# Comment on “Trapped flux in a small crystal of CaKFe4As4 at ambient pressure
and in a diamond anvil pressure cell” by S. L. Bud’ko et al.

J. E. Hirscha and F. Marsigliob aDepartment of Physics, University of
California, San Diego, La Jolla, CA 92093-0319
bDepartment of Physics, University of Alberta, Edmonton, Alberta, Canada T6G
2E1

###### Abstract

In their paper arXiv:2405.08189, Supercond. Sci. Technol. 37 (2024) 065010
tiny , Bud’ko et al. present experimental results for trapped magnetic flux
for a tiny sample of a type II superconductor. The paper aims to provide
evidence in support of the interpretation that similar measurements performed
in samples of hydrogen-rich materials under high pressure by Minkov, Bud’ko
and coauthors etrappedp  are conclusive evidence ensr2024  for
superconductivity in hydrides under pressure. Here we point out that the new
evidence presented by Bud’ko et al. tiny  further supports our interpretation
hmtrapped ; hmtrapped2  that the reported measurements of trapped flux on
hydrides under pressure etrappedp  are evidence that the samples are $not$
superconducting.

##  I introduction

In the paper Supercond. Sci. Technol. 37 (2024) 065010 tiny , Bud’ko et al.
present results of flux trapping measurements for a tiny sample of
superconducting CaKFe4As4. The aim of the work is to provide support for the
interpretation that similar measurements on hydrides under high pressure
etrappedp  are evidence that hydrides under high pressure are superconductors.
Figure 1 shows measurements of a tiny sample of the superconducting material
on the top panel, and on the bottom panel of a sample of similar size of H3S.
Superficially, the two panels may look qualitatively similar. In this paper we
point out that in fact they are qualitatively different. But first some
historical context.

Minkov, Bud’ko and coauthors have published results of experiments on hydrides
under high pressure etrappedp , that showed that when a magnetic field is
applied to the material and subsequently removed, a remnant magnetization is
detected, that they interpret as originating in “trapped flux”. They inferred
from their measurements that the remnant magnetization is produced by electric
currents flowing in the material that don’t decay with time, providing
“conclusive evidence” ensr2024  that the material is a superconductor.

Figure 1: Top panel: trapped magnetic moment for a sample of superconducting
CaKFe4As4 after applied field is removed in zero field cooling (ZFC, black
points) and field cooling (FC, red points) protocols, from Ref. tiny . Bottom
panel: same for a sample of H3S under high pressure, from Ref. etrappedp . In
both panels, lines were drawn through the points to guide the eye by the
authors of Refs. tiny ; etrappedp .

In Ref. hmtrapped , we argued that the measurements of Minkov, Bud’ko et al.
etrappedp  do not support superconductivity because the observed dependence of
trapped moment under zero field cooling (ZFC) versus magnetic field was linear
rather than quadratic. We showed results of our calculations with a simple
model showing that the expected behavior due to superconducting currents
should be quadratic hmtrapped .

Subsequently, Bud’ko, Xu and Canfield canfield  did similar measurements as
Minkov, Bud’ko et al. etrappedp  on known superconducting materials, using
samples that were about 1000 times larger in volume than the samples used by
Minkov, Bud’ko et al canfield . Bud’ko, Xu and Canfield concluded canfield
that their measurements supported the interpretation of the experiments by
Minkov, Bud’ko et al etrappedp .

Subsequently, Talantsev, Bud’ko et al. talan  and Bud’ko budko  criticized our
paper Ref. hmtrapped , arguing that our paper relied on the “wrong model”,
that it used “selective manipulations (hide/delete) of calculated datasets”,
that we did not perform a proper fit to the Minkov, Bud’ko et al. measurements
etrappedp , that our model was “unphysical” and was “incorrect”, and that its
predictions were incompatible with the measurements on superconducting samples
by Bud’ko, Xu and Canfield canfield , hence that our conclusions in Ref.
hmtrapped  were invalid. In these criticisms, they did not address the main
point of our paper Ref. hmtrapped , namely that the linear dependence of ZFC
moment versus field observed in Ref. etrappedp  for H3S is evidence that the
signal is not due to superconductivity.

Subsequently we showed in Refs. hmtrapped2 , myresponse  that (i) the
criticism by Ref. talan  was unfounded, (ii) the criticism by Ref. budko  was
unfounded, and (iii) the reported results by Bud’ko, Xu and Canfield Ref.
canfield  in fact strongly supported our original analysis and interpretation
hmtrapped  of the Minkov, Bud’ko et al. results on hydrides etrappedp .

Subsequently, in the paper that we are commenting on here tiny , Bud’ko et al.
report results on a sample of volume comparable to that of the hydride
materials, i.e. 1000 times smaller than in their previous work canfield ,
under pressure and at ambient pressure, and argue that the results on their
tiny sample provide further support to the interpretation that the
measurements of Minkov, Bud’ko et al. on hydrides etrappedp  are evidence for
superconductivity. In Ref. tiny , Bud’ko et al also repeat their claim budko
that their new results contradict the “unphysical” and “incorrect” results of
Ref. hmtrapped .

##  II analysis

Our analysis of flux trapping experiments in Refs. hmtrapped ; hmtrapped2  was
based on the Bean model, under the simplifying assumption that the critical
current is uniform over the superconducting region. We recognize that this is
a simplifying assumption and that real superconducting samples could show some
variation of critical current as function of field magnitude and position in
the sample, so we don’t expect exact agreement of calculated results with
measured results. Still, we expect that our model will capture the essential
physics and yield approximately correct results for measurements of trapped
flux in superconducting samples. In particular, an important aspect of the
physics emphasized in Refs. hmtrapped ; hmtrapped2  is that under ZFC the
trapped magnetic moment predicted by our model and mandated on physical
grounds should be quadratic and not linear as function of applied field.

The model, quantitatively described in Refs. hmtrapped ; hmtrapped2 , has the
following adjustable parameters: (i) $H^{*}$, the value of the magnetic field
for which an applied magnetic field reaches the center of the sample, which is
proportional to the critical current density. (ii) $H_{p}$, the threshold
value of the magnetic field below which an applied field does not penetrate,
which is the lower critical field corrected for demagnetization, and (iii) the
value of the saturation trapped magnetic moment $m_{s}$.

In this Comment, we show that in fact the most recent results of Bud’ko et al.
tiny  provide further confirmation of the validity of our analysis hmtrapped ;
hmtrapped2  of the Minkov, Bud’ko et al. experiments etrappedp  and of our
interpretation that the signals measured in hydrides are not due to
superconductivity.

Figure 2:  Theoretical fit to the experimental results of Ref. tiny  for
trapped flux in CaKFe4As4 (red and black lines) using the theoretical model of
Refs. hmtrapped ; hmtrapped2 . Parameters used for the fit are given in the
figure.

Figure 2 shows comparison of the predictions of our model with the measured
results on the superconducting sample studied in Ref. tiny , with the values
of the adjustable parameters shown in the figure. These values were chosen so
as to give the best possible fit to the low field values of the ZFC trapped
moment for low fields, which clearly shows supralinear behavior (quadratic) as
seen in the inset of Fig. 2. It can be seen that these parameters also give a
good fit to the FC moment at low field (inset of Fig. 2). Furthermore, they
provide a reasonable fit to the behavior over the entire field range, as shown
in the main body of Fig. 2. There are some deviations from the measured values
at intermediate field values, but it can be seen that the values where the
magnetic moment reaches saturation, which in our model is $H^{*}$ and
$2H^{*}+H_{p}$ for FC and ZFC protocols respectively, are not far from the
values indicated by the experimental points.

Figure 3:  Theoretical fit to the experimental results of Ref. etrappedp  for
trapped flux in H3S (red and black lines) using the theoretical model of Refs.
hmtrapped ; hmtrapped2 . Parameters used for the fit are given in the figure.
They were chosen to approximately match the field values where the magnetic
moment reaches saturation.

Figure 4:  Theoretical fit to the experimental results of Ref. etrappedp  for
trapped flux in H3S (red and black lines) using the theoretical model of Refs.
hmtrapped ; hmtrapped2 . Parameters used for the fit are given in the figure.
They were chosen to approximately match the low field behavior of the FC
moment.

Figure 5:  Theoretical fit to the experimental results of Ref. etrappedp  for
trapped flux in H3S (red and black lines) using the theoretical model of Refs.
hmtrapped ; hmtrapped2 . Parameters used for the fit are given in the figure.
They were chosen to approximately match the low field behavior of the ZFC
moment.

In Figs. 3,4,5 we attempt such fits for the hydride sample $H_{3}S$ using the
measured values reported in Ref. etrappedp . The low field behavior of the ZFC
moment gives a value of $H_{p}=0.042T$, as determined in Ref. etrappedp ,
which we use in the three figures. We start in Fig. 3 by choosing $H^{*}$ to
approximately fit the field values where the moment reaches saturation, which
was determined to be $H^{*}=0.7T$ in Ref. etrappedp . The moment under ZFC
reaches saturation at magnetic field approximately twice as large as the FC,
as our model predicts. It can be seen in Fig. 3 that the red and black lines
approximately fit the measured points near the values where the moments reach
saturation. However, it can be seen in the inset of Fig. 3 that there is
drastic disagreement with the behavior of the moments at small field, the
theoretical curves giving values substantially lower than the measured ones,
particularly for the ZFC protocol.

In Fig. 4 we pick $H^{*}$ so as to fit the low field behavior of the FC
moment. This requires a much smaller value of $H^{*}$, namely $H^{*}=0.3T$. It
can be seen that the low field values of the ZFC moment are still
substantially underestimated. Furthermore, the overall behavior seen in the
main body of the figure is in drastic disagreement with experiment, since the
curves reach saturation substantially before the measured moments reach
saturation, both for FC and ZFC protocols.

Finally, in Fig. 5 we pick an even smaller value of $H^{*}$, $H^{*}=0.2$, to
attempt a better match to the low field ZFC moments. It can be seen that the
ZFC moments are still not matched by the theory, because they don’t follow the
same functional form as the theory predicts, namely quadratic in field,
nevertheless we try to fit the behavior on average, with some points falling
below, some above, the theoretical curve. The small field FC moment is now
overestimated by the theoretical values, and as seen in the main panel of the
figure the overall disagreement with the measured values is stark both for FC
and ZFC moments.

Figure 6:  Top panels: Linear regression fits of square root of trapped moment
versus magnetic field for H3S etrappedp  (top left panel) and for the tiny
sample of CaKFe4As4 tiny  (top right panel). It can be seen that for the
superconductor (top right panel) the fit is very good, indicating quadratic
behavior of trapped moment versus field as predicted by our model and by
general physical arguments. Instead, for H3S (top left panel) the fit is not
good. Bottom panels: Linear regression fit of trapped moment versus magnetic
field for H3S (bottom left panel) and for CaKFe4As4 (bottom right panel). It
can be seen that for the superconductor (bottom right panel) the fit is not
very good, indicating that the trapped moment under ZFC is not linear with
magnetic field, as predicted by our model and general physical arguments.
Instead, for H3S (bottom left panel) the linear fit is excellent.

Finally, as we also did in Ref. hmtrapped2 , in order to quantify the
deviation of the low-field ZFC data for H3S from the expected behavior of a
superconductor, we show in Fig. 6 linear regression fits regression  to the
square root and first power of the trapped moment versus magnetic field
respectively for the two materials considered here. The coefficient $R^{2}$
measures how well the data fit the regression model, with $R^{2}=1$ being a
perfect fit. It can be seen in the top panels that $R^{2}$ is larger than 0.99
for the superconductor indicating an excellent fit to quadratic behavior of
magnetic moment versus magnetic field, while for H3S the $R^{2}$ differs
substantially from unity, indicating the observed behavior is not consistent
with quadratic dependence of moment on field. Conversely, as seen in the
bottom panels, a linear fit of magnetic moment vs field fits the H3S data
extremely well, with $R^{2}=0.9979$, and does substantially worse for the
standard superconductor. This provides strong evidence that for the standard
superconductor the behavior of magnetic moment versus field under ZFC is
quadratic while it is linear for H3S, just as we found in Ref. hmtrapped2  for
several other superconductors as well.

##  III effect of pressure and background subtraction

For the sample under pressure, Ref. tiny  does not report magnetic field
dependence of the trapped moment with sufficient detail in their Fig. 6 to
make it possible for us to fit the results as done in Fig. 2. Nevertheless, we
can get some information from the temperature-dependent trapped moments
reported in Fig. 5(b) of Ref. tiny , shown here in Fig. 7(a). It is very clear
from Fig. 7(a) that the behavior of the ZFC moment with field in the DAC under
pressure is very non-linear, just as it was for the sample at ambient
pressure: in increasing the magnetic field from 500 Oe to 1 kOe, i.e. by 500
Oe, the increase in trapped moment seen in Fig. 7(a) is a factor of 5 times
larger than in increasing the magnetic field from 100 to 500 Oe, i.e. by 400
Oe. This is in clear disagreement with what would be expected for a sample
whose trapped moment behaved linearly like the hydride sample shown in the
inset of Fig. 1 lower panel, which would show an increase in moment when going
from 500 Oe to 1 kOe that is only a factor 1.25 larger than in going from 100
to 500 Oe. For comparison, Fig. 7(b) shows similar curves for H3S from Fig.
1(c) of Ref. etrappedp . Here, the increase in the low temperature moment in
going from 125 mT to 250 mT, i.e. an increase of 125 mT, is approximately
twice the increase in going from 60 mT to 125 mT, i.e. an increase of 65 mT,
as expected for linear behavior.

Furthermore, while the purpose of Ref. tiny  is “to address concerns about
sample size and signal associated with trapped flux measurements in a DAC,”
i.e. under pressure, the authors of Ref. tiny  have actually managed to raise
concerns instead. They themselves have noted the large discrepancy in signal
size in the normal state under 2.2 GPa pressure in the DAC (their Fig. 4(a))
compared to that at zero pressure without the DAC (their Fig. 1), and the
accompanying increase in noise, possibly attributing these discrepancies to
the presence of “a fairly large background” in the DAC (see the discussion on
p. 4-5 of Ref. tiny ). The authors include an Appendix B with figures produced
from a commercial DAC — see the description of this commercial DAC plus
accompanying software in Ref. [16] of Ref. tiny , — but do not show background
and sample + background results as a function of temperature or applied
magnetic field. It would also have been useful to see more detailed results
for the trapped field magnetization as a function of applied field in the DAC
under pressure — only four unsaturated values are shown in their Fig. 6.

Figure 7:  (a) From Fig. 5(b) of Ref. etrappedp : temperature dependent
trapped flux magnetization of CaKFe4As4 measured in H =0 using ZFC protocol in
DAC under pressure of 2.2 GPa. The magnetic field was applied at base
temperature $T=1.8K$, then removed. (b) From Fig. 1(c) of Ref. etrappedp :
temperature-dependent trapped flux magnetization of H3S measured in ${\rm
H}=0$ using ZFC protocol in DAC under pressure of 155 GPa. The magnetic field
was applied at base temperature $T=10K$, then removed.

As one of the purposes of Ref. tiny  is to convince the reader of the validity
of hydride results presented in Ref. etrappedp  (their Ref. [6]), it is
unfortunate that there are “important differences in protocols for trapped
flux magnetization measurements between this work and [6].” In particular the
presence of a “remnant magnetic field…which depends on the superconducting
magnet design and geometry as well as the history of magnetic field applied
prior to the measurement” is a disturbing source of uncertainty. One is left
wondering about the possibility of spurious effects in both these measurements
and those in the hydrides, Ref. etrappedp , that could depend significantly on
the background subtraction procedure that is applied, that is not clearly
explained in Ref. etrappedp .

Given that this group has considerable experience with CaKFe4As4 at even
higher pressures (above 4 GPa) where this material is not superconducting
kaluarachchi2017 , it would be of considerable interest to test for trapped
flux under these higher pressures using the same background subtraction
procedure as used for the lower pressures where it is superconducting. Another
test to determine the role of the apparatus alone would be to redo the
experiments at zero pressure inside the DAC, to better differentiate the roles
of the actual pressure versus the presence of the DAC itself.

Ultimately, however, for a convincing demonstration of the utility of
magnetization measurements on hydride materials in DACs, one would need to
follow the same protocol in a known superconductor as was used in the
hydrides, and properly document the background subtractions that are required.

##  IV conclusion

In Ref. tiny , the authors conducted trapped flux experiments on a sample of a
known superconducting material, attempting to mimic the conditions in the
experiments for hydrides under pressure etrappedp , by picking a tiny sample
of volume comparable to the hydride samples, and subjecting it to pressure in
a diamond anvil cell.

Contrary to the statements in Refs. budko ; canfield ; talan ; tiny  by Bud’ko
and coauthors that our model is unphysical and incorrect and wrong and that we
selectively hide/delete data, our model reproduces the observed behavior in
this superconducting tiny sample remarkably well, as shown in Fig. 2. Upon
picking the parameters in the model to fit the low field behavior of the ZFC
moment, our model predicts the behavior of the small field FC moment as well
as the overall behavior of the ZFC and FC moments including the values where
the moments reach saturation remarkably well. The maximum discrepancy in the
values of calculated moment vs measured moment in Fig. 2 is $25\%$. This
indicates that the model captures the essential physics of trapped moments in
type II superconductors even for tiny samples such as those used in the
hydride experiments.

In contrast, we showed that our model cannot describe the behavior of the
moments reported for the hydride sample H3S etrappedp  for any set of
parameters in the model. Choosing $H^{*}=0.7T$ as was done in Ref. etrappedp ,
to fit approximately the fields where the moments reach saturation, gives rise
to major discrepancies in the calculated versus measured values for lower
fields, as shown in Fig. 3; the discrepancy in the values of calculated moment
vs measured moment in Fig. 3 is up to $800\%$, 30 times larger than for the
superconducting sample in Fig. 2. If $H^{*}$ is instead chosen to
approximately fit the low field behavior, there are major differences for the
high field behavior, as shown in Figs. 4 and 5. Furthermore, the quadratic
behavior of ZFC moment with field at small field that is seen in this tiny
sample tiny , as well as all the superconductor samples studied by these
authors earlier canfield , is perfectly fitted by our model. The linear
behavior seen in the hydrides that is not fitted by our model is in stark
contrast with the behavior of all the superconducting samples, including the
sample studied in Ref. tiny  in the DAC under pressure 2.2 GPa, as we
discussed in Sect. III.

Furthermore, Ref. tiny  showed that when the tiny sample is measured in a DAC
under pressure significant other issues come into play. The magnetization
under an applied field as well as the noise show large differences compared to
the measurements without the DAC. To be relevant to the interpretation of
hydride results, the protocols used, in particular background subtraction,
should be the same. In particular, Ref. tiny  emphasized the need for point-
by-point background subtraction, it is not clear what was done in Ref.
etrappedp .

We conclude that the remnant moments measured in hydrides etrappedp , shown
here in Fig. 1 lower panel, do not show behavior that is consistent with the
behavior seen in the tiny superconducting sample studied in Ref. tiny ,
neither at ambient pressure nor under pressure. The most likely possibility is
that the remnant moments measured in Ref. etrappedp  originate in other
phenomena or experimental artifacts unrelated to superconductivity, in
accordance with the conclusions reached in our earlier papers on this subject
hmtrapped ; hmtrapped2 .

###### Acknowledgements.

FM was supported in part by the Natural Sciences and Engineering Research
Council of Canada (NSERC) and by Alberta Innovates.

## References

  * (1) S. L. Bud’ko, S. Huyan, M. Xu and P. C. Canfield, “Trapped flux in a small crystal of $CaKFe_{4}As_{4}$ at ambient pressure and in a diamond anvil pressure cell”, Supercond. Sci. Technol. 37, 065010 (2024).
  * (2) V. S. Minkov, V. Ksenofontov, S. L. Budko, E. F. Talantsev and M. I. Eremets, “Magnetic flux trapping in hydrogen-rich high-temperature superconductors”, Nat. Phys. 19, 1293 (2023).
  * (3) M. I. Eremets, “The current status and future development of high-temperature conventional superconductivity”,  National Science Review, nwae047 (2024).
  * (4) J. E. Hirsch and F. Marsiglio, “Evidence Against Superconductivity in Flux Trapping Experiments on Hydrides Under High Pressure”, J Supercond Nov Magn 35, 3141–3145 (2022).
  * (5) J. E. Hirsch and F. Marsiglio, “Further analysis of flux trapping experiments on hydrides under high pressure”, Physica C 620, 1354500 (2024).
  * (6) S. L. Bud’ko, M. Xu and P. C. Canfield, “Trapped flux in pure and Mn-substituted $CaKFe_{4}As_{4}$ and $MgB_{2}$ superconducting single crystals”, Supercond. Sci. Technol. 36, 115001 (2023).
  * (7) E.F. Talantsev, V.S. Minkov, V. Ksenofontov, S.L. Bud’ko, M.I. Eremets “Is $MgB_{2}$ a superconductor?”, arXiv:2312.04495v3 (2023).
  * (8) S. L. Bud’ko, “Comment on “Evidence Against Superconductivity in Flux Trapping Experiments on Hydrides Under High Pressure”,  J Supercond Nov Magn 37, 473 (2024).
  * (9) J. E. Hirsch, “Comment on arXiv:2312.04495 by M. I. Eremets and coauthors”, OSF preprint https://doi.org/10.31219/osf.io/p29ht (2023),  arXiv:2401.08927 (2024).
  * (10) These fits were performed using the linear regression calculator at https://www.graphpad.com/quickcalcs/linear1/.
  * (11) Udhara S. Kaluarachchi, Valentin Taufour, Aashish Sapkota, Vladislav Borisov, Tai Kong, William R. Meier, Karunakar Kothapalli, Benjamin G. Ueland, Andreas Kreyssig, Roser Valentí, Robert J. McQueeney, Alan I. Goldman, Sergey L. Bud’ko, and Paul C. Canfield, “Pressure-induced half-collapsed-tetragonal phase in ${\mathrm{CaKFe}}_{4}{\mathrm{As}}_{4}$,” Phys. Rev. B96, 140501 (2017).


for any $u\in{\cal U}$. Recall
$\widetilde{\Sigma}_{jk}^{\varepsilon}(u,v)=(n-1)^{-1}\sum_{i=1}^{n}\\{\hat{\varepsilon}_{i,jk}(u)-\bar{\hat{\varepsilon}}_{jk}(u)\\}\\{\hat{\varepsilon}_{i,kj}(v)-\bar{\hat{\varepsilon}}_{kj}(v)\\}$.
Hence it holds that

$\displaystyle\\{\hat{\varepsilon}_{i,jk}(u)-\bar{\hat{\varepsilon}}_{jk}(u)\\}\\{\hat{\varepsilon}_{i,kj}(v)-\bar{\hat{\varepsilon}}_{kj}(v)\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\bigg{[}\tilde{\varepsilon}_{i,jk}(u)-\int_{{\cal
U}}\\{{\bf Y}_{i,-j,-k}(t)-\widebar{\bf
Y}_{-j,-k}(t)\\}^{{\mathrm{\scriptscriptstyle
T}}}\\{\widehat{\boldsymbol{\beta}}_{jk}(u,t)-\boldsymbol{\beta}_{jk}(u,t)\\}\,{\rm
d}t\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\bigg{[}\tilde{\varepsilon}_{i,kj}(v)-\int_{{\cal
U}}\\{{\bf Y}_{i,-j,-k}(s)-\widebar{\bf
Y}_{-j,-k}(s)\\}^{{\mathrm{\scriptscriptstyle
T}}}\\{\widehat{\boldsymbol{\beta}}_{kj}(v,s)-\boldsymbol{\beta}_{kj}(v,s)\\}\,{\rm
d}s\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\tilde{\varepsilon}_{i,jk}(u)\tilde{\varepsilon}_{i,kj}(v)-\tilde{\varepsilon}_{i,jk}(u)\int_{{\cal
U}}\\{{\bf Y}_{i,-j,-k}(s)-\widebar{\bf
Y}_{-j,-k}(s)\\}^{{\mathrm{\scriptscriptstyle
T}}}\\{\widehat{\boldsymbol{\beta}}_{kj}(v,s)-\boldsymbol{\beta}_{kj}(v,s)\\}\,{\rm
d}s$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\tilde{\varepsilon}_{i,kj}(v)\int_{{\cal
U}}\\{{\bf Y}_{i,-j,-k}(t)-\widebar{\bf
Y}_{-j,-k}(t)\\}^{{\mathrm{\scriptscriptstyle
T}}}\\{\widehat{\boldsymbol{\beta}}_{jk}(u,t)-\boldsymbol{\beta}_{jk}(u,t)\\}\,{\rm
d}t$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int_{{\cal
U}}\int_{{\cal
U}}\\{\widehat{\boldsymbol{\beta}}_{jk}(u,t)-\boldsymbol{\beta}_{jk}(u,t)\\}^{{\mathrm{\scriptscriptstyle
T}}}\\{{\bf Y}_{i,-j,-k}(t)-\widebar{\bf Y}_{-j,-k}(t)\\}\\{{\bf
Y}_{i,-j,-k}(s)-\widebar{\bf Y}_{-j,-k}(s)\\}^{{\mathrm{\scriptscriptstyle
T}}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\\{\widehat{\boldsymbol{\beta}}_{kj}(v,s)-\boldsymbol{\beta}_{kj}(v,s)\\}\,{\rm
d}t{\rm d}s$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overset{\triangle}{=}{\rm
I}_{i}(j,k,u,v)-{\rm II}_{i}(j,k,u,v)-{\rm III}_{i}(j,k,u,v)+{\rm
IV}_{i}(j,k,u,v)\,.$ (S.14)

Notice that
$\widetilde{T}_{n,jk}^{\varepsilon}=n\|\widetilde{\Sigma}_{jk}^{\varepsilon}\|_{{\cal
S}}^{2}$ and $\widetilde{\Sigma}_{jk}^{\varepsilon}(u,v)$ exhibits a similar
decomposition as $\widetilde{\Sigma}_{jk}(u,v)$ in (10), where
$(n-1)^{-1}\sum_{i=1}^{n}\\{{\rm II}_{i}(j,k,u,v)+{\rm III}_{i}(j,k,u,v)\\}$
and $(n-1)^{-1}\sum_{i=1}^{n}{\rm IV}_{i}(j,k,u,v)$ corresponds to
$\widehat{\Sigma}_{jk}^{Xe}(u,v)$ and $\widehat{\Sigma}_{jk}^{ee}(u,v)$ in
Section 3, respectively. In the sequel, we will consider the convergence rates
of $\int_{{\cal U}}\int_{{\cal U}}\\{(n-1)^{-1}\sum_{i=1}^{n}{\rm
IV}_{i}(j,k,u,v)\\}^{2}\,{\rm d}u{\rm d}v$, $\int_{{\cal U}}\int_{{\cal
U}}\\{(n-1)^{-1}\sum_{i=1}^{n}{\rm II}_{i}(j,k,u,v)\\}^{2}\,{\rm d}u{\rm d}v$
and $\int_{{\cal U}}\int_{{\cal U}}\\{(n-1)^{-1}\sum_{i=1}^{n}{\rm
III}_{i}(j,k,u,v)\\}^{2}\,{\rm d}u{\rm d}v$, respectively, which will in turn
specify the rates defined in (11).

Convergence rate of $\int_{{\cal U}}\int_{{\cal
U}}\\{(n-1)^{-1}\sum_{i=1}^{n}{\rm IV}_{i}(j,k,u,v)\\}^{2}\,{\rm d}u{\rm d}v$.
By Inequality 1, Propositions 2 and 3,

$\displaystyle\int_{{\cal U}}\int_{{\cal
U}}\bigg{\\{}\frac{1}{n-1}\sum_{i=1}^{n}{\rm
IV}_{i}(j,k,u,v)\bigg{\\}}^{2}\,{\rm d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}=\int_{{\cal U}}\int_{{\cal U}}\bigg{[}\int_{{\cal
U}}\int_{{\cal
U}}\\{\widehat{\boldsymbol{\beta}}_{jk}(u,t)-\boldsymbol{\beta}_{jk}(u,t)\\}^{{\mathrm{\scriptscriptstyle
T}}}\widehat{\boldsymbol{\Xi}}_{-j,-k}(t,s)\\{\widehat{\boldsymbol{\beta}}_{kj}(v,s)-\boldsymbol{\beta}_{kj}(v,s)\\}\,{\rm
d}t{\rm d}s\bigg{]}^{2}\,{\rm d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}\lesssim\big{\|}\int_{{\cal
U}}\widehat{\boldsymbol{\Xi}}_{-j,-k}(\cdot,s)\\{\widehat{\boldsymbol{\beta}}_{kj}(\cdot,s)-\boldsymbol{\beta}_{kj}(\cdot,s)\\}{\rm
d}s\big{\|}_{{\cal
S},\max}^{2}\|\widehat{\boldsymbol{\beta}}_{jk}-\boldsymbol{\beta}_{jk}\|_{{\cal
S},1}^{2}$ $\displaystyle~{}~{}~{}~{}=O_{\rm
p}(\delta_{1n}^{2}\delta_{2n}^{2})\,.$

Since the convergence rate does not depend on $(j,k)$ in the above inequality,
then for any $1\leq j\neq k\leq p$,

$\displaystyle\int_{{\cal U}}\int_{{\cal
U}}\bigg{\\{}\frac{1}{n-1}\sum_{i=1}^{n}{\rm
IV}_{i}(j,k,u,v)\bigg{\\}}^{2}\,{\rm d}u{\rm d}v=O_{\rm
p}(\delta_{1n}^{2}\delta_{2n}^{2})\,.$

Hence,

$\delta_{1n}\delta_{2n}\asymp n^{-a_{2}}.$ (S.15)

Convergence rates of $\int_{{\cal U}}\int_{{\cal
U}}\\{(n-1)^{-1}\sum_{i=1}^{n}{\rm II}_{i}(j,k,u,v)\\}^{2}\,{\rm d}u{\rm d}v$
. Recall that $\varepsilon_{i,jk}(\cdot)$ and ${\bf Y}_{i,-j,-k}(\cdot)$ are
independent. By Theorem 2 in Fang et al. (2022), we obtain that
$\|(n-1)^{-1}\sum_{i=1}^{n}\tilde{\varepsilon}_{i,jk}({\bf
Y}_{i,-j,-k}-\widebar{\bf Y}_{-j,-k})\|_{{\cal S},\max}=O_{\rm p}(\sqrt{\log
p/n})$. This together with Inequality 1 and Proposition 2 yields that

$\displaystyle\int_{{\cal U}}\int_{{\cal
U}}\bigg{\\{}\frac{1}{n-1}\sum_{i=1}^{n}{\rm
II}_{i}(j,k,u,v)\bigg{\\}}^{2}\,{\rm d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}\leq\bigg{\|}\frac{1}{n-1}\sum_{i=1}^{n}\tilde{\varepsilon}_{i,jk}({\bf
Y}_{i,-j,-k}-\widebar{\bf Y}_{-j,-k})\bigg{\|}_{{\cal
S},\max}^{2}\|{\boldsymbol{\beta}}_{jk}-\boldsymbol{\beta}_{jk}\|_{{\cal
S},1}^{2}=O_{\rm p}(\delta_{1n}^{2}n^{-1}\log p)\,.$

Notice that the convergence rate does not depend on $(j,k)$ in the above
formula. Then for any $1\leq j\neq k\leq p$,

$\displaystyle\int_{{\cal U}}\int_{{\cal
U}}\bigg{\\{}\frac{1}{n-1}\sum_{i=1}^{n}{\rm
II}_{i}(j,k,u,v)\bigg{\\}}^{2}\,{\rm d}u{\rm d}v=O_{\rm
p}(\delta_{1n}^{2}n^{-1}\log p)\,.$

Analogously, we also have for any $1\leq j\neq k\leq p$,

$\displaystyle\int_{{\cal U}}\int_{{\cal
U}}\bigg{\\{}\frac{1}{n-1}\sum_{i=1}^{n}{\rm
III}_{i}(j,k,u,v)\bigg{\\}}^{2}\,{\rm d}u{\rm d}v=O_{\rm
p}(\delta_{1n}^{2}n^{-1}\log p)\,.$

Hence,

$\delta_{1n}\sqrt{\log p/n}\asymp n^{-a_{1}}.$ (S.16)

Recall
$\widetilde{T}_{n,jk}^{\varepsilon}=n\|\widetilde{\Sigma}_{jk}^{\varepsilon}\|_{{\cal
S}}^{2}$. By (A.6) and Cauchy–Schwarz inequality,

$\displaystyle\widetilde{T}_{n,jk}^{\varepsilon}=$
$\displaystyle~{}n\int_{{\cal U}}\int_{{\cal
U}}\bigg{[}\frac{1}{n-1}\sum_{i=1}^{n}\\{\varepsilon_{i,jk}(u)-\bar{\varepsilon}_{jk}(u)\\}\\{\varepsilon_{i,kj}(v)-\bar{\varepsilon}_{kj}(v)\\}\bigg{]}^{2}\,{\rm
d}u{\rm d}v$ $\displaystyle~{}+O_{\rm
p}\\{n\delta_{1n}^{2}(\delta_{2n}^{2}+n^{-1}\log p)\\}\,.$ (S.17)

Following the similar arguments as the proof of Theorem 2 with $X_{ij}(\cdot)$
and $X_{ik}(\cdot)$ replaced by $\varepsilon_{i,jk}(\cdot)$ and
$\varepsilon_{i,kj}(\cdot)$, respectively, we complete the proof of Theorem 5.
$\hfill\Box$

###  A.7 Proof of Proposition 2

The proof follows from Theorem 4 in Fang et al. (2022). Note that since ${\bf
Y}_{1}(\cdot),\dots,{\bf Y}_{n}(\cdot)$ are i.i.d. sampled from a multivariate
Gaussian process, the stability measure takes the value of $1$ and hence does
not show up in the final rate. $\hfill\Box$

###  A.8 Proof of Proposition 3

By Lemma B2, we have that

$\displaystyle\bigg{\|}\int_{\cal
U}\widehat{\boldsymbol{\Xi}}_{-j,-k}(\cdot,s)(\widehat{\boldsymbol{\beta}}_{kj}-\boldsymbol{\beta}_{kj})(\cdot,s)\,{\rm
d}s\bigg{\|}_{{\cal S},\max}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\bigg{\|}\int_{\cal
U}\\{\widehat{\boldsymbol{\Xi}}_{-j,-k}(\cdot,s)\widehat{\boldsymbol{\beta}}_{kj}(\cdot,s)-{\boldsymbol{\Xi}}_{-j,-k}(\cdot,s)\boldsymbol{\beta}_{kj}(\cdot,s)\\}\,{\rm
d}s\bigg{\|}_{{\cal
S},\max}+\|\widehat{\boldsymbol{\Xi}}_{-j,-k}-{\boldsymbol{\Xi}}_{-j,-k}\|_{{\cal
S},\max}\|\boldsymbol{\beta}_{kj}\|_{{\cal S},1}$
$\displaystyle~{}~{}~{}~{}~{}~{}\lesssim\bigg{\|}\int_{\cal
U}\\{\widehat{\boldsymbol{\Xi}}_{-j,-k}(\cdot,s)\widehat{\boldsymbol{\beta}}_{kj}(\cdot,s)-{\boldsymbol{\Xi}}_{-j,-k}(\cdot,s)\boldsymbol{\beta}_{kj}(\cdot,s)\\}\,{\rm
d}s\bigg{\|}_{{\cal S},\max}+O_{p}(s\sqrt{{\log p}/{n}}).$ (S.18)

Recall that
$\hat{\beta}_{kj,\ell}(v,s)=\sum_{m_{1},m_{2}\in[d]}\hat{b}_{kj,\ell
m_{1}m_{2}}\hat{\phi}_{km_{1}}(v)\hat{\phi}_{\ell m_{2}}(s)$ and for $\ell\in
S_{kj},$ $\beta_{kj,\ell}(v,s)=\sum_{m_{1},m_{2}=1}^{\infty}b_{kj,\ell
m_{1}m_{2}}\phi_{km_{1}}(v)\phi_{\ell m_{2}}(s).$ Rewriting this problem under
a FPCA framework leads to

$\displaystyle\int_{\cal
U}\\{\widehat{\Xi}^{(-j,-k)}_{\ell^{\prime}\ell}(t,s)\hat{\beta}_{kj,\ell}(v,s)-\Xi^{(-j,-k)}_{\ell^{\prime}\ell}(t,s)\beta_{kj,\ell}(v,s)\\}\,{\rm
d}s$ $\displaystyle~{}~{}~{}=\sum_{m_{1},m_{2}\in[d]}\bigg{[}\hat{b}_{kj,\ell
m_{1}m_{2}}\hat{\phi}_{km_{1}}(v)\bigg{\\{}\sum_{m_{2}^{\prime}=1}^{\infty}\bigg{(}\frac{\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}\hat{\xi}_{i\ell
m_{2}}}{n}\bigg{)}\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-b_{kj,\ell
m_{1}m_{2}}\phi_{km_{1}}(v)\bigg{\\{}\sum_{m_{2}^{\prime}=1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\}}\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\underbrace{\sum_{m_{1}=d+1}^{\infty}\sum_{m_{2}\in[d]}b_{kj,\ell
m_{1}m_{2}}\phi_{km_{1}}(v)\bigg{\\{}\sum_{m_{2}^{\prime}=1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\}}}_{R_{1,kj\ell^{\prime}\ell}(t,v)}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\underbrace{\sum_{m_{1}\in[d]}\sum_{m_{2}=d+1}^{\infty}b_{kj,\ell
m_{1}m_{2}}\phi_{km_{1}}(v)\bigg{\\{}\sum_{m_{2}^{\prime}=1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\}}}_{R_{2,kj\ell^{\prime}\ell}(t,v)}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\underbrace{\sum_{m_{1}=d+1}^{\infty}\sum_{m_{2}=d+1}^{\infty}b_{kj,\ell
m_{1}m_{2}}\phi_{km_{1}}(v)\bigg{\\{}\sum_{m_{2}^{\prime}=1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\}}}_{R_{3,kj\ell^{\prime}\ell}(t,v)}$
$\displaystyle=\underbrace{\sum_{m_{1},m_{2}\in[d]}\bigg{[}(\hat{b}_{kj,\ell
m_{1}m_{2}}-b_{kj,\ell
m_{1}m_{2}})\hat{\phi}_{km_{1}}(v)\bigg{\\{}\sum_{m_{2}^{\prime}=1}^{\infty}\bigg{(}\frac{\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}\hat{\xi}_{i\ell
m_{2}}}{n}\bigg{)}\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\}}\bigg{]}}_{I_{1,kj\ell^{\prime}\ell}(t,v)}$
$\displaystyle~{}~{}~{}+\underbrace{\sum_{m_{1},m_{2}\in[d]}\bigg{(}b_{kj,\ell
m_{1}m_{2}}\hat{\phi}_{km_{1}}(v)\bigg{[}\sum_{m_{2}^{\prime}=1}^{\infty}\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\{}\frac{\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}\hat{\xi}_{i\ell
m_{2}}}{n}-\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\bigg{\\}}\bigg{]}\bigg{)}}_{I_{2,kj\ell^{\prime}\ell}(t,v)}$
$\displaystyle~{}~{}~{}+\underbrace{\sum_{m_{1},m_{2}\in[d]}\bigg{[}b_{kj,\ell
m_{1}m_{2}}\\{\hat{\phi}_{km_{1}}(v)-\phi_{km_{1}}(v)\\}\bigg{\\{}\sum_{m_{2}^{\prime}=1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\}}\bigg{]}}_{I_{3,kj\ell^{\prime}\ell}(t,v)}-{R_{kj\ell^{\prime}\ell}(t,v)},$

where
$R_{kj\ell^{\prime}\ell}(t,v)=\sum_{r\in[3]}R_{r,kj\ell^{\prime}\ell}(t,v)$.
Hence, we obtain that

$\displaystyle\bigg{\|}\int_{\cal
U}\\{\widehat{\boldsymbol{\Xi}}_{-j,-k}(\cdot,s)\widehat{\boldsymbol{\beta}}_{kj}(\cdot,s)-{\boldsymbol{\Xi}}_{-j,-k}(\cdot,s)\boldsymbol{\beta}_{kj}(\cdot,s)\\}\,{\rm
d}s\bigg{\|}_{{\cal S},\max}$
$\displaystyle~{}~{}~{}=\max_{\ell^{\prime}\in[p-2]}\bigg{\|}\sum_{\ell\in[p-2]}\int_{\cal
U}\\{\widehat{\Xi}^{(-j,-k)}_{\ell^{\prime}\ell}(\cdot,s)\hat{\beta}_{kj,\ell}(\cdot,s)-\Xi^{(-j,-k)}_{\ell^{\prime}\ell}(\cdot,s)\beta_{kj,\ell}(\cdot,s)\\}\,{\rm
d}s\bigg{\|}_{\cal S}$
$\displaystyle~{}~{}~{}\leq\max_{\ell^{\prime}\in[p-2]}\big{\|}\sum_{\ell\in[p-2]}I_{1,kj\ell^{\prime}\ell}\big{\|}_{\cal
S}+s\cdot\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|I_{2,kj\ell^{\prime}\ell}\|_{\cal
S}+s\cdot\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|I_{3,kj\ell^{\prime}\ell}\|_{\cal
S}+s\cdot\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|R_{kj\ell^{\prime}\ell}\|_{\cal
S}.$

Let $\omega_{0}=\max_{\ell\in[p]}\int_{\cal U}\Xi_{\ell\ell}(u,u){\rm d}u.$ By
Cauchy–Schwarz inequality, the orthonormality of $\phi_{jm}(\cdot)$ for
$j\in[p]$ and $m\in[d]$ and Condition 12, we obtain that

$\displaystyle\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|R_{1,kj\ell^{\prime}\ell}\|_{\cal
S}^{2}=\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\bigg{\|}\sum_{m_{1}=d+1}^{\infty}\sum_{m_{2}\in[d]}b_{kj,\ell
m_{1}m_{2}}\phi_{km_{1}}\bigg{\\{}\sum_{m_{2}^{\prime}=1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\phi_{\ell^{\prime}m_{2}^{\prime}}\bigg{\\}}\bigg{\|}_{\cal S}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}=\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\sum_{m_{2}\in[d]}\sum_{m_{2}^{\prime\prime}\in[d]}\sum_{m_{1}=d+1}^{\infty}b_{kj,\ell
m_{1}m_{2}}b_{kj,\ell
m_{1}m_{2}^{\prime\prime}}\sum_{m_{2}^{\prime}=1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}^{\prime\prime}})$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\sum_{m_{2}\in[d]}\sum_{m_{2}^{\prime\prime}\in[d]}\sum_{m_{1}=d+1}^{\infty}b_{kj,\ell
m_{1}m_{2}}b_{kj,\ell
m_{1}m_{2}^{\prime\prime}}\sum_{m_{2}^{\prime}=1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}^{2})\sqrt{\mathbb{E}(\xi_{i\ell^{\prime}m_{2}}^{2})\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime\prime}}^{2})}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\omega_{0}\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\sqrt{\sum_{m_{2}\in[d]}\sum_{m_{2}^{\prime\prime}\in[d]}\bigg{(}\sum_{m_{1}=d+1}^{\infty}b_{kj,\ell
m_{1}m_{2}}b_{kj,\ell
m_{1}m_{2}^{\prime\prime}}\bigg{)}^{2}}\sqrt{\sum_{m_{2}\in[d]}\sum_{m_{2}^{\prime\prime}\in[d]}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}}^{2})\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime\prime}}^{2})}$
$\displaystyle~{}~{}~{}~{}~{}~{}\lesssim\max_{\ell\in[p-2]}\sum_{m_{2}\in[d]}\sum_{m_{1}=d+1}^{\infty}b_{kj,\ell
m_{1}m_{2}}^{2}\lesssim\sum_{m_{2}\in[d]}\sum_{m_{1}=d+1}^{\infty}(m_{1}+m_{2})^{-2\upsilon-1}=O(d^{-2\upsilon+1}).$

Following the same argument, we can also have
$\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|R_{kj\ell^{\prime}\ell}\|_{\cal
S}=O(d^{-\upsilon+1/2}).$ We next focus on the terms
$I_{r,kj\ell^{\prime}\ell}$ for $r\in[3]$. By the orthonormality of
$\phi_{jm}(\cdot)$ for $j\in[p]$ and $m\in[d]$ and Condition 12,

$\displaystyle\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|I_{3,kj\ell^{\prime}\ell}\|_{\cal
S}^{2}=\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\bigg{\|}\sum_{m_{1},m_{2}\in[d]}\bigg{[}b_{kj,\ell
m_{1}m_{2}}\\{\hat{\phi}_{km_{1}}-\phi_{km_{1}}\\}\bigg{\\{}\sum_{m_{2}^{\prime}=1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\phi_{\ell^{\prime}m_{2}^{\prime}}\bigg{\\}}\bigg{]}\bigg{\|}_{\cal
S}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}=\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\sum_{m_{2}^{\prime}=1}^{\infty}\int_{\cal
U}\bigg{[}\sum_{m_{1}\in[d]}\\{\hat{\phi}_{km_{1}}(v)-\phi_{km_{1}}(v)\\}\cdot\bigg{\\{}\sum_{m_{2}\in[d]}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})b_{kj,\ell m_{1}m_{2}}\bigg{\\}}\bigg{]}^{2}{\rm d}v$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq
d\max_{m_{1}\in[d]}\|\hat{\phi}_{km_{1}}-\phi_{km_{1}}\|^{2}\cdot\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\sum_{m_{2}^{\prime}=1}^{\infty}\sum_{m_{1}\in[d]}\bigg{\\{}\sum_{m_{2}\in[d]}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})b_{kj,\ell m_{1}m_{2}}\bigg{\\}}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}\lesssim
d\max_{m_{1}\in[d]}\|\hat{\phi}_{km_{1}}-\phi_{km_{1}}\|^{2}.$

This, together with Lemma B2, implies that
$\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|I_{3,kj\ell^{\prime}\ell}\|_{\cal
S}=O_{p}(d^{\varpi+3/2}\sqrt{{\log d}/{n}}).$ Recall that
$X_{i\ell}(\cdot)=\sum_{m=1}^{\infty}\xi_{i\ell m}\phi_{\ell m}(\cdot),$ where
the coefficients $\xi_{i\ell m}=\langle X_{i\ell},\phi_{\ell m}\rangle,$ and
the corresponding estimated FPC scores $\hat{\xi}_{i\ell m}=\langle
X_{i\ell},\hat{\phi}_{\ell m}\rangle.$ By the orthonormality of
$\hat{\phi}_{jm}(\cdot)$ for $j\in[p]$ and $m\in[d],$ Condition 12 and the
fact that $\|\Xi_{\ell^{\prime}\ell}\|_{{\cal S}}\leq\omega_{0},$ we obtain
that

$\displaystyle\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|I_{2,kj\ell^{\prime}\ell}\|_{\cal
S}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}=\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\bigg{\|}\sum_{m_{1},m_{2}\in[d]}\bigg{(}b_{kj,\ell
m_{1}m_{2}}\hat{\phi}_{km_{1}}\bigg{[}\sum_{m_{2}^{\prime}=1}^{\infty}\phi_{\ell^{\prime}m_{2}^{\prime}}\bigg{\\{}\frac{\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}\hat{\xi}_{i\ell
m_{2}}}{n}-\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}\xi_{i\ell
m_{2}})\bigg{\\}}\bigg{]}\bigg{)}\bigg{\|}_{\cal S}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}=\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\sum_{m_{1}\in[d]}\bigg{\|}\sum_{m_{2}\in[d]}\bigg{(}b_{kj,\ell
m_{1}m_{2}}\cdot\bigg{[}\int_{\cal
U}\big{\\{}\widehat{\Xi}_{\ell^{\prime}\ell}(\cdot,s)\hat{\phi}_{\ell
m_{2}}(s)-\Xi_{\ell^{\prime}\ell}(\cdot,s)\phi_{\ell m_{2}}(s)\big{\\}}{\rm
d}s\bigg{]}\bigg{)}\bigg{\|}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}\lesssim\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\sum_{m_{2}\in[d]}\int_{\cal
U}\bigg{[}\int_{\cal
U}\big{\\{}\widehat{\Xi}_{\ell^{\prime}\ell}(\cdot,s)\hat{\phi}_{\ell
m_{2}}(s)-\Xi_{\ell^{\prime}\ell}(\cdot,s)\phi_{\ell m_{2}}(s)\big{\\}}{\rm
d}s\bigg{]}^{2}{\rm d}t$ $\displaystyle~{}~{}~{}~{}~{}~{}\lesssim
d\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|\widehat{\Xi}_{\ell^{\prime}\ell}-\Xi_{\ell^{\prime}\ell}\|_{\cal
S}^{2}+d\max_{\ell\in[p-2],\,m_{2}\in[d]}\|\hat{\phi}_{\ell m_{2}}-\phi_{\ell
m_{2}}\|^{2}.$

Then by Lemma B2, we have that
$\max_{(\ell^{\prime},\ell)\in[p-2]^{2}}\|I_{2,kj\ell^{\prime}\ell}\|_{\cal
S}=O_{p}(d^{\varpi+3/2}\sqrt{{\log(pd)}/{n}}).$ We now focus on the term
$I_{1,kj\ell^{\prime}\ell}(t,v)$. We further decompose
$I_{1,kj\ell^{\prime}\ell}(t,v)$ as

$\displaystyle I_{1,kj\ell^{\prime}\ell}(t,v)$
$\displaystyle=\underbrace{\sum_{m_{1},m_{2}\in[d]}\bigg{[}(\hat{b}_{kj,\ell
m_{1}m_{2}}-b_{kj,\ell
m_{1}m_{2}})\hat{\phi}_{km_{1}}(v)\bigg{\\{}\sum_{m_{2}^{\prime}\in[d]}\bigg{(}\frac{\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}\hat{\xi}_{i\ell
m_{2}}}{n}\bigg{)}\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\}}\bigg{]}}_{I_{1,kj\ell^{\prime}\ell}^{(1)}(t,v)}$
$\displaystyle~{}~{}+\underbrace{\sum_{m_{1},m_{2}\in[d]}\bigg{[}(\hat{b}_{kj,\ell
m_{1}m_{2}}-b_{kj,\ell
m_{1}m_{2}})\hat{\phi}_{km_{1}}(v)\bigg{\\{}\sum_{m_{2}^{\prime}=d+1}^{\infty}\bigg{(}\frac{\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}\hat{\xi}_{i\ell
m_{2}}}{n}\bigg{)}\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\bigg{\\}}\bigg{]}}_{I_{1,kj\ell^{\prime}\ell}^{(2)}(t,v)}.$

We next introduce some matrix forms for the convenience of theoretical
development regarding $I_{1,jk\ell^{\prime}\ell}^{(1)}$. For a block matrix
${\bf B}=({\bf B}_{jk})_{j,k\in[p]}\in\mathbb{R}^{pd\times pd}$ with the
$(j,k)$th block ${\bf B}_{jk}\in{\mathbb{R}}^{d\times d},$ we define its
$d$-block versions of elementwise $\ell_{\infty}$ and matrix $\ell_{1}$ norms
by $\|{\bf B}\|_{\max}^{(d)}=\max_{j,k}\|{\bf B}_{jk}\|_{\text{F}}$ and
$\|{\bf B}\|_{1}^{(d)}={\max}_{k}\sum_{j}\|{\bf B}_{jk}\|_{\text{F}},$
respectively. Define ${\bf
V}_{j}=(\boldsymbol{\xi}_{1j},\dots,\boldsymbol{\xi}_{nj})^{{\mathrm{\scriptscriptstyle
T}}}\in\mathbb{R}^{n\times d},$ ${\bf V}=({\bf V}_{1},\dots,{\bf
V}_{p})\in\mathbb{R}^{n\times pd}$, and
$\boldsymbol{\Psi}_{jk}=(\boldsymbol{\Psi}_{jk,\ell}^{{\mathrm{\scriptscriptstyle
T}}})_{\ell\in[p]\setminus\\{j,k\\}}^{{\mathrm{\scriptscriptstyle
T}}}\in\mathbb{R}^{(p-2)d\times d}$. We can rewrite (14) as

$\displaystyle{\bf V}_{j}={\bf V}_{-j,-k}\boldsymbol{\Psi}_{jk}+{\bf
R}_{jk}+{\bf E}_{jk},$ (S.19)

where ${\bf V}_{-j,-k}$ denotes the submatrix of ${\bf V}$ by removing the
$j$th and $k$th blocks, namely ${\bf V}_{j}$ and ${\bf V}_{k}$. Note ${\bf
R}_{jk}$ and ${\bf E}_{jk}$ are $n\times d$ matrices whose row vectors are
formed by truncation errors
$\\{{\mathbf{r}}_{i,jk}\in\mathbb{R}^{d}:i\in[n]\\}$ and random errors
$\\{\boldsymbol{\epsilon}_{i,jk}\in\mathbb{R}^{d}:i\in[n]\\}$, where
${\mathbf{r}}_{i,jk}=(r_{i1,jk},\dots,r_{id,jk})^{{\mathrm{\scriptscriptstyle
T}}}$ with
$r_{im_{1},jk}=\sum_{\ell\in[p]\setminus\\{j,k\\}}\sum_{m_{2}=d+1}^{\infty}\langle\langle\phi_{jm_{1}},\beta_{jk,\ell}\rangle,\phi_{\ell
m_{2}}\rangle\xi_{i\ell m_{2}}$ and
$\boldsymbol{\epsilon}_{i,jk}=(\epsilon_{i1,jk},\dots,\epsilon_{id,jk})^{{\mathrm{\scriptscriptstyle
T}}}$ with
$\epsilon_{im_{1},jk}=\langle\phi_{jm_{1}},\varepsilon_{i,jk}\rangle$ for
$m_{1}\in[d]$, respectively.

Let $\widehat{{\bf Q}}_{\ell}=\\{(n)^{-1}\widehat{{\bf
V}}_{\ell}^{{\mathrm{\scriptscriptstyle T}}}\widehat{{\bf
V}}_{\ell}\\}^{1/2}\in\mathbb{R}^{d\times d}$ with $\widehat{\bf
V}_{\ell}=(\widehat{\boldsymbol{\xi}}_{1\ell},\dots,\widehat{\boldsymbol{\xi}}_{n\ell})^{{\mathrm{\scriptscriptstyle
T}}}\in\mathbb{R}^{n\times d}$ for $\ell\in[p].$ Define $\widehat{\bf
V}=(\widehat{\bf V}_{1},\dots,\widehat{\bf V}_{p})\in\mathbb{R}^{n\times pd}$
and $\widehat{{\bf D}}=\text{diag}(\widehat{{\bf Q}}_{1},\dots,\widehat{{\bf
Q}}_{p})\in\mathbb{R}^{pd\times pd}$. Let $\widehat{\bf V}_{-j,-k}$ denotes
the submatrix of $\widehat{\bf V}$ by removing the $j$th and $k$th blocks,
namely $\widehat{\bf V}_{j}$ and $\widehat{\bf V}_{k}$ and $\widehat{{\bf
D}}_{-j,-k}$ denotes the submatrix of $\widehat{{\bf D}}$ by removing the
$j$th and $k$th diagonal blocks, namly $\widehat{{\bf Q}}_{j}$ and
$\widehat{{\bf Q}}_{k}$.
$\\{\widehat{\boldsymbol{\Psi}}_{jk,\ell}\\}_{\ell\in[p]\setminus\\{j,k\\}}$
can be equivalently obtained by computing
$\widehat{\boldsymbol{\Psi}}_{jk}=\widehat{{\bf D}}_{-j,-k}^{-1}\widehat{{\bf
B}}_{jk}=(\widehat{\boldsymbol{\Psi}}_{jk,\ell})_{\ell\in[p]\setminus\\{j,k\\}},$
where

$\widehat{\bf B}_{jk}=\underset{{\bf B}_{jk}\in\mathbb{R}^{(p-2)d\times
d}}{\text{arg min}}\left\\{\frac{1}{2n}\|\widehat{{\bf V}}_{j}-\widehat{{\bf
V}}_{-j,-k}\widehat{{\bf D}}_{-j,-k}^{-1}{\bf
B}_{jk}\|_{\text{F}}^{2}+\tau_{n}\|{\bf B}_{jk}\|^{(d)}_{1}\right\\}.$

Let ${{\bf D}}=\text{diag}({{\bf Q}}_{1},\dots,{{\bf
Q}}_{p})\in\mathbb{R}^{pd\times pd}$ with ${{\bf Q}}_{j}=\\{(n)^{-1}{{\bf
V}}_{j}^{{\mathrm{\scriptscriptstyle T}}}{{\bf
V}}_{j}\\}^{1/2}\in\mathbb{R}^{d\times d}$ for $j\in[p]$ and ${{\bf
B}}_{jk}={{\bf D}}_{-j,-k}{\boldsymbol{\Psi}}_{jk},$ where ${{\bf D}}_{-j,-k}$
denotes the submatrix of ${{\bf D}}$ by removing the $j$th and $k$th diagonal
blocks, namely ${{\bf Q}}_{j}$ and ${{\bf Q}}_{k}$. Notice that

$\displaystyle\max_{\ell^{\prime}\in[p-2]}\big{\|}\sum_{\ell\in[p-2]}I_{1,kj\ell^{\prime}\ell}^{(1)}\big{\|}_{\cal
S}=n^{-1}\|{\bf D}_{-k,-j}{\bf D}_{-k,-j}^{-1}{\bf
V}_{-k,-j}^{{\mathrm{\scriptscriptstyle T}}}\widehat{\bf
V}_{-k,-j}(\widehat{\boldsymbol{\Psi}}_{kj}-\boldsymbol{\Psi}_{kj})\|_{\max}^{(d)}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq n^{-1}\|{\bf D}_{-k,-j}\|_{\max}\|{\bf
D}_{-k,-j}^{-1}{\bf V}_{-k,-j}^{{\mathrm{\scriptscriptstyle T}}}\widehat{\bf
V}_{-k,-j}(\widehat{\boldsymbol{\Psi}}_{kj}-\boldsymbol{\Psi}_{kj})\|_{\max}^{(d)}$
$\displaystyle~{}~{}~{}~{}~{}~{}\lesssim n^{-1}\|\widehat{\bf
D}_{-k,-j}^{-1}\widehat{\bf V}_{-k,-j}^{{\mathrm{\scriptscriptstyle
T}}}\widehat{\bf V}_{-k,-j}\widehat{\bf D}_{-k,-j}^{-1}(\widehat{\bf
B}_{kj}-{\bf B}_{kj})\|_{\max}^{(d)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\|{\bf D}_{-k,-j}^{-1}{\bf
V}_{-k,-j}^{{\mathrm{\scriptscriptstyle T}}}\widehat{\bf
V}_{-k,-j}\widehat{\bf D}_{-k,-j}^{-1}\|_{\max}^{(d)}\|{\bf
D}_{-k,-j}-\widehat{\bf
D}_{-k,-j}\|_{\max}\|\boldsymbol{\Psi}_{kj}\|_{1}^{(d)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+n^{-1}\|{\bf D}_{-k,-j}^{-1}{\bf
V}_{-k,-j}^{{\mathrm{\scriptscriptstyle T}}}\widehat{\bf
V}_{-k,-j}\widehat{\bf D}_{-k,-j}^{-1}-\widehat{\bf
D}_{-k,-j}^{-1}\widehat{\bf V}_{-k,-j}^{{\mathrm{\scriptscriptstyle
T}}}\widehat{\bf V}_{-k,-j}\widehat{\bf
D}_{-k,-j}^{-1}\|_{\max}^{(d)}\|\widehat{\bf B}_{kj}-{\bf
B}_{kj}\|_{1}^{(d)}.$

By the KKT condition, we have $n^{-1}\|\widehat{{\bf
D}}_{-k,-j}^{-1}\widehat{{\bf V}}_{-k,-j}^{{\mathrm{\scriptscriptstyle
T}}}(\widehat{{\bf V}}_{k}-\widehat{{\bf V}}_{-k,-j}\widehat{{\bf
D}}_{-k,-j}^{-1}\widehat{\bf B}_{kj})\|_{\max}^{(d)}\leq\tau_{n}$. This,
together with Lemma B5 implies that $n^{-1}\|\widehat{\bf
D}_{-k,-j}^{-1}\widehat{\bf V}_{-k,-j}^{{\mathrm{\scriptscriptstyle
T}}}\widehat{\bf V}_{-k,-j}\widehat{\bf D}_{-k,-j}^{-1}(\widehat{\bf
B}_{kj}-{\bf B}_{kj})\|_{\max}^{(d)}\leq 3\tau_{n}/2$. Then, by Lemmas B2–B4
and the fact that $\|\boldsymbol{\Psi}_{kj}\|_{1}^{(d)}=O(s)$, we obtain that
$\max_{\ell^{\prime}\in[p-2]}\|\sum_{\ell\in[p-2]}I_{1,kj\ell^{\prime}\ell}^{(1)}\|_{\cal
S}=O_{p}(\tau_{n})$. Furthermore, by Cauchy–Schwarz inequality,

$\displaystyle\sum_{m_{2}^{\prime}=d+1}^{\infty}\bigg{(}\frac{\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}\hat{\xi}_{i\ell
m_{2}}}{n}\bigg{)}\phi_{\ell^{\prime}m_{2}^{\prime}}(t)=\frac{\sum_{i=1}^{n}\\{\hat{\xi}_{i\ell
m_{2}}(\sum_{m_{2}^{\prime}=d+1}^{\infty}\xi_{i\ell^{\prime}m_{2}^{\prime}}\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\\}}{n}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\sqrt{\hat{\omega}_{\ell
m_{2}}}\sqrt{\frac{\sum_{i=1}^{n}\\{\sum_{m_{2}^{\prime}=d+1}^{\infty}\xi_{i\ell^{\prime}m_{2}^{\prime}}\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\\}^{2}}{n}}.$

Combining this with the orthonormality of $\hat{\phi}_{jm}(\cdot)$ for
$j\in[p]$ and $m\in[d]$ thus leads to

$\displaystyle\max_{\ell^{\prime}\in[p-2]}\big{\|}I_{1,kj\ell^{\prime}\ell}^{(2)}\big{\|}_{\cal
S}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}=\max_{\ell^{\prime}\in[p-2]}\bigg{\|}\sum_{m_{1},m_{2}\in[d]}\bigg{[}(\hat{b}_{kj,\ell
m_{1}m_{2}}-b_{kj,\ell
m_{1}m_{2}})\hat{\phi}_{km_{1}}\bigg{\\{}\sum_{m_{2}^{\prime}=d+1}^{\infty}\bigg{(}\frac{\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}\hat{\xi}_{i\ell
m_{2}}}{n}\bigg{)}\phi_{\ell^{\prime}m_{2}^{\prime}}\bigg{\\}}\bigg{]}\bigg{\|}_{\cal
S}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\max_{\ell^{\prime}\in[p-2]}\sum_{m_{1}\in[d]}\bigg{\|}\sum_{m_{2}\in[d]}\bigg{\\{}(\hat{b}_{kj,\ell
m_{1}m_{2}}-b_{kj,\ell m_{1}m_{2}})\cdot\sqrt{\hat{\omega}_{\ell
m_{2}}}\sqrt{\frac{\sum_{i=1}^{n}\\{\sum_{m_{2}^{\prime}=d+1}^{\infty}\xi_{i\ell^{\prime}m_{2}^{\prime}}\phi_{\ell^{\prime}m_{2}^{\prime}}(t)\\}^{2}}{n}}\bigg{\\}}\bigg{\|}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq
d\sum_{m_{1},m_{2}\in[d]}\bigg{\\{}(\hat{b}_{kj,\ell m_{1}m_{2}}-b_{kj,\ell
m_{1}m_{2}})\sqrt{\hat{\omega}_{\ell
m_{2}}}\bigg{\\}}^{2}\cdot\max_{\ell^{\prime}\in[p-2]}\frac{\sum_{m_{2}^{\prime}=d+1}^{\infty}\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}^{2}}{n}$

By the fact that
${n^{-1}\sum_{i=1}^{n}\xi_{i\ell^{\prime}m_{2}^{\prime}}^{2}}-\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}^{2})=\langle\phi_{\ell^{\prime}m_{2}^{\prime}},\langle\widehat{\Xi}_{\ell^{\prime}\ell^{\prime}}-\Xi_{\ell^{\prime}\ell^{\prime}},\phi_{\ell^{\prime}m_{2}^{\prime}}\rangle\rangle\leq\|\widehat{\Xi}_{\ell^{\prime}\ell^{\prime}}-\Xi_{\ell^{\prime}\ell^{\prime}}\|_{\cal
S}=O_{p}\\{n^{-1/2}\\}$,
$\sum_{m_{2}^{\prime}=d+1}^{\infty}\mathbb{E}(\xi_{i\ell^{\prime}m_{2}^{\prime}}^{2})=O(d^{-\varpi+1})$
and Lemma B3, we thus have
$\max_{\ell^{\prime}\in[p-2]}\big{\|}\sum_{\ell\in[p-2]}I_{1,kj\ell^{\prime}\ell}^{(2)}\big{\|}_{\cal
S}=O_{p}(d^{-\varpi/2+1}s\tau_{n}/\underline{\mu})$. Combining the above
results, we obtain that

$\bigg{\|}\int_{\cal
U}\widehat{\boldsymbol{\Xi}}_{-j,-k}(\cdot,s)(\widehat{\boldsymbol{\beta}}_{kj}-\boldsymbol{\beta}_{kj})(\cdot,s)\,{\rm
d}s\bigg{\|}_{{\cal
S},\max}=O_{p}(\tau_{n})+O_{p}(d^{-\varpi/2+1}s\tau_{n}/\underline{\mu}).$

##  Appendix B Additional technical proofs

###  B.1 Proof of Lemma A1

For each $j\in[p]$, let
$Z_{ij}(\cdot)=\sum_{m=1}^{\infty}\vartheta_{ijm}\psi_{jm}(\cdot),$ where the
coefficients $\vartheta_{ijm}=\langle Z_{ij},\psi_{jm}\rangle,$ namely FPC
scores, are uncorrelated random variables with mean zero and
$\text{Cov}(\vartheta_{ijm},\vartheta_{ijm^{\prime}})=\varrho_{jm}I(m=m^{\prime}).$
Recall that $q\in\mathcal{H}_{0}$, we have $\|\Sigma_{j_{q}k_{q}}\|_{{\cal
S}}=0$, implying that
$\text{Cov}(\vartheta_{ij_{q}m},\vartheta_{ik_{q}m^{\prime}})=0$ for any
$m,m^{\prime}\geq 1$. We thus obtain that

$\displaystyle\Gamma_{j_{q}k_{q}}(u_{1},v_{1},u_{2},v_{2})$
$\displaystyle~{}~{}~{}=\text{Cov}\\{Z_{ij_{q}}(u_{1})Z_{ik_{q}}(v_{1}),Z_{ij_{q}}(u_{2})Z_{ik_{q}}(v_{2})\\}$
$\displaystyle~{}~{}~{}=\text{Cov}\bigg{\\{}\sum_{m_{1}=1}^{\infty}\sum_{m_{2}=1}^{\infty}\vartheta_{ij_{q}m_{1}}\vartheta_{ik_{q}m_{2}}\psi_{j_{q}m_{1}}(u_{1})\psi_{k_{q}m_{2}}(v_{1}),\sum_{m_{3}=1}^{\infty}\sum_{m_{4}=1}^{\infty}\vartheta_{ij_{q}m_{3}}\vartheta_{ik_{q}m_{4}}\psi_{j_{q}m_{3}}(u_{2})\psi_{k_{q}m_{4}}(v_{2})\bigg{\\}}$
$\displaystyle~{}~{}~{}=\sum_{m_{1}=1}^{\infty}\sum_{m_{2}=1}^{\infty}\text{Cov}\\{\vartheta_{ij_{q}m_{1}}\vartheta_{ik_{q}m_{2}},\vartheta_{ij_{q}m_{1}}\vartheta_{ik_{q}m_{2}}\\}\psi_{j_{q}m_{1}}(u_{1})\psi_{k_{q}m_{2}}(v_{1})\psi_{j_{q}m_{1}}(u_{2})\psi_{k_{q}m_{2}}(v_{2})$
$\displaystyle~{}~{}~{}=\sum_{m_{1}=1}^{\infty}\sum_{m_{2}=1}^{\infty}\varrho_{j_{q}m_{1}}\varrho_{k_{q}m_{2}}\psi_{j_{q}m_{1}}(u_{1})\psi_{k_{q}m_{2}}(v_{1})\psi_{j_{q}m_{1}}(u_{2})\psi_{k_{q}m_{2}}(v_{2}),$
(S.20)

for $q\in\mathcal{H}_{0}$. Since
$\iint\psi_{j_{q}m_{1}}(u)\psi_{k_{q}m_{2}}(v)\psi_{j_{q}m_{3}}(u)\psi_{k_{q}m_{4}}(v){\rm
d}u{\rm d}v=I(m_{1}=m_{3})I(m_{2}=m_{4})$,
$\\{\psi_{j_{q}m_{1}}\otimes\psi_{k_{q}m_{2}}\\}_{m_{1},m_{2}=1}^{\infty}$
follows an orthonormal basis of $\mathbb{S}$. Hence, (S.20) gives the spectral
decomposition of $\Gamma_{j_{q}k_{q}}(\cdot,\cdot,\cdot,\cdot)$. Then for each
$q\in\mathcal{H}_{0}$, there always exist a bijection
$\tilde{\pi}_{q}:\mathbb{N}_{+}\times\mathbb{N}_{+}\to\mathbb{N}_{+}$ such
that
$a_{i\tilde{\pi}_{q}(m_{1},m_{2})}^{(q)}=\vartheta_{ij_{q}m_{1}}\vartheta_{ik_{q}m_{2}},$
$\lambda_{\tilde{\pi}_{q}(m_{1},m_{2})}^{(q)}=\varrho_{j_{q}m_{1}}\varrho_{k_{q}m_{2}},$
and
$\phi_{j_{q}k_{q}\tilde{\pi}_{q}(m_{1},m_{2})}^{(q)}(u,v)=\psi_{j_{q}m_{1}}(u)\psi_{k_{q}m_{2}}(v).$

This, together with Condition 3 and Lemma 4 in Fang et al. (2022), implies
that there exist a constant $\tilde{M}$ such that
$\mathbb{E}\\{\exp(\tilde{M}\vartheta_{ijm}^{2}/{\varrho_{jm}})\\}\leq 2$ for
$j\in[p]$ and $m\geq 1$. By Young’s inequality, there exists a constant $M$,
for any $m_{1},m_{2}\geq 1$,
$\mathbb{E}\exp[Ma_{i\tilde{\pi}_{q}(m_{1},m_{2})}^{(q)}/\\{\lambda_{\tilde{\pi}_{q}(m_{1},m_{2})}^{(q)}\\}^{1/2}]\leq\mathbb{E}\\{\exp(\tilde{M}\vartheta_{ij_{q}m_{1}}^{2}/{\varrho_{j_{q}m_{1}}})+\exp(\tilde{M}\vartheta_{ik_{q}m_{2}}^{2}/{\varrho_{k_{q}m_{2}}})\\}/2\leq
2$. Hence, the proof is completed. $\hfill\Box$

###  B.2 Inequality 1 and its proof

###### Inequality 1

Let $\boldsymbol{{\cal F}}=({\cal F}_{1},\ldots,{\cal
F}_{p})^{{\mathrm{\scriptscriptstyle T}}}$ with each ${\cal
F}_{i}\in\mathbb{S}$ and $\boldsymbol{{\cal G}}=({\cal G}_{1},\ldots,{\cal
G}_{p})^{{\mathrm{\scriptscriptstyle T}}}$ with each ${\cal
G}_{i}\in\mathbb{S}$. Then $\iint\\{\int\boldsymbol{{\cal
F}}(u,s)^{{\mathrm{\scriptscriptstyle T}}}\boldsymbol{{\cal G}}(v,s)\,{\rm
d}s\\}^{2}{\rm d}u{\rm d}v\leq\|\boldsymbol{{\cal F}}\|_{{\cal
S},\max}^{2}\|\boldsymbol{{\cal G}}\|_{{\cal S},1}^{2}$.

Proof. By Cauchy–Schwarz inequality, we obtain that

$\displaystyle\iint\bigg{\\{}\int\boldsymbol{{\cal
F}}(u,s)^{{\mathrm{\scriptscriptstyle T}}}\boldsymbol{{\cal G}}(v,s)\,{\rm
d}s\bigg{\\}}^{2}{\rm d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}~{}~{}=\iint\bigg{\\{}\sum_{i=1}^{p}\int{\cal
F}_{i}(u,s){\cal G}_{i}(v,s)\,{\rm d}s\bigg{\\}}^{2}{\rm d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}~{}~{}=\iint{\sum_{i,j=1}^{p}\iint{\cal
F}_{i}(u,s_{1}){\cal G}_{i}(v,s_{1}){\cal F}_{j}(u,s_{2}){\cal
G}_{j}(v,s_{2})\,{\rm d}s_{1}{\rm d}s_{2}}{\rm d}u{\rm d}v$
$\displaystyle~{}~{}~{}~{}~{}~{}=\sum_{i,j=1}^{p}\iint\bigg{\\{}\int{\cal
F}_{i}(u,s_{1}){\cal F}_{j}(u,s_{2})\,{\rm d}u\bigg{\\}}\bigg{\\{}\int{\cal
G}_{i}(v,s_{1}){\cal G}_{j}(v,s_{2})\,{\rm d}v\bigg{\\}}\,{\rm d}s_{1}{\rm
d}s_{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\sum_{i,j=1}^{p}\bigg{[}\iint\bigg{\\{}\int{\cal
F}_{i}(u,s_{1}){\cal F}_{j}(u,s_{2})\,{\rm d}u\bigg{\\}}^{2}\,{\rm d}s_{1}{\rm
d}s_{2}\iint\bigg{\\{}\int{\cal G}_{i}(v,s_{1}){\cal G}_{j}(v,s_{2})\,{\rm
d}v\bigg{\\}}^{2}\,{\rm d}s_{1}{\rm d}s_{2}\bigg{]}^{1/2}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\sum_{i,j=1}^{p}\|{\cal F}_{i}\|_{{\cal
S}}\|{\cal F}_{j}\|_{{\cal S}}\|{\cal G}_{i}\|_{{\cal S}}\|{\cal
G}_{j}\|_{{\cal S}}=\big{(}\sum_{i=1}^{p}\|{\cal F}_{i}\|_{{\cal S}}\|{\cal
G}_{i}\|_{{\cal S}}\big{)}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}\leq\big{(}\max_{i}\|{\cal F}_{i}\|_{{\cal
S}}\sum_{i=1}^{p}\|{\cal G}_{i}\|_{{\cal S}}\big{)}^{2}=\|\boldsymbol{{\cal
F}}\|_{{\cal S},\max}^{2}\|\boldsymbol{{\cal G}}\|_{{\cal S},1}^{2}.$

Hence, the proof is complete. $\hfill\Box$

###  B.3 Lemma B2 and its proof

###### Lemma B2

Recall that ${\bf Y}_{i}(\cdot)$ follows a mean zero multivariate Gaussian
process with covariance function matrix
$\boldsymbol{\Xi}(u,v)=\text{Cov}\\{{\bf Y}_{i}(u),{\bf Y}_{i}(v)\\}$ for
$(u,v)\in{\cal U}^{2}.$ Then there exist some positive constants $c_{1}$ and
$c_{2}$ such that, (i) for $n\gtrsim\log p$, the estimate
$\widehat{\boldsymbol{\Xi}}$ satisfies
$\|\widehat{\boldsymbol{\Xi}}-{\boldsymbol{\Xi}}\|_{{\cal
S},\max}\lesssim\sqrt{\log p/n}$ with probability greater than
$1-c_{1}(pd)^{-c_{2}};$ (ii) Suppose Condition 10 hold. Then for
$n\gtrsim\log(pd)d^{4\varpi+2}$, the estimates $\\{\hat{\phi}_{\ell
m}(\cdot)\\}$ satisfies $\max_{\ell\in[p],\,m\in[d]}\|\hat{\phi}_{\ell
m}-\phi_{\ell m}\|^{2}\lesssim d^{\varpi+1}\sqrt{\log(pd)/n}$, with
probability greater than $1-c_{1}(pd)^{-c_{2}}$.

Proof. The proof follows from Theorems 2 and 3 in Guo and Qiao (2023) and
hence is omitted here. $\hfill\Box$

###  B.4 Lemma B3 and its proof

###### Lemma B3

Suppose Conditions 10–11 hold. Then there exist some positive constants
$c_{1}$ and $c_{2}$ such that, for $n\gtrsim\log(pd)d^{4\varpi+2}$, with
probability greater than $1-c_{1}(pd)^{-c_{2}},$ the estimate $\widehat{\bf
B}_{kj}$ satisfies $\|\widehat{\bf B}_{kj}-{\bf B}_{kj}\|_{1}^{(d)}\lesssim
s\tau_{n}/\underline{\mu}.$

Proof. The proof follows similarly to that of Theorem 4 in Fang et al. (2022)
and hence is omitted here. $\hfill\Box$

###  B.5 Lemma B4 and its proof

###### Lemma B4

Suppose Conditions 10–11 hold. Then there exist some positive constants
$c_{1}$ and $c_{2}$ such that, for $n\gtrsim\log(pd)d^{4\varpi+2}$, with
probability greater than $1-c_{1}(pd)^{-c_{2}},$ $n^{-1}\|{\bf
D}_{-k,-j}^{-1}{\bf V}_{-k,-j}^{{\mathrm{\scriptscriptstyle T}}}\widehat{\bf
V}_{-k,-j}\widehat{\bf D}_{-k,-j}^{-1}-\widehat{\bf
D}_{-k,-j}^{-1}\widehat{\bf V}_{-k,-j}^{{\mathrm{\scriptscriptstyle
T}}}\widehat{\bf V}_{-k,-j}\widehat{\bf D}_{-k,-j}^{-1}\|_{\max}\lesssim
d^{\varpi+1}\sqrt{\log(pd)/n}.$

Proof. The proof follows similarly to that of Lemma 17 in Fang et al. (2022)
and hence is omitted here. $\hfill\Box$

###  B.6 Lemma B5 and its proof

###### Lemma B5

Suppose Conditions 10–11 hold. Then there exist some positive constants
$c_{1}$ and $c_{2}$ such that, for $n\gtrsim\log(pd)d^{4\varpi+2}$, with
probability greater than $1-c_{1}(pd)^{-c_{2}},$ $n^{-1}\|\widehat{{\bf
D}}_{-k,-j}^{-1}\widehat{{\bf V}}_{-k,-j}^{{\mathrm{\scriptscriptstyle
T}}}(\widehat{{\bf V}}_{k}-\widehat{{\bf V}}_{-k,-j}\widehat{{\bf
D}}_{-k,-j}^{-1}{\bf B}_{kj})\|_{\max}^{(d)}\leq\tau_{n}/2$.

Proof. The proof follows from Proposition 7 in Fang et al. (2022) and hence is
omitted here. $\hfill\Box$

##  Appendix C Additional empirical results

###  C.1 Simulation studies

Table S2 and S2 present the empirical FDRs and powers of the proposed and BC
methods over $100$ replications at target FDR level $\alpha=0.05$ under Models
1–2, and Models 3–4, respectively. The same patterns as those from Tables
LABEL:table.sim.12–LABEL:table.sim.34 can be observed. Note that the results
of hard and soft thresholding methods do not depend on $\alpha$, and hence are
not reported here.

Table S1: Empirical FDRs ($\%$) and powers ($\%$) of MFC and BH methods at the
$5\%$ nominal level for Models 1 and 2 over 100 simulation runs.

$n$ | $p$ | Scenario | Model 1 |  | Model 2
---|---|---|---|---|---
MFC | BC |  | MFC | BC
FDR | Power | FDR | Power |  | FDR | Power | FDR | Power
100 | 30 | Fully | 4.10 | 88.00 | 0.11 | 64.75 |  | 3.88 | 83.50 | 0.10 | 59.48
|  |  $T_{ij}$ = 51 | 4.37 | 85.79 | 0.11 | 62.68 |  | 4.12 | 82.00 | 0.22 | 55.32
|  |  $T_{ij}$ = 25 | 4.28 | 83.35 | 0.14 | 60.81 |  | 3.85 | 79.95 | 0.24 | 52.55
| 60 | Fully | 4.59 | 82.63 | 0.07 | 58.05 |  | 4.00 | 77.32 | 0.03 | 48.94
|  |  $T_{ij}$ = 51 | 4.84 | 79.75 | 0.12 | 56.06 |  | 4.48 | 74.91 | 0.09 | 45.16
|  |  $T_{ij}$ = 25 | 4.86 | 77.20 | 0.09 | 54.26 |  | 4.76 | 72.96 | 0.10 | 41.68
200 | 30 | Fully | 4.56 | 99.67 | 0.07 | 93.32 |  | 4.77 | 98.11 | 0.12 | 90.07
|  |  $T_{ij}$ = 51 | 4.68 | 99.4 | 0.11 | 90.95 |  | 5.00 | 97.36 | 0.15 | 88.18
|  |  $T_{ij}$ = 25 | 4.49 | 98.93 | 0.10 | 87.81 |  | 4.57 | 96.77 | 0.11 | 86.11
| 60 | Fully | 4.59 | 99.34 | 0.06 | 87.44 |  | 4.71 | 95.04 | 0.08 | 80.85
|  |  $T_{ij}$ = 51 | 4.65 | 98.65 | 0.05 | 83.75 |  | 4.96 | 93.98 | 0.06 | 78.33
|  |  $T_{ij}$ = 25 | 4.50 | 97.77 | 0.03 | 80.29 |  | 4.92 | 93.09 | 0.09 | 75.92

Table S2: Empirical FDRs ($\%$) and powers ($\%$) of MFG and BH methods at the
$5\%$ nominal level for Models 3 and 4 over 100 simulation runs.

$n$ | $p$ | Scenario | Model 3 |  | Model 4
---|---|---|---|---|---
MFG | BC |  | MFG | BC
FDR | Power | FDR | Power |  | FDR | Power | FDR | Power
200 | 30 | Fully | 4.58 | 60.49 | 0.48 | 26.04 |  | 4.26 | 69.56 | 0.29 | 46.31
|  |  $T_{ij}$ = 51 | 4.55 | 59.05 | 0.38 | 24.51 |  | 4.57 | 68.61 | 0.31 | 44.86
|  |  $T_{ij}$ = 25 | 4.46 | 57.61 | 0.17 | 24.12 |  | 4.15 | 68.00 | 0.30 | 44.42
| 60 | Fully | 4.94 | 49.09 | 0.44 | 15.27 |  | 4.86 | 65.32 | 0.27 | 37.44
|  |  $T_{ij}$ = 51 | 4.62 | 47.83 | 0.50 | 14.48 |  | 4.74 | 64.28 | 0.19 | 36.29
|  |  $T_{ij}$ = 25 | 5.22 | 46.14 | 0.29 | 13.72 |  | 4.91 | 63.57 | 0.28 | 35.43
400 | 30 | Fully | 4.44 | 91.61 | 0.05 | 62.25 |  | 4.77 | 94.25 | 0.30 | 80.19
|  |  $T_{ij}$ = 51 | 4.01 | 90.47 | 0.06 | 59.49 |  | 4.79 | 93.50 | 0.28 | 78.39
|  |  $T_{ij}$ = 25 | 4.13 | 90.11 | 0.03 | 58.68 |  | 5.16 | 93.53 | 0.24 | 77.61
| 60 | Fully | 4.95 | 88.57 | 0.06 | 52.99 |  | 4.55 | 92.03 | 0.09 | 71.29
|  |  $T_{ij}$ = 51 | 4.82 | 87.50 | 0.08 | 50.38 |  | 4.66 | 90.95 | 0.06 | 69.36
|  |  $T_{ij}$ = 25 | 5.02 | 85.92 | 0.08 | 48.44 |  | 4.70 | 90.31 | 0.08 | 67.85

###  C.2 Real data analysis

We present in Figure S1 some randomly selected entries of
$\widehat{\boldsymbol{\Sigma}}(\cdot,\cdot)$ in Figure 2 of the HCP data. The
signal of the estimated covariance surfaces is observable over the entire
interval ${\cal U}$ in the sense that the values of $\widehat{\Sigma}_{jk}$
change from positive to negative (or from negative to positive) as a function
of $(u,v)$ smoothly along some directions. Figure S2 displays some randomly
selected elements of
$\\{\widetilde{\Sigma}_{jk}^{\varepsilon}(\cdot,\cdot),1\leq j<k\leq p\\}$ in
Figure 3 of the EEG data, highlighting again the smoothly varying pattern of
signals for the brain imaging dataset.

Figure S1:  Selected significant elements of
$\\{\widehat{\Sigma}_{jk}(\cdot,\cdot):1\leq j<k\leq p\\}$ in Figure 2. Figure
S2:  Selected significant elements of
$\\{\widetilde{\Sigma}_{jk}^{\varepsilon}(\cdot,\cdot):1\leq j<k\leq p\\}$ in
Figure 3.

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