent from panels c,d. As a consistency check $\Delta P(j)$ of panels
c,d was finally calculated and is shown in panel f. Comparing the result shown
in panel f with that in panel e (shown with the same vertical scale to
facilitate comparison) it can be seen that there are no shadow curves in panel
f, demonstrating that not only the temperature dependence of panels c,d is
smooth, the differential shown in panel f is, surprisingly for an experimental
quantity, also completely smooth. The remarkable smoothness of $P(T)$ is
further illustrated by Fig. 3 where this curve is displayed together with the
first, second, and third temperature derivate. The noise level is the lowest
for the offset in the range $0.16555\pm 0.00005$ nV.

In the second derivative graph, shown in more detail in the lower panel of
Fig. 3, all segments are straight lines. This clearly demonstrates that $P(T)$
is a chain of 14 polynomials of order 3:

$\chi(T)=a_{n}+b_{n}(T-T_{n})+c_{n}(T_{-}T_{n})^{2}+d_{n}(T-T_{n})^{3}.$ (2)

The corresponding coefficients for each segment are provided in table 1. The
curve, and its 1st, 2nd and 3rd derivatives fits the profile of a cubic spline
Denk (2017) with 15 nodes. The nodes are shown as crosses in the top panels of
Fig. 3. The second derivatives extrapolate to zero at the extremal nodes,
which corresponds to the so-called “natural” boundary conditions of the cubic
spline Denk (2017) 111A cubic spline smoothly connects $n$ nodes with
coordinates $(x_{j},y_{j})$ ($1\leq j\leq n$) by applying the condition of
continuity of first and second derivatives at the $n-2$ internal nodes. Two
additional boundary condition are needed to fully define the spline function.
Ref. Denk, 2017 specifies the following types of additional boundary
conditions: (i) Natural ($d^{2}y/dx^{2}=0$ at the two ends), (ii) Not-a-knot
($d^{3}y/dx^{3}$ are continuous for $j=2$ and $j=N-1$) (iii) Periodic ($dy/dx$
and $d^{2}y/dx^{2}$ are equal for $j=1$ and $j=N$) (iv) Quadratic (the first
and the last segment are quadratic).. This is the standard spline option of
commercial plotting software. Applying the cubic spline with natural boundary
conditions of different versions of commercial plotting software to the 15
nodes, and exporting on the temperature range defined by the two extremal
nodes, gives the result in the middle and righthand panels of Fig. 3. The
smaller noise of the middle column as compared to the right hand column
(especially the third derivative) signals an improvement over time of the
spline accuracy in subsequent versions of this software.

The behavior of the data shown in Fig. 2c,d, together with the fact that the
segments can be joined by vertical shifts that are all of the same form
$0.16555n$, indicates that the disconnected segments are portions of a
continuous curve that has been broken up by quantized steps. The sequence of
steps form together a quantized component $q(T)$ which is entirely responsible
for the steep rise of $\chi^{\prime}_{sc}(T)$ at 170 K in Fig. 2a,b. The
“superconducting signal” $\chi^{\prime}_{sc}(T)$ shown in Fig. 2a,b can be
expressed as:

$\chi^{\prime}_{sc}(T)=q(T)+P(T)$ (3)

where the unwrapped curve $P(T)$ is given in Fig. 2c,d. Fig. 4 shows the same
information as Fig. 2 for the quantized component $q(T)$. The connected
segments in Fig. 2a,b are now horizontal, and the $y$-values in panel c are
integers.

According to Ref. Snider  _et al._ , 2020, a background signal
$\chi^{\prime}_{bg}(T)$ measured at 108 GPa was subtracted from
$\chi^{\prime}_{mv}(T)$ in obtaining $\chi^{\prime}_{sc}(T)$ that was
published in Ref. Snider  _et al._ , 2020. In other words,

$\chi^{\prime}_{sc}(T)=\chi^{\prime}_{mv}(T)-\chi^{\prime}_{bg}(T)$ (4)

Comparison of Eq. (4) and Eq. (3) would suggest that $\chi^{\prime}_{mv}(T)$
corresponds to $q(T)$ and $\chi^{\prime}_{bg}(T)$ to $-P(T)$ respectively.
Assuming that’s the case, we have to understand (a) why the measured voltage
is a series of flat steps separated by jumps of a fixed magnitude
$0.16555\>nV$, and (b) why the background signal is a smooth curve without
experimental noise.

(a) A digital lock-in amplifier will yield discrete values for the measured
voltages, where the size of the step between neighboring values of measured
voltages is given by the instrumental resolution. If $q(T)$ is the raw data
(“measured voltage”), this would indicate that the resolution of the
instrument in this measurement was of order 0.2 nV. Such a low resolution
could result from setting the digitizer range of the lock-in amplifier to a
large value, approximately $100\mu$ V Ramshaw (2022).

(b) The smooth behavior of the background signal $-P(T)$ ($-1\times$ panels
c,d of Fig. 2) could be explained if, rather than measured values of the
background signal, a polynomial fit to the measured values was subtracted from
the raw data (“measured voltage”). However, Ref. Snider  _et al._ , 2020 does
not mention such a procedure.

We have thus arrived at a possible way to understand the very unusual nature
of the susceptibility data for 160 GPa reported in Ref. Snider  _et al._ ,
2020, namely the conjecture that measured raw data correspond to the quantized
component of the superconducting signal shown in panels a,b of Fig. 4, and the
background signal to the negative of the unwrapped curve displayed in Fig.
2c,d.

On the other hand, the superconducting signal as well as the measured raw data
were reported in Ref. Dias and Salamat, Dec. 28,2021 Table 5, and we can infer
from them the background signal simply by subtraction. Therefore we compare,
in Figs. 5a and b, the reported raw data and the background signal inferred
from the reported raw data (“measured voltage”) and the reported data
(“superconducting signal”)Dias and Salamat (Dec. 28,2021) with our
hypothesized raw data and background signal $q(T)$ and $-P(T)$. It can be seen
in Fig. 5 that there is a complete disconnect between the raw data (“measured
voltage”) and the background signal inferred from the numbers reported in Ref.
Dias and Salamat, Dec. 28,2021, and the raw data and background signal
inferred from the analysis of the superconducting signal Snider  _et al._
(2020) (numerical values given in Ref. Dias and Salamat, Dec. 28,2021). In
particular, there is certainly no way that a polynomial fit of the black
points in Fig. 5b would have any resemblance to the red curve shown in Fig.
5b, and there is a significant difference between the black and red curves in
Fig. 5a.

$n$ | $T_{n}$ | $T_{n+1}$ | $a_{n}$ | $b_{n}$ | $c_{n}$ | $d_{n}$
---|---|---|---|---|---|---
| K | K | nV | nVK-1 | nVK-2 | nVK-3
$1$ | $173.0403$ | $172.9601$ | $-0.5341667$ | $-0.831967885$ | $-1.49856942$ | $+91.64778904$
$2$ | $172.9601$ | $172.5577$ | $-0.5232667$ | $+1.109787341$ | $-23.37545976$ | $-36.48274797$
$3$ | $172.5577$ | $171.9764$ | $-2.3770167$ | $+2.067066472$ | $+22.01346341$ | $+19.04094347$
$4$ | $171.9764$ | $171.6215$ | $+0.1218833$ | $-4.22534782$ | $-10.91711562$ | $-15.11550441$
$5$ | $171.6215$ | $171.3609$ | $+0.9189167$ | $-2.05715895$ | $+7.615487151$ | $+32.88542602$
$6$ | $171.3609$ | $171.2379$ | $+1.3920167$ | $+0.525694384$ | $-26.80460669$ | $-144.7971468$
$7$ | $171.2379$ | $170.9758$ | $+1.1875666$ | $+0.594273457$ | $+24.45470318$ | $+45.53297675$
$8$ | $170.9758$ | $170.9066$ | $+1.8941667$ | $-2.851417507$ | $-14.19295181$ | $-68.69964351$
$9$ | $170.9066$ | $170.2829$ | $+2.0458833$ | $-1.872857591$ | $-2.192399589$ | $-1.056277196$
$10$ | $170.2829$ | $168.7992$ | $+2.6164$ | $-0.355780736$ | $-0.099419915$ | $-0.122458732$
$11$ | $168.7992$ | $168.0085$ | $+3.32481$ | $-0.856801015$ | $+0.365108139$ | $+0.853754437$
$12$ | $168.0085$ | $167.6593$ | $+3.808773$ | $+0.16991766$ | $-1.59661518$ | $+1.657661377$
$13$ | $167.6593$ | $167.3225$ | $+3.484567$ | $+1.86328885$ | $-3.976717692$ | $-10.72445083$
$14$ | $167.3225$ | $166.9137$ | $+2.814148$ | $+0.911730129$ | $+6.406661165$ | $+5.764676053$
Table 1:  Coefficients of Eq. 2 for the unwrapped component of the
susceptibility, $P(T)$, at 160 GPa. The ordering of the temperatures in Table
5 of Ref. Dias and Salamat, Dec. 28,2021 (from high to low) has been
maintained. The first column gives the segment number, second and third
columns give the upper and lower temperature of the two nodes defining the
segment, the fourth column gives $a_{n}=P(T_{n})$. The coefficients in columns
5, 6 and 7 were calculated from the temperature derivatives shown in Fig. 3
using, in this order, $d_{n}=\langle d^{3}P/dT^{3}\rangle_{n}^{n+1}/6$,
$c_{n}=\langle d^{2}P/dT^{2}-6d_{n}(T-T_{n})\rangle_{n}^{n+1}/2$,
$b_{n}=\langle dP/dT-2c_{n}(T-T_{n})-3d_{n}(T-T_{n})^{2}\rangle_{n}^{n+1}$,
where $\langle..\rangle_{n}^{n+1}$ is the average in the temperature interval
$\\{T_{n},...,T_{n+1}\\}$. The quantities inside the brackets are, by
construction and apart from noise, constant within each interval. The Excel
file with this calculation can be downloaded from Ref. Hirsch and van der
Marel, 2022b.

Although we will demonstrate below using correlation maps that the raw data in
Ref. Dias and Salamat, Dec. 28,2021 for 138 and 160GPa contain a quantized
component, and using correlation functions that this is in fact the case for
all reported pressures, this component is superimposed on a noisy background.
Consequently the raw data are not themselves integer multiples of a fixed
step-size. The reported values of the raw data are given in Table 5 of Ref.
Dias and Salamat, Dec. 28,2021 with 11 significant digits. This is not
necessarily the experimental resolution. The experimental resolution is set by
the complete analogue and digital chain of which the analogue-to-digital
converter is the last element. The smallest step between neighboring
temperatures in Table 5 of Ref. Dias and Salamat, Dec. 28,2021 is of order
0.0001 nV. Hence the resolution of the experimental setup is 0.0001 nV or a
smaller number. This is about three orders of magnitude higher resolution than
the resolution of the measuring device that would yield the quantized
component (red curve in Fig. 5a) as measured raw data.

It can also be seen in Fig. 5 that there is much larger noise in the raw data
and background signal reported in Ref. Dias and Salamat, Dec. 28,2021 than
there is in the red curves that were deduced from the reported superconducting
signal in section II. The fact that the reported superconducting signal is
significantly less noisy than the reported raw data was already noted in Ref.
Hirsch, 2022, not only for pressure 160 GPa but for all other pressures as
well. This is independent of the unwrapping analysis discussed above.

##  III Protocols 1 and 2

Ref. Snider  _et al._ , 2020 described the following protocol (hereafter
Protocol 1) for background correction: “The background signal, determined from
a non-superconducting C-S-H sample at 108 GPa, has been subtracted from the
data”. Later in Ref. Dias and Salamat, 2022 the authors stated “We note here
that we did not use the measured voltage values of 108 GPa as the background”
and provided a different protocol (hereafter Protocol 2): “We use the
temperature dependence of the measured voltage above and below the $T_{c}$ of
each pressure measurement and scale to determine a user defined background.”
The method illustrated in Fig. 2a of Ref. Dias and Salamat, 2022 does not lead
to the “superconducting signal” reported for 160 GPa, i.e. the superposition
of a 15-node cubic spline and a quantized component. To introduce those two
features in the “superconducting signal” requires that they are already
present in the raw data, in the “user defined background”, or in both.

Figure 6: Simulation of Eq. 5. “Superconducting signal” ($\chi^{\prime}_{sc}$,
dark blue symbols, obtained from the numerical values reported in Table 5 of
Ref. Dias and Salamat, Dec. 28,2021), “background signal” (dark yellow open
symbols, simulated data) and “Measured voltage”
($\chi^{\prime}_{mv}=\chi^{\prime}_{sc}+\chi^{\prime}_{mv}$, green symbols,
shifted vertically to avoid clutter). To facilitate comparison vertical dotted
lines have been drawn for each of the steps of $\chi^{\prime}_{sc}$.

Figure 7: “Measured voltage” ($\chi^{\prime}_{mv}$, green symbols),
“superconducting signal” ($\chi^{\prime}_{sc}$, dark blue symbols) obtained
from the numerical values reported in Table 5 of Ref. Dias and Salamat, Dec.
28,2021, and “background signal”
($\chi^{\prime}_{bg}=\chi^{\prime}_{mv}-\chi^{\prime}_{sc}$, dark yellow open
symbols, shifted vertically to avoid clutter). To facilitate comparison
vertical dotted lines have been drawn for each of the steps of
$\chi^{\prime}_{sc}$.

Figure 8: Superconducting signal $\chi^{\prime}_{sc}(j)$ for 138 GPa (top) and
160 GPa (bottom), the first discrete differential
$\Delta\chi^{\prime}_{sc}(j)=\chi^{\prime}_{sc}(j+1)-\chi^{\prime}_{sc}(j)$
and second discrete differential
$\Delta^{2}\chi^{\prime}_{sc}(j)=\Delta\chi^{\prime}_{sc}(j+1)-\Delta\chi^{\prime}_{sc}(j)$.

Figure 9: Superconducting signal $\chi^{\prime}_{sc}(j)$ for 166 GPa (top) and
178 GPa (bottom), the first discrete differential
$\Delta\chi^{\prime}_{sc}(j)=\chi^{\prime}_{sc}(j+1)-\chi^{\prime}_{sc}(j)$
and second discrete differential
$\Delta^{2}\chi^{\prime}_{sc}(j)=\Delta\chi^{\prime}_{sc}(j+1)-\Delta\chi^{\prime}_{sc}(j)$.

Figure 10: Superconducting signal $\chi^{\prime}_{sc}(j)$ for 182 GPa (top)
and 189 GPa (bottom), the first discrete differential
$\Delta\chi^{\prime}_{sc}(j)=\chi^{\prime}_{sc}(j+1)-\chi^{\prime}_{sc}(j)$
and second discrete differential
$\Delta^{2}\chi^{\prime}_{sc}(j)=\Delta\chi^{\prime}_{sc}(j+1)-\Delta\chi^{\prime}_{sc}(j)$.
Figure 11: Histograms of $\Delta^{2}\chi^{\prime}_{sc}$ of the
“superconducting signal” at all pressures. This representation of the data was
introduced by Dukwon Dukwon (2022). Figure 12: Histograms of
$\Delta^{2}\chi^{\prime}_{sc}$ and $\Delta^{2}\chi^{\prime}_{mv}$ of the
“measured voltage” at 160 GPa.  Figure 13: Histograms of
$\Delta^{2}\chi^{\prime}_{sc}$ and $\Delta^{2}\chi^{\prime}_{mv}$ of the
“measured voltage” at 138 GPa.  Figure 14: Histogram of
$\Delta^{2}\chi^{\prime}_{mv}$, corresponding to Fig.7 Figure 15: Left:
$\Delta^{2}\chi^{\prime}_{sc}$ versus $\Delta^{2}\chi^{\prime}_{mv}$ at 138
GPa. Right: Histograms of $\Delta^{2}\chi^{\prime}_{mv}$ in the slots
corresponding to the horizontal stripes. Figure 16: Left:
$\Delta^{2}\chi^{\prime}_{sc}$ versus $\Delta^{2}\chi^{\prime}_{mv}$ at 160
GPa. Right: Histograms of $\Delta^{2}\chi^{\prime}_{mv}$ in the slots
corresponding to the horizontal stripes.

##  IV Protocol 3

The inconsistencies pointed out in section III do not occur if, instead of
assuming that $\chi^{\prime}_{mv}(T)$ and $\chi^{\prime}_{bg}(T)$ are
independent signals, we assume that $\chi^{\prime}_{sc}(T)$ and
$\chi^{\prime}_{bg}(T)$ are independent and rearrange Eq. 4 as follows 222This
equation was previously inferred in Ref. Hirsch, 2021 from the fact that the
noise in the data is significantly smaller than in the raw data:

$\chi^{\prime}_{mv}(T)=\chi^{\prime}_{sc}(T)+\chi^{\prime}_{bg}(T)$ (5)

This also removes the incongruences of the noise: The noise of
$\chi^{\prime}_{mv}(T)$ is now the sum of the noises of
$\chi^{\prime}_{sc}(T)$ and $\chi^{\prime}_{bg}(T)$ and therefore bigger than
the noise of each of the latter two. A simulation is shown in Fig. 7, where
for the background an arbitrary curve was taken having a noise structure
similar to that shown in Fig. 2a of Ref. Dias and Salamat, 2022. In Fig. 7 we
show the “measured voltage” data points for 160 GPa corresponding to connected
segments of the “superconducting signal” points. It is apparent that several
of the jumps in the data points are reflected in the raw data. Globally the
features of $\chi^{\prime}_{mv}(T)$ are similar to those in Fig. 7. Despite
differences in the noise structure, points in common of Fig. 7 and Fig. 7 are
that (i) the $\chi^{\prime}_{mv}(T)$ is noisier than $\chi^{\prime}_{sc}(T)$
and $\chi^{\prime}_{bg}(T)$ and (ii) several (but not all) of the steps
present in the $\chi^{\prime}_{sc}(T)$ also show up in
$\chi^{\prime}_{mv}(T)$. In section VI we return to this point using a
correlation method, and find that the quantized steps of $\chi^{\prime}_{sc}$
and those of $\chi^{\prime}_{mv}$ are correlated in statistically significant
manner. This indicates that whatever the explanation is for the existence of
these jumps in the data points has to also account for their existence in the
raw data.

##  V “superconducting signal” at all pressures

The data at 160 GPa stood out because of the striking sawtooth shape of the
“superconducting signal”, which is in fact the superposition of a smooth curve
composed of third order polynomials (15-node cubic spline with natural
boundary conditions) and a quantized component in increments of 0.16555 nV.
For the other pressures the “superconducting signal” does not have the
conspicuous sawtooth appearance. However, just like for 160 GPa the reported
data Dias and Salamat (Dec. 28,2021) show much larger noise in the
$\chi^{\prime}_{mv}(T)$ and $\chi^{\prime}_{bg}(T)$ than in
$\chi^{\prime}_{sc}(T)$ obtained by subtraction Hirsch (2021, 2022). From
Figs. 8, 9, 10 showing the quantities obtained from Snider  _et al._ (2020);
Dias and Salamat (Dec. 28,2021); Hirsch and van der Marel (2022b)

$\displaystyle\Delta\chi^{\prime}_{sc}(j)$ $\displaystyle=$
$\displaystyle\chi^{\prime}_{sc}(j)-\chi^{\prime}_{sc}(j-1)$
$\displaystyle\Delta^{2}\chi^{\prime}_{sc}(j)$ $\displaystyle=$
$\displaystyle\Delta\chi^{\prime}_{sc}(j)-\Delta\chi^{\prime}_{sc}(j-1)$ (6)

and Fig. 11 showing the distribution of $\Delta^{2}\chi^{\prime}_{sc}$ in the
form of histograms (this method was suggested by Dukwon Dukwon (2022)), we see
that for all pressures there are underlying steps present in the temperature
traces. The size of the steps differs from one pressure to another. The
quantization steps (Figs. 8, 9, 10 and 11,) are the smallest for 189 GPa. The
corresponding quantization values are summarized in Table 2

Pressure |  $q(T)$ in ScSig ? | $\Delta_{q}$ | $\delta\chi^{\prime}_{mv}$ | $\delta\chi^{\prime}_{mv}/\Delta_{q}$ |  $q(T)$ in MV ?
---|---|---|---|---|---
GPa |  | nV | nV |  |
138 | yes | 0.025 | 0.03 | 1.2 | yes
160 | yes | 0.17 | 0.14 | 0.8 | yes
166 | yes | 0.015 | 0.2 | 13 | yes
178 | yes | 0.007 | 0.1 | 14 | yes
182 | yes | 0.006 | 0.08 | 13 | yes
189 | yes | 0.003 | 0.12 | 40 | yes/no
Table 2: Properties of the “superconducting signal” and the “measured voltage”
at all pressures. The second (last) column is the result of the present
section (section VI.2). In the fourth column $\delta\chi^{\prime}_{mv}$ is the
variance of the distribution of $\Delta^{2}\chi^{\prime}_{mv}$.

This suggests that for all pressures the “superconducting signal” is the sum
of a quantized component $q(T)$ and a smooth function $P(T)$. In all cases the
distribution of $\Delta^{2}\chi^{\prime}_{sc}$ has some broadening, and this
effect is most important when the step size is smallest. That there is no -or
neglible- random noise in the superconducting signals for all pressures is
born out by Figs. 8, 9, 10, where numerous smooth sections show up prominently
in $\Delta\chi^{\prime}_{sc}$ and $\Delta^{2}\chi^{\prime}_{sc}$. It is
further illustrated by the simulation of Fig. 23 in Appendix B, where the
“superconducting signal” has steps $\Delta_{q}=$0.03 nV, and the effect of
random noise $\in\\{-0.03;0.03\\}$ nV is to completely obliterate the aliasing
structure in $\Delta\chi^{\prime}_{sc}$ and $\Delta^{2}\chi^{\prime}_{sc}$.

In all cases the aliasing effect is most clearly seen in
$\Delta^{2}\chi^{\prime}_{sc}$, demonstrating that for all pressures
$\chi^{\prime}_{sc}(T)$ is a superposition of a quantized component $q(T)$ and
a smooth function $P(T)$, i.e. random noise -if present- has smaller amplitude
than the step-size.

Summarizing this section (i) for all pressures the “superconducting signal” is
the superposition of a quantized component and a smooth component, (ii) for
the 160 GPa data the smooth component is a 15 point cubic spline with natural
boundary conditions.

##  VI measured voltage at all pressures

###  VI.1 Correlation maps

The temperature traces of the “measured voltage” at 160 GPa shown in Fig. 7
and the corresponding histogram of $\Delta^{2}\chi^{\prime}_{mv}$ displayed in
Fig. 12, show clear signatures of the quantized steps in the “measured
voltage” reported in Snider  _et al._ (2020); Dias and Salamat (Dec. 28,2021)
at $\pm 0.2$ nV. We have verified that all steps contributing to the maxima at
$\pm 0.2$ nV coincide with steps of the “superconducting signal” 333The fact
that the measured voltage peaks at $\pm 0.2$ nV instead of $\pm 0.165$ nV of
the superconducting signal is a statistical fluctuation. To verify this we
computed $\sqrt{\left<(\Delta^{2}\chi^{\prime}_{mv})^{2}\right>_{{i.s.}}}$ for
the set of temperatures corresponding to isolated steps in
$\chi^{\prime}_{sc}(T)$. The total number of isolated steps is 57. The result
is $\sqrt{\left<(\Delta^{2}\chi^{\prime}_{mv})^{2}\right>_{{i.s.}}}=0.185\pm
0.075~{}nV$. We see that the maximum at $\pm 0.2$ nV falls comfortably within
this range of values.. For 138 GPa the histogram of
$\Delta^{2}\chi^{\prime}_{mv}$ shown in Fig. 13, does not provide a clear
indication of quantization, which is expected given the smaller step-size.
Below we will show using correlation maps that in fact the “measured voltage”
at 138 GPa has clear signatures of the same quantized steps as the
“superconducting signal” at that pressure, and using correlation functions
that this is the case for all 6 reported pressures. Despite their smaller
size, these steps are nonetheless manifested by the aliasing effect in the
temperature traces and histograms of $\Delta^{2}\chi^{\prime}_{sc}$. In the
“measured voltage” $\chi^{\prime}_{mv}$ the steps are obscured by the larger
amplitude of the noise present in the background signal $\chi^{\prime}_{bg}$
that, as we will argue below, has been added to $\chi^{\prime}_{sc}$. The
exception is 160 GPa, where the steps do show up in the “measured voltage” due
to the fact that their size (0.17 nV) is much larger than for the other
pressures ($<0.025$ nV).

A simulation is shown in Fig. 7, where for the background signal an arbitrary
curve was taken having a noise structure similar to that shown in Fig. 2a of
Ref. Dias and Salamat, 2022. (i) The “measured voltage” is noisier than
“superconducting signal” and “background signal” and (ii) several (but not
all) of the steps present in the “superconducting signal” also show up in
“measured voltage”. In the corresponding histogram of
$\Delta^{2}\chi^{\prime}_{mv}$ (Fig. 14) the peaks due to the quantized steps
around $\pm 0.17$ nV remain visible despite the broadening due to the noise of
the added “background” signal, just as in Fig. 12.

Additional insight is provided from correlations between
$\Delta^{2}\chi^{\prime}_{mv}$ and $\Delta^{2}\chi^{\prime}_{sc}$. In Figs. 15
and 16 correlation maps in the
$(\Delta^{2}\chi^{\prime}_{mv},\Delta^{2}\chi^{\prime}_{sc})$ plane are
displayed for 138 and 160 GPa. These maps are parametric plots of the
($\Delta^{2}\chi^{\prime}_{mv}(T_{j})$,$\Delta^{2}\chi^{\prime}_{sc}(T_{j})$)
coordinate for all values of $T_{j}$. The scale of the maps has been adjusted
according to the quantization step size $\Delta_{q}$ deduced from
$\Delta^{2}\chi^{\prime}_{sc}$ (Figs. 8, 11, Table 2). For 138 and 160 GPa the
noise in the “measured voltage” is sufficiently small to reveal that
$\Delta^{2}\chi^{\prime}_{mv}$ has the same step structure as
$\Delta^{2}\chi^{\prime}_{sc}$. This shows up in the map for 138 GPa as a
pile-up of points around $(-0.025,-0.025)$ and $(0.025,0.025)$ and in the map
for 160 GPa around $(-0.17,-0.17)$ and $(0.17,0.17)$. In the right hand panels
this is shown in the form of histograms of $\Delta^{2}\chi^{\prime}_{mv}$
corresponding to the horizontal stripes in the corresponding correlations
maps. Applying the same analysis to
($\Delta^{2}\chi^{\prime}_{mv}(T_{j})$,$\Delta^{2}\chi^{\prime}_{sc}(T_{j})$)
we see (Figs. 24 and 25 of Appendix B) that the degree of correlation between
“background signal” and “superconducting signal” is zero. Using these
distributions we obtain the average values and variances summarized in Fig.
17. This figure demonstrates unequivocally that the $q(T)$ is 100 percent
present in the “measured voltage” and totally absent from the “background
signal”.

Figure 17: Average values of
($\Delta^{2}\chi^{\prime}_{mv},\Delta^{2}\chi^{\prime}_{sc})$ (left) and
($\Delta^{2}\chi^{\prime}_{bg},\Delta^{2}\chi^{\prime}_{sc})$ (right) with the
corresponding variances obtained by averaging over the horizontal stripes in
Figs. 15, 16, 24, 25.

This result for 138 and 160 GPa is puzzling for the following reasons: Since
the “superconducting signal” is claimed to be obtained from the raw data
(“measured voltage”) by a background-correction method Dias and Salamat
(2022), such a method would have to distill the pathological feature
$q(T)+P(T)$ out of the noisy “measured voltage” signal. Even if that were
technically possible, there is no obvious motivation to interpret $q(T)+P(T)$
as the “superconducting signal”.

From Table 2 we see that for 166, 178, and 182 GPa the noise amplitude in
$\Delta^{2}\chi^{\prime}_{mv}\sim 13\Delta_{q}$, and for 189 GPa even
$40\Delta_{q}$, which is too large compared to the step size to display a
statistically significant correlation with $\Delta^{2}\chi^{\prime}_{sc}$.
This is illustrated by the case of 166 GPa in Fig. 26 in Appendix B.

###  VI.2 Correlation functions

Figure 18: “Superconducting signal” at 160 GPa reproduced from the tables in
Refs. Dias and Salamat (Dec. 28,2021), and the first 4 discrete derivatives
normalized by $\Delta_{q}=0.16555$ nV.

Figure 19: Correlation functions
$g_{\alpha;\beta}^{(n)}=<\Delta^{n}\chi^{\prime}_{\alpha};\Delta^{n}\chi^{\prime}_{\beta}>=<\Delta^{n}{\alpha};\Delta^{n}{\beta}>$
(where the latter is the shorthand notation along the ordinate) at 160 GPa.
From left to right: order $n=0$,1,2,3 and 4. From top to bottom:
$g_{bg;mv}^{(n)}(j)$, $g_{sc;mv}^{(n)}(j)$, $g_{sc;bg}^{(n)}(j)$,
$g_{sc;sc}^{(n)}(j)$. All values have been normalized by $g_{sc;sc}^{(n)}(0)$.
The computer codes for obtaining $g_{\alpha;\beta}^{(n)}(j)$ can be downloaded
from Ref. Hirsch and van der Marel, 2022b.

Figure 20: From left to right: correlation functions $g_{\alpha;\beta}^{(4)}$
at 138, 166, 178, 182 and 189 GPa. From top to bottom: $g_{bg;mv}^{(4)}(j)$,
$g_{sc;mv}^{(4)}(j)$, $g_{sc;bg}^{(4)}(j)$, $g_{sc;sc}^{(4)}(j)$.

We now turn to an analysis of the degree of correlation between
$\chi^{\prime}_{mv}$, $\chi^{\prime}_{bg}$ and $\chi^{\prime}_{sc}$. To that
end we consider the nth discrete differential

$\Delta^{n}\chi^{\prime}_{\alpha}(j)=\Delta^{n-1}\chi^{\prime}_{\alpha}(j)-\Delta^{n-1}\chi^{\prime}_{\alpha}(j-1)$
(7)

and we define the correlation functions

$g_{\alpha;\beta}^{(n)}(j)=\sum_{k}^{N-1}\Delta^{n}\chi^{\prime}_{\alpha}(k)\Delta^{n}\chi^{\prime}_{\beta}(j+k)$
(8)

where $j$, $k$ and $j+k$ are taken modulo $N$. Before we decide which order
$n$ we will use for the correlation functions, we take a look at
$\Delta^{n}\chi^{\prime}_{sc}(j)$ for $n=0,1,2,3,4$ for the case of 160 GPa,
shown in Fig. 18. For $n\geq 1$ the values are normalized by
$\Delta_{q}=0.16555$ nV. It is immediately clear that $n=0$ is not a good
choice, since it results in a large vertical offset of the correlation
functions. This is offset is absent for $n\geq 1$, but for $n=1$ there is
considerable warping. The warping is smaller -but still finite- for $n=2$. For
$n=3$ there is no warping, but since
$\chi^{\prime}_{sc}(T)=P(T_{j})+q(T_{j})$, and $P(T_{j})$ is a third order
spline, the function $\Delta^{3}P(T_{j})$ is almost 444$\Delta^{3}P(T_{j})$ is
not exactly constant because (i) the discrete derivative is not an exact
derivative, (ii) there are slight variations of the temperature steps, and
(iii) $P(T)$ is non-analytical at the nodes a constant, but unequal to zero.
The fourth derivative $\Delta^{4}\chi^{\prime}_{sc}(j)$ contains no
contribution from $P(T_{j})$ other than noise, and is therefore fully
representative of the quantized component $q(T_{j})$. In Fig. 19 the
autocorrelation functions are displayed of the “superconducting signal”
$g_{sc;sc}^{(n)}(j)$ at 160 GPa for order $n=0,1,2,3,4$. From this example we
see, that for all orders $n\geq 1$ the autocorrelation function
$g_{sc;sc}^{(n)}(j)$ has a narrow peak at $j=0$. Going from lower to higher
order, the number of satellites left and right of the central peak increases.
This peak is exactly reproduced, with the same amplitude and same satellite
structure, in $g_{sc;mv}^{(n)}(j)$, and it is completely absent in
$g_{sc;bg}^{(n)}(j)$. We furthermore observe (i) a peak centered at $j=0$ in
$g_{bg;mv}^{(n)}(j)$, which (ii) is significantly broader than the one of
$g_{sc;sc}^{(n)}(j)$,. This indicates that (i) measured voltage and background
signal are interdepend, and (ii) there is some correlation between adjacent
data points. Point (i) contradicts the statement of Ref. Snider  _et al._ ,
2020 that the background signal was obtained in a separate experiment using a
different pressure. Point (ii) is a natural consequence of smoothing, as is
detailed in Appendix A.

From here on we use $n=4$ and apply the same analysis to the other 5
pressures. The result is shown in Fig. 20 for $g_{sc;sc}$, $g_{sc;bg}$
$g_{sc;mv}$, and $g_{bg;mv}$. In Appendix B Figs. 29 and 30 these functions
and also $g_{bg;bg}$ and $g_{mv;mv}$ are shown in a broader range.

  1. 1.

For all pressures the measured voltage strongly correlates with the background
signal.

  2. 2.

For none of the pressures the background signal correlates with the
superconducting signal.

  3. 3.

For 138, 160, 166, 178 and 182 GPa the measured voltage correlates with the
superconducting signal and at $j=0$ the MV-SC correlation has the same
amplitude as the SC-SC autocorrelation function.

  4. 4.

For 189 GPa the measured voltage appears to have a reduced correlation with
the superconducting signal i.e. the amplitude for $j=0$ is about half that of
the autocorrelation function of the superconducting signal. Most likely this
is the consequence of noise: The ratio
$\delta\chi^{\prime}_{mv}/\Delta_{q}\sim 40$ for this pressure, where
$\delta\chi^{\prime}_{mv}$ is the noise of $\Delta^{2}\chi^{\prime}_{mv}$ and
$\Delta_{q}$ is the quantized step size. In comparison for 166, 178 and 182
GPa $\delta\chi^{\prime}_{mv}/\Delta_{q}\sim 13$, and
$\delta\chi^{\prime}_{mv}/\Delta_{q}\sim 1$ for 138 and 160 GPa. 189 GPa being
the only pressure out six where the correlation with the superconducting
signal is less clear, we will assume from here on that the weaker correlation
in this case is due to noise.

To arrive at a crisp interpretation, we will now do two simulations calculated
according to protocol 1 and 3. No simulation is attempted for protocol 2,
because we do not know how to obtain the “user defined background” (UDB-1) of
Ref. Dias and Salamat, 2022.

Simulation using Protocol 1

  * •

Determine “measured voltage” including random noise loc

  * •

Determine “background signal” including random noise loc

  * •

Calculate “superconducting signal” = “measured voltage” - “background signal”

In left panel of Fig. 22 we show the measured voltage and superconducting
signal according to this protocol. The corresponding correlation functions,
shown in the left panel of Fig. 22, have the following features:

  1. 1.

The measured voltage has zero correlation with the background signal.

  2. 2.

The superconducting signal correlates with the background signal.

  3. 3.

The superconducting signal correlates with the measured voltage.

Comparing this to the above-mentioned data of Ref. Snider  _et al._ , 2020 we
see that points 1 and 2 are in disagreement. Consequently the data of Ref.
Snider  _et al._ , 2020 are not obtained with Protocol 1, in contrast to what
was stated in this paper.

Protocol 2, no simulation possible

  * •

In Ref. Dias and Salamat, 2022 the authors negate Protocol 1 and say instead
that the background signal was not measured but somehow extracted from the
measured raw data through a UDB-1 procedure that is left undefined.

  * •

Then, the “superconducting signal” is obtained from

“superconducting signal” = “measured voltage” - “background signal”

Protocol 2 is invalidated by the fact that there is no correlation between
“superconducting signal” and “background signal”.

Simulation using Protocol 3

  * •

Determine “superconducting signal” = $P(T)+q(T)$

  * •

Determine “background signal” including random noise loc ,

  * •

Calculate “measured voltage” = “superconducting signal” + “background signal”

In Fig. 22 we show the measured voltage and superconducting signal according
to this protocol. The corresponding correlation functions, shown in the right
panel of Fig 22, have the following features:

  1. 1.

The measured voltage has strong correlation with the background signal. The
central peak of $g_{bg;mv}(i)$ has satellites in for $i=\pm 7$, which are a
consequence of the smoothing of the background data, for which we used 7-point
adjacent averaging loc .

  2. 2.

The superconducting signal has zero correlation with the background signal.

  3. 3.

The superconducting signal correlates with the measured voltage and at $j=0$
the SC-MV correlation has the same amplitude as the SC-SC autocorrelation
function.

Figure 21: Simulated “measured voltage” and “superconducting” using protocol
1(left panel) and 3 (right panel). An excel file with the calculation of the
simulation shown in this figure can be downloaded from Ref. Hirsch and van der
Marel, 2022b

Figure 22: Correlation functions for $n=4$ of simulated data. Left: see left
panel of Fig. 22, Right: see right panel of Fig. 22. From top to bottom :
“Measured voltage” with “background”. “Measured voltage” with “superconducting
signal”. “Background” with “superconducting signal”. Autocorrelation of
“superconducting signal”. All values have been normalized by
$g_{sc;sc}^{(4)}(0)$. To facilitate comparison of the lower three panels the
same vertical range is used.

Comparing this to the above-mentioned data of Ref. Snider  _et al._ , 2020 we
see that there is a match for all three points. Consequently the data of Ref.
Snider  _et al._ , 2020; Dias and Salamat, Dec. 28,2021 have been obtained
with Protocol 3, in contrast to what was stated in this paper. An immediate
consequence is, that for none of the reported pressures the ‘measured voltage”
data provided in Ref. Dias and Salamat, Dec. 28,2021 are raw data. For the
case of 178 GPa, the $g_{sc;sc}^{(4)}(i)$ shows satellites of the central peak
for $i=\pm 13$. As we can see from simulation shown in Appendix B Fig. 28 a
possible explanation is, that 13-point adjacent average smoothing has been
applied to $g_{sc;sc}^{(4)}(i)$ at 178 GPa.

##  VII Summary

We have analyzed in detail the published “superconducting signal” and
“measured voltage” for all pressures reported in Refs. Snider  _et al._
(2020); Dias and Salamat (Dec. 28,2021). The following Protocol 3 is to the
best of our understanding the only possible scenario compatible with all
reported properties of the data.

  1. 1.

The “superconducting signal” $\chi^{\prime}_{sc}(T)$ is generated as the sum
of a quantized component $q(T)$ and a smooth function $P(T)$:
$\chi^{\prime}_{sc}(T)=q(T)+P(T)$.

  2. 2.

A “background signal” $\chi^{\prime}_{bg}(T)$ containing noise is determined.

  3. 3.

The “measured voltage” $\chi^{\prime}_{mv}(T)$ (alternatively labeled “raw
data”) is obtained by adding the “background signal” to the “superconducting
signal”: $\chi^{\prime}_{mv}(T)=\chi^{\prime}_{sc}(T)+\chi^{\prime}_{bg}(T)$.

We have demonstrated that

  * •

For 138, 160, 166, 178, 182 and 189 GPa the “superconducting signal” exhibits
the features expected for point 1 of the aforementioned protocol.

  * •

For 160 GPa the smooth function $P(T)$ is a cubic spline with 15 nodes having
“natural” boundary conditions at the two extremal nodes, and (ii)
$q(T)=\Delta_{q}n(T)$ where $\Delta_{q}=0.16555$ nV and $n(T)$ is an integer.
This component cannot be identified with the background signal, since it
departs strongly from the difference “measured voltage”-“superconducting
signal” according to the data reported in Ref. Dias and Salamat, Dec. 28,2021.

  * •

The quantized component of the 160 GPa data cannot be identified with the raw
data since, other than containing the steep rise at 170 K, it departs strongly
from the raw data (“measured voltage”) reported in Ref. Dias and Salamat, Dec.
28,2021.

  * •

For 138 and 160, 166, 178, 182 and possibly also 189 GPa the “measured
voltage” has been obtained according to point 3 of the protocol.

Recap:

Protocol 1: In Ref. Snider  _et al._ , 2020 the authors say the “measured
voltage” and “background signal” were independently measured. Then, data is
obtained from “superconducting signal”=“measured voltage” - “background
signal”
This is invalidated by the facts that:
(i) there is no correlation between “superconducting signal” and “background
signal”
(ii) there is correlation between “measured voltage” and “background signal”
(iii) “superconducting signal” has much less noise than “background signal”
and “measured voltage”

Protocol 2: In Ref. Dias and Salamat, 2022, the authors negate protocol 1 and
say instead that the background signal was not measured but somehow extracted
from the measured raw data (“measured voltage”) through a “User Defined
Background” (UDB) procedure. Three UDB variants were given in Ref. Dias and
Salamat, 2022.
Then, data is obtained from “superconducting signal”=“measured voltage” -
“background signal”
The variant UDB-1 is invalidated on the same grounds that there is no
correlation between “superconducting signal” and “background signal”. For
UDB-2 and UDB-3 the absence of correlation is trivial since for these variants
the background is a straight line. These variants are invalidated by the fact
that the “data” have much less noise than the “raw data”: they would have the
same noise with these variants.

Protocol 3: a protocol that is consistent with “measured voltage” and
“superconducting signal” posted in Ref. Dias and Salamat, Dec. 28,2021 is:
(a) “superconducting signal” is generated by adding a quantized component and
a smooth curve.
(b) “background signal” containing noise is generated somehow.
(c) “measured voltage” is obtained from: “measured voltage” = “superconducting
signal” + “background signal”.
This is consistent with all the facts, in particular that:
(i) there is no correlation between “superconducting signal” and “background
signal”.
(ii) there is correlation between “measured voltage” and “background signal”.
(iii) “measured voltage” contains the same steps as the ones of
“superconducting signal”.

Our analysis leads to the conclusion that the only protocol consistent with
the facts is protocol 3. In this protocol the “measured voltage” are not
actually measured raw data that reflect the properties of a physical system,
contrary to what Refs. Snider  _et al._ (2020), Dias and Salamat (Dec.
28,2021) and Dias and Salamat (2022) claim.

Data availability. The datasets generated and analyzed during the current
study are available in Ref. Hirsch and van der Marel, 2022b.

Acknowledgements. We thank P. Armitage, B. Ramshaw, J. Zaanen and G.
Grisonnanche for stimulating discussions and Dukwon for a useful suggestion.

##  Appendix A Effect of smoothing

If $\chi^{\prime}_{raw}(T)$ ($\chi^{\prime}_{sm}(T)$) are the susceptibility
without (with) smoothing, and the smoothing profile is $f(T)$, the relation
between unsmoothed and smoothed function is

$\displaystyle\chi^{\prime}_{sm}(T_{i})=\sum_{-\infty}^{\infty}\chi^{\prime}_{raw}(T_{j})f_{i,j}$
(9)

The first possibility that comes to mind is the smoothing that results from
having the integration time of the lock-in amplifier longer than the sampling
time per temperature during the temperature scan. Assuming that the data are
measured during a cool-down ramp, the corresponding smoothing profile is

$\displaystyle f_{i,j}=CH(T_{j}-T_{i})e^{(i-j)/n}$ (10)

where $H(x)$ is the Heaviside step function and $C$ a normalization factor.
However, from simulations (see Appendix B Fig. 27) we can see that
$g_{bg;mv}^{(1)}(j)$ does not have the “wavy” profile present in the 160 GPa
data. If instead we assume gaussian profile

$\displaystyle f_{i,j}=Ce^{-[(i-j)/n]^{2}}$ (11)

the wavy profile is exactly reproduced (see Appendix B Fig. 27).

## References

  * Snider  _et al._ (2020) E. Snider, N. Dasenbrock-Gammon, R. McBride, M. Debessai, H. Vindana, K. Vencatasamy, K. V. Lawler, A. Salamat, and R. P. Dias, “Room-temperature superconductivity in a carbonaceous sulfur hydride,” Nature 586, 373 (2020).
  * Dias and Salamat (Dec. 28,2021) R. P. Dias and A. Salamat, “Standard Superconductivity in Carbonaceous Sulfur Hydride,” arXiv:2111.15017 (Dec. 28,2021).
  * Hirsch (2022) J. E. Hirsch, “Comment on “Room temperature superconductivity of carbonaceous sulfur hydride”,” Europhysics Letters 137, 36001 (2022).
  * Hirsch and van der Marel (2022a) J. E. Hirsch and D. van der Marel, “Incompatibility of published ac magnetic susceptibility of a room temperature superconductor with measured raw data,” Matter and Radiation at Extremes 7, 048401 (2022a).
  * Dias and Salamat (2022) R. P. Dias and A. Salamat, “Reply to “Comment on Nature 586, 373 (2020) by E. Snider et al.”,” arXiv:2201.11883v1 (2022).
  * Hirsch and van der Marel (2022b) J. E. Hirsch and D. van der Marel, Open data to arXiv220107686v7 (2022b).
  * Denk (2017) Timo Denk, “Cubic Spline Interpolation,” https://timodenk.com (2017).
  * Note (1) A cubic spline smoothly connects $n$ nodes with coordinates $(x_{j},y_{j})$ ($1\leq j\leq n$) by applying the condition of continuity of first and second derivatives at the $n-2$ internal nodes. Two additional boundary condition are needed to fully define the spline function. Ref. denk2017 specifies the following types of additional boundary conditions: (i) Natural ($d^{2}y/dx^{2}=0$ at the two ends), (ii) Not-a-knot ($d^{3}y/dx^{3}$ are continuous for $j=2$ and $j=N-1$) (iii) Periodic ($dy/dx$ and $d^{2}y/dx^{2}$ are equal for $j=1$ and $j=N$) (iv) Quadratic (the first and the last segment are quadratic).
  * Ramshaw (2022) Brad Ramshaw, private communication (2022).
  * Dukwon (2022) Dukwon, “Discussion forum,” reddit (2022).
  * Note (2) This equation was previously inferred in Ref. preprint1 from the fact that the noise in the data is significantly smaller than in the raw data.
  * Hirsch (2021) J. E. Hirsch, “Disconnect between published ac magnetic susceptibility of a room temperature superconductor and measured raw data,” Preprints , 202112.0115 (2021).
  * Note (3) The fact that the measured voltage peaks at $\pm 0.2$ nV instead of $\pm 0.165$ nV of the superconducting signal is a statistical fluctuation. To verify this we computed $\sqrt{\left<(\Delta^{2}\chi^{\prime}_{mv})^{2}\right>_{{i.s.}}}$ for the set of temperatures corresponding to isolated steps in $\chi^{\prime}_{sc}(T)$. The total number of isolated steps is 57. The result is $\sqrt{\left<(\Delta^{2}\chi^{\prime}_{mv})^{2}\right>_{{i.s.}}}=0.185\pm 0.075~{}nV$. We see that the maximum at $\pm 0.2$ nV falls comfortably within this range of values.
  * Note (4) $\Delta^{3}P(T_{j})$ is not exactly constant because (i) the discrete derivative is not an exact derivative, (ii) there are slight variations of the temperature steps, and (iii) $P(T)$ is non-analytical at the nodes.
  * (15) In the simulations shown in Fig. 22 the adjacent average over 7 neighboring temperature points was taken, as a crude approximation of the effect of the finite integration time of a lock-in amplifier .

##  Appendix B Extended Data

Figure 23: Simulated data. Top line, from left to right: Smooth function
$P(T_{j})$, quantized component $q(T_{j})=n_{j}\Delta_{q}$ with
$\Delta_{q}=0.03$ nV, superconducting signal
$\chi^{\prime}_{sc}(j)=P(T_{j})+q(T_{j})$ and superconducting signal with
random noise $\chi^{\prime}_{sc}(j)+r(j)$ with $r(j)\in\\{-0.03;0.03\\}$ nV.
Second and third line: first discrete differential and second discrete
differential. Figure 24: Left: $\Delta^{2}\chi^{\prime}_{sc}$ versus
$\Delta^{2}\chi^{\prime}_{bg}$ at 138 GPa. Right: Histograms of
$\Delta^{2}\chi^{\prime}_{bg}$ in the slots corresponding to the horizontal
stripes. Figure 25: Left: $\Delta^{2}\chi^{\prime}_{sc}$ versus
$\Delta^{2}\chi^{\prime}_{bg}$ at 160 GPa. Right: Histograms of
$\Delta^{2}\chi^{\prime}_{bg}$ in the slots corresponding to the horizontal
stripes.

Figure 26: Top: Histograms of $\Delta^{2}\chi^{\prime}_{sc}$ and
$\Delta^{2}\chi^{\prime}_{mv}$ of the “measured voltage” at 166 GPa. Bottom:
Left: $\Delta^{2}\chi^{\prime}_{sc}$ versus $\Delta^{2}\chi^{\prime}_{mv}$.
Right: Histograms of $\Delta^{2}\chi^{\prime}_{mv}$ in the slots corresponding
to the horizontal stripes. Simulation with exponential smoothing of BG: ac
susceptibility and correlation functions

160 GPa: ac susceptibility and correlation functions

Simulation with gaussian smoothing of BG: ac susceptibility and correlation
functions

Figure 27: Simulation of the effect of an exponential smoothing of the
background of data (BG) obtained using protocol 3. BG is artificially
generated as a smooth function + random noise. A smoothing with $1/e$ distance
of 20 data points is applied to BG. The superconducting signal (ScSig) is
generated as a smooth function plus quantized function. The measured voltage
(MV) is generated using MV = BG + ScSig. Left: $\chi^{\prime}_{mv}$,
$\chi^{\prime}_{bg}$, $\chi^{\prime}_{sc}$. Middle: correlation functions of
the 6 combinations of $\Delta\chi^{\prime}_{mv}$, $\Delta\chi^{\prime}_{bg}$,
$\Delta\chi^{\prime}_{sc}$. Right: correlation functions of the 6 combinations
of $\Delta^{4}\chi^{\prime}_{mv}$, $\Delta^{4}\chi^{\prime}_{bg}$,
$\Delta^{4}\chi^{\prime}_{sc}$. Middle row of panels: Same correlation
functions as above for the 160 GPa data reported in Refs. Snider  _et al._
(2020); Dias and Salamat (Dec. 28,2021). Bottom row of panels: Same
correlation functions as above for a simulation of the effect of gaussian
smoothing of BG of data obtained using protocol 3. BG is artificially
generated as a smooth function + random noise. A smoothing with $1/e$ distance
of 3 data points is applied to BG. ScSig is generated as a smooth function
plus quantized function. MV is generated using MV = BG + ScSig. Simulation
with smoothing of BG and ScSig: ac susceptibility and correlation functions

178 GPa: ac susceptibility and correlation functions

Figure 28: Top panels: Simulation of the effect of a gaussian smoothing of the
background of data (BG) obtained using protocol 3 and Adjacent Averaging of
ScSig. BG is artificially generated as a smooth function + random noise. A
smoothing with $1/e$ distance of 3 data points is applied to BG. The
superconducting signal (ScSig) is generated as a smooth function plus
quantized function, and then Adjacent Averaged over 13 points. The measured
voltage (MV) is generated using MV = BG + ScSig. Left: $\chi^{\prime}_{mv}$,
$\chi^{\prime}_{bg}$, $\chi^{\prime}_{sc}$. Middle: correlation functions of
the 6 combinations of $\Delta\chi^{\prime}_{mv}$, $\Delta\chi^{\prime}_{bg}$,
$\Delta\chi^{\prime}_{sc}$. Right: correlation functions of the 6 combinations
of $\Delta^{4}\chi^{\prime}_{mv}$, $\Delta^{4}\chi^{\prime}_{bg}$,
$\Delta^{4}\chi^{\prime}_{sc}$. Bottom row of panels: Same correlation
functions as above for the 178 GPa data reported in Refs. Snider  _et al._
(2020); Dias and Salamat (Dec. 28,2021). 138 GPa: ac susceptibility and
correlation functions

160 GPa: ac susceptibility and correlation functions

166 GPa: ac susceptibility and correlation functions

Figure 29: Left: $\chi^{\prime}_{mv}$, $\chi^{\prime}_{bg}$,
$\chi^{\prime}_{sc}$. Middle: correlation functions of the 6 combinations of
$\Delta\chi^{\prime}_{mv}$, $\Delta\chi^{\prime}_{bg}$,
$\Delta\chi^{\prime}_{sc}$. Right: correlation functions of the 6 combinations
of $\Delta^{4}\chi^{\prime}_{mv}$, $\Delta^{4}\chi^{\prime}_{bg}$,
$\Delta^{4}\chi^{\prime}_{sc}$. From top to bottom:138 GPa (top), 160 GPa
(middle), 166 GPa (bottom) data reported in Refs. Snider  _et al._ (2020);
Dias and Salamat (Dec. 28,2021). 178 GPa: ac susceptibility and correlation
functions

182 GPa: ac susceptibility and correlation functions

189 GPa: ac susceptibility and correlation functions

Figure 30: Left: $\chi^{\prime}_{mv}$, $\chi^{\prime}_{bg}$,
$\chi^{\prime}_{sc}$. Middle: correlation functions of the 6 combinations of
$\Delta\chi^{\prime}_{mv}$, $\Delta\chi^{\prime}_{bg}$,
$\Delta\chi^{\prime}_{sc}$. Right: correlation functions of the 6 combinations
of $\Delta^{4}\chi^{\prime}_{mv}$, $\Delta^{4}\chi^{\prime}_{bg}$,
$\Delta^{4}\chi^{\prime}_{sc}$. From top to bottom:178 GPa (top), 182 GPa
(middle), 189 GPa (bottom) data reported in Refs. Snider  _et al._ (2020);
Dias and Salamat (Dec. 28,2021).


| 0.5 | 4.93 | 6.78 | 1.98 | 87.55 | 12.68 | 0.10 | 5.10 | 6.99 | 2.00 | 86.19 | 13.05
| 0.7 | 5.23 | 7.21 | 2.10 | 88.22 | 13.45 | 0.15 | 5.19 | 7.13 | 2.04 | 86.76 | 13.24
| 0.9 | 5.44 | 7.51 | 2.19 | 89.64 | 13.78 | 0.20 | 5.24 | 7.21 | 2.07 | 87.13 | 13.28
0.50 | 0.0 | 4.97 | 6.84 | 1.99 | 87.52 | 12.84 | 0.0 | 5.38 | 7.30 | 2.08 | 83.89 | 13.84
| 0.3 | 5.04 | 6.94 | 2.01 | 87.29 | 13.01 | 0.05 | 5.19 | 7.15 | 2.05 | 86.78 | 13.14
| 0.5 | 5.18 | 7.14 | 2.07 | 87.55 | 13.37 | 0.10 | 5.22 | 7.21 | 2.06 | 87.61 | 13.09
| 0.7 | 5.09 | 7.07 | 2.05 | 91.29 | 12.70 | 0.15 | 5.21 | 7.21 | 2.07 | 88.85 | 12.95
| 0.9 | 5.24 | 7.38 | 2.15 | 95.26 | 12.77 | 0.20 | 5.23 | 7.27 | 2.09 | 91.04 | 12.92
0.75 | 0.0 | 5.22 | 7.22 | 2.09 | 88.26 | 13.43 | 0.0 | 5.23 | 7.09 | 2.03 | 83.98 | 13.34
| 0.3 | 5.06 | 7.04 | 2.03 | 89.46 | 12.83 | 0.05 | 5.19 | 7.17 | 2.05 | 88.33 | 12.92
| 0.5 | 4.95 | 6.92 | 1.99 | 92.28 | 12.24 | 0.10 | 5.17 | 7.16 | 2.04 | 89.25 | 12.73
| 0.7 | 4.94 | 6.96 | 2.03 | 96.10 | 11.91 | 0.15 | 5.09 | 7.09 | 2.04 | 92.26 | 12.46
| 0.9 | 4.86 | 6.81 | 1.99 | 95.67 | 11.68 | 0.20 | 5.16 | 7.20 | 2.08 | 93.35 | 12.65

Table S.5:  RRR values obtained for the period 01/03/2017 through 12/30/2020
for select efficient frontier portfolios optimized under MCVaR0.99 and MV.
Note the difference between mCVaR0.99 and MV in the range of $\alpha$
provided.

Model | Tot. Ret | Ann. Ret | AvgTO | ETL95 | ETR95 | MDD | ESG* |  |
---|---|---|---|---|---|---|---|---|---
| (%) | (%) | (%) | (%) | (%) | (%) | avg | std |  |
EWBH | 90.14 | 17.69 | 0.00 | -3.41 | 2.86 | 32.87 | 63.25 | 1.81 |  |
$\lambda$ |  mCVaR0.99 |  |
0.00 | 95.01 | 18.45 | 0.40 | -3.48 | 2.92 | 34.06 | 61.50 | 1.85 |  |
0.25 | 128.28 | 23.27 | 0.35 | -3.78 | 3.54 | 29.49 | 85.82 | 8.48 |  |
0.50 | 132.40 | 23.83 | 0.35 | -3.82 | 3.62 | 29.66 | 87.68 | 8.67 |  |
0.75 | 135.19 | 24.21 | 0.34 | -3.85 | 3.69 | 29.84 | 88.97 | 8.85 |  |
$\lambda$ | MV |  |
0.00 | 93.80 | 18.26 | 0.40 | -3.76 | 3.13 | 37.23 | 63.10 | 2.92 |  |
0.25 | 139.50 | 24.78 | 0.33 | -3.90 | 3.64 | 29.08 | 85.98 | 8.94 |  |
0.50 | 148.41 | 25.94 | 0.32 | -3.93 | 3.72 | 29.38 | 87.85 | 8.73 |  |
0.75 | 147.24 | 25.79 | 0.33 | -3.90 | 3.73 | 29.43 | 88.70 | 8.77 |  |
Model | Mean | Median | Std | Skew | ExKurt | Mean | Median | Std | Skew | ExKurt
| $\times 10^{-4}$ | $\times 10^{-4}$ | $\times 10^{-2}$ |  | $\times 10^{-4}$ | $\times 10^{-4}$ | $\times 10^{-2}$ |
EWBH | 6.4 | 11.1 | 1.3 | -1.02 | 21.9 |
$\lambda$ |  mCVaR0.99 | MV
0.00 | 6.7 | 11.4 | 1.4 | -1.28 | 23.9 | 6.6 | 11.6 | 1.471 | -1.36 | 23.7
0.25 | 8.2 | 12.0 | 1.5 | -0.76 | 19.6 | 8.7 | 12.7 | 1.586 | -0.71 | 18.5
0.50 | 8.4 | 11.5 | 1.6 | -0.75 | 19.6 | 9.0 | 12.4 | 1.607 | -0.71 | 18.5
0.75 | 8.5 | 10.9 | 1.6 | -0.72 | 19.1 | 9.0 | 11.8 | 1.607 | -0.68 | 18.2
Model | SR | Sortino | STAR | Rachev | Gini | SR | Sortino | STAR | Rachev | Gini
| (%) | (%) | (%) | (%) | (%) | (%) | (%) | (%) | (%) | (%)
EWBH | 4.86 | 6.59 | 1.88 | 83.86 | 12.26 |
$\lambda$ |  mCVaR0.99 | MV
0.00 | 4.87 | 6.57 | 1.91 | 83.82 | 12.19 | 4.48 | 6.00 | 1.75 | 83.05 | 11.02
0.25 | 5.32 | 7.43 | 2.17 | 93.70 | 13.03 | 5.48 | 7.66 | 2.23 | 93.54 | 13.34
0.50 | 5.36 | 7.50 | 2.20 | 94.67 | 13.08 | 5.63 | 7.90 | 2.31 | 94.65 | 13.74
0.75 | 5.35 | 7.50 | 2.21 | 95.86 | 12.97 | 5.60 | 7.88 | 2.31 | 95.62 | 13.62

Table S.6: Performance measures, return moment values and RRR results obtained
for the tangent portfolios under mCVaR0.99 and MV optimization during the
period 01/03/2017 to 12/30/2020.