# Chemical mapping of temperate sub-Neptune atmospheres:
Constraining the deep-interior H2O/H2 using the atmospheric CO2/CH4

Jeehyun Yang Jet Propulsion Laboratory, California Institute of Technology,
Pasadena, CA 91109, USA  Renyu Hu Jet Propulsion Laboratory, California
Institute of Technology, Pasadena, CA 91109, USA  Division of Geological and
Planetary Sciences, California Institute of Technology, Pasadena, CA 91125,
USA

###### Abstract

Understanding the envelope composition of sub-Neptune-type exoplanets is
challenging due to the inherent degeneracy in their interior composition
scenarios. Particularly, the H2O/H2 ratio, or can be expressed as the O/H
ratio, in the planetary envelope provides crucial insights into the origin of
these exoplanets relative to the ice line during formation. Using self-
consistent radiative transfer modeling and a rate-based automatic chemical
network generator combined with 1D photochemical kinetic-transport atmospheric
modeling, we investigate atmospheres of temperate sub-Neptunes, ranging from
H2-dominated to H2O-dominated scenarios with Teq = 250–400 K, using K2-18 b
(Teq = 255 K), LP 791-18 c (T${eq}$ = 324 K), and TOI-270 d (Teq = 354 K) as
examples. Our models indicate that using the atmospheric CO2/CH4 ratio to
infer the deep-interior H2O/H2 ratio. Applying to recent JWST observations,
our findings suggest K2-18 b likely has an interior highly enriched in water
(approximately 50% H2O), exceeding the amount of water in a
100$\times$Z$\odot$ scenario and suggesting a formation history that involved
substantial accretion of ices. In contrast, TOI-270 d has an interior
composition of approximately 25% H2O, which is comparable to the conventional
metallicity framework with a metallicity higher than 100$\times$Z$\odot$.
Furthermore, our models identify carbonyl sulfide (OCS) and sulfur dioxide
(SO2) as strong indicators of at least a 10% water-rich envelope in temperate
sub-Neptunes. These results provide a method to delineate the internal
composition and formation mechanisms of sub-Neptunes with $T_{\rm eq}<\sim$
500 K via atmospheric characterization through transmission spectroscopy.

Astrochemistry (75) — Exoplanet atmospheres (487) — Planetary atmospheres
(1244) — Exoplanet atmospheric composition (2021) — Theoretical models (2107)

††software: EPACRIS (Hu et al., 2012, 2013; Hu & Seager, 2014; Hu, 2019; Yang
& Hu, 2024; Scheucher & Hu, 2024), RMG (Gao et al., 2016; Liu et al., 2021;
Johnson et al., 2022; RMG, 2023)

##  1 Introduction

Sub-Neptune-sized planets larger than 1.6–1.8 R${\oplus}$ generally have a
volatile-rich envelope and are the most common types of planets detected so
far. These planets are separated from super-Earth-sized planets by the radius
valley near 1.8 R${\oplus}$ (Fulton et al., 2017; Fulton & Petigura, 2018; Van
Eylen et al., 2018; Hardegree-Ullman et al., 2019). Despite their prevalence,
our understanding of the envelope composition of sub-Neptunes is very limited
due to the inherent envelope mass-metallicity degeneracy in their interior
composition. The radius and mass of sub-Neptunes can be explained by either
(i) a massive, high-molecular-weight (e.g., H2O) volatile layer or (ii) a
lighter H2/He-dominated envelope on top of a rock/iron core (Rogers & Seager,
2010; Valencia, D. et al., 2010; Luque & Pallé, 2022).

For this reason, the precise characterization of the internal composition and
chemical regimes of sub-Neptunes has become a growing interest from a
formation and evolution perspective. This characterization might help
discriminate among multiple pathways to form sub-Neptunes. For example,
‘H2O-rich’ sub-Neptunes are believed to form farther from their star, beyond
the ice line, where they can efficiently accrete large amounts of water and
volatiles in the form of solid material before migrating to their current
close-in orbits (Lambrechts, M. et al., 2014; Morbidelli et al., 2015). In
contrast, ‘H2-rich’ sub-Neptunes with lower volatile content are thought to
originate within the ice line, where volatiles are not present as ice and most
of their mass is accumulated through the accretion of drifting rocky pebbles
(Johansen & Lambrechts, 2017). Therefore, determining the deep-interior H2O/H2
ratio of sub-Neptunes spanning the radius valley is essential for enhancing
our understanding of planetary formation and evolution.

Among various sub-Neptunes, temperate sub-Neptunes (Teq $\leq$ 400 K) serve as
excellent laboratories for studying planetary interior compositions using JWST
observations. Cold exoplanets could have liquid water oceans, and the chemical
composition of their atmospheres would be primarily regulated by interactions
between the atmosphere and the liquid water and thus not represent the
envelope composition. Many hotter exoplanets are predicted to contain CO and
CO2 in their atmospheres due to high temperatures, and thus these gases do not
exclusively represent the deep-interior composition (Moses et al., 2011; Line
et al., 2011; Venot et al., 2012; Moses et al., 2016; Tsai et al., 2018;
Fortney et al., 2020).

Additionally, previous studies showed that exoplanets with Teq between 500-800
K exhibit the most attenuated spectral features, probably obscured by cloud
and hazes (Morley et al., 2015; Brande et al., 2024). Therefore, temperate
sub-Neptuens with Teq lower than 400 make them favorable observation targets.
However, it is important to note that in the cooler end of the temperate range
(i.e., Teq $\sim$ 250 K), water may condense below the photosphere (the region
probed by JWST), complicating the direct inference of internal H2O content
from atmospheric observations.

Recent advancements in observational techniques as well have made sub-Neptunes
abundant laboratories with a high enough signal-to-noise ratio. Numerous JWST
observations of various sub-Neptunes are already available such as K2-18 b
(Madhusudhan et al., 2023) and TOI-270 d (Benneke et al., 2024) or will be
available soon (LP 791-18 c). Since high-altitude clouds or hazes that form at
Teq $\geq$ 400 K are unlikely to obscure the transmission spectra (Morley et
al., 2015), JWST observations can provide valuable information about the
molecular species present in these atmospheres, if available.

Several efforts have been made to infer the elemental composition of the deep
atmospheres of sub-Neptunes using both theoretical and observational studies
(Thorngren et al., 2016; Thorngren & Fortney, 2019; Luque & Pallé, 2022; Burn
et al., 2024; Benneke et al., 2024). There has been an attempt to model the
atmospheres of water-rich sub-Neptunes (from solar to 100% H2O) using 1D
radiative-convective equilibrium modeling coupled with their associated
transmission and thermal emission spectra (Kempton et al., 2023). However,
this approach lacks disequilibrium chemistry and thus may miss infrared
absorbers beyond those abundant in thermochemical equilibrium. To our
knowledge, no attempts have been made to infer the envelope composition,
particularly the H2O/H2 ratio in the full range from gas accretion to ice
accretion, of potentially water-rich temperate sub-Neptunes and their
chemically coupled atmospheres by fully utilizing 1D photochemical modeling.
CO2 is a carbon-bearing species indicative of high temperatures and oxidative
atmospheres, while CH4 indicates low temperatures and reducing atmospheres
(Moses et al., 2011, 2016). Consequently, the ratio between these two
molecules can provide insights into the bulk envelope H2O/H2 ratio of
temperate sub-Neptunes, potentially helping us understand their formation
locations relative to the ice line within the protoplanetary disk (Johansen &
Lambrechts, 2017; Burn et al., 2024).

In this work, we employ self-consistent radiative transfer modeling and a
rate-based automatic chemical network generator combined with 1D photochemical
kinetic-transport atmospheric modeling to investigate various atmospheric
scenarios, ranging from H2-dominated to H2O-dominated atmospheres of temperate
sub-Neptune-type exoplanets. We introduce a new method for inferring the deep-
interior (or bulk envelope) H2O/H2 ratio using the atmospheric CO2/CH4 ratio
and a new framework to characterize temperate sub-Neptunes using this O/H
ratio, different from conventional multipliers of solar metallicity framework.
This new variable (i.e., O/H ratio) not only guides the interpretation of JWST
observations of sub-Neptunes but also provides us with new insights into their
origin and location within the protoplanetary disk during planetary formation.

##  2 Methods

###  2.1 elemental parameterization of planetary envelope accretion

To investigate the diverse envelope accretion scenarios of sub-Neptunes, we
established seven distinct H2O/H2 accretion scenarios, represented by
corresponding oxygen-to-hydrogen (O/H) ratio scenarios, ranging from H2-rich
to H2O-rich envelopes. This variability was numerically achieved by adjusting
the O/H ratio beyond the framework of conventional solar metallicity,
$Z_{\odot}$ (Lodders, 2020). As detailed in Table A1 in Appendix A, the third
column lists standard multipliers of solar metallicity ($Z_{\odot}$) at
1$\times$, 10$\times$, 100$\times$, 1000$\times$, and 10000$\times$ (denoted
as H2:H2O accretion ratio =100:0 for simplicity). Based on this and retaining
the original abundances of carbon, nitrogen, and sulfur (i.e., each elemental
abundance follows its solar elemental abundance multiplied by an integer, n),
we systematically varied the O to H ratio. From now on, maintaining C, N, and
S at each of their solar elemental abundances multiplied by n is denoted as
[C+N+S]=n$\times Z_{\odot}$. We began the O/H variation using the following
equation:

$\displaystyle[\ce{H}+\ce{He}+\ce{O}]_{n\times Z_{\odot}}=a$
$\displaystyle\frac{[\ce{H2}]}{[\ce{H2}]+[\ce{H2O}]}=x_{\text{acc}},$

where [H2O] and [H2] represent water and hydrogen abundance in the envelope,
respectively. Therefore, $x_{\text{acc}}$ represents the accretion ratio for
H2. If planetary formation takes place inside the ice line and closer to its
parent star, $x_{\text{acc}}$ will approach 1 (i.e., H2-rich). Conversely, if
planetary formation occurs beyond the ice line and farther from its parent
star, $x_{\text{acc}}$ will approach 0 (i.e., H2O-rich). Consequently, each
elemental abundance [H], [O], and [He] will be as the following equation:

$\begin{split}[\ce{H}]&=2([\ce{H2}]+[\ce{H2O}])\\\
[\ce{He}]&=\frac{[\ce{H}]}{11.91}=\frac{2([\ce{H2}]+[\ce{H2O}])}{11.91}\\\
[\ce{O}]&=[\ce{H2O}]=(1-x_{\text{acc}})([\ce{H2}]+[\ce{H2O}]),\\\ \end{split}$

where the H/He ratio was maintained at 11.91 to determine the helium abundance
(Lodders, 2020). Finally, the absolute elemental abundance for the
$x_{\text{acc}}$ scenarios would be

$\begin{split}[\ce{H}]_{x_{\text{acc}}}&=a\times\frac{[\ce{H}]}{[\ce{H}]+[\ce{He}]+[\ce{O}]}\\\
&=\frac{2a}{2+(1-x_{\text{acc}})+\frac{2}{11.91}}\\\
[\ce{He}]_{x_{\text{acc}}}&=a\times\frac{[\ce{He}]}{[\ce{H}]+[\ce{He}]+[\ce{O}]}\\\
&=\frac{2a}{\left(2+(1-x_{\text{acc}})+\frac{2}{11.91}\right)\times 11.91}\\\
[\ce{O}]_{x_{\text{acc}}}&=a\times\frac{[\ce{O}]}{[\ce{H}]+[\ce{He}]+[\ce{O}]}\\\
&=\frac{a(1-x_{\text{acc}})}{2+(1-x_{\text{acc}})+\frac{2}{11.91}}\\\
\end{split}$ (1)

$[\ce{H}+\ce{He}+\ce{O}]_{x_{\text{acc}}}=[\ce{H}+\ce{He}+\ce{O}]_{n\times
Z_{\odot}}=a$

Overall, the mixing ratio of [H+He+O] is unchanged, thus retaining the
original ratios of carbon, nitrogen, and sulfur fixed.

As an example, for the H2:H2O=90:10 accretion scenario at 1$\times Z_{\odot}$
(the 2nd row and the 4th column in Table A1),

$\displaystyle a=0.921775+0.077379+0.00495=1.004104$ $\displaystyle
x_{\text{acc}}=0.9$

This results in a total H-to-O ratio of 20.00 (can be approximated using
90$\times$2+10$\times$2 versus 10). Following the same procedure, we
progressed through 75:25, 50:50, 25:75, and 10:90, culminating in a completely
water-rich envelope at a 0:100 ratio. It should be noted that in the case of
100$\times$solar metallicity, if we assume all oxygen to be in the form of
water, then the amount of water already exceeds the abundance of water in the
H2:H2O=90:10 scenario. Consequently, the oxygen abundance at 100$\times$solar
metallicity (i.e.,H2:H2O=100:0) is slightly higher than the oxygen abundance
at H2:H2O=90:10, as seen by comparing the third and fourth columns in the 20th
row in Table A1.

This spectrum of accretion scenarios reflects the variations in planetary
formation locations, particularly concerning the ice line, as depicted in
Figure A1 in Appendix A. We applied these 21 scenarios (3 different solar
metallicity cases $\times$ 7 different O/H ratio or H2-to-H2O accretion ratio)
across three temperate sub-Neptunes: K2-18 b, LP791-18 c, and TOI-270 d,
resulting in a total of 63 scenarios. We investigated 17 additional scenarios
to understand model sensitivities. For K2-18 b, we tested the sensitivity of
the H2:H2O=50:50 and [C+N+S]=100$\times$ solar metallicity scenario to the
eddy diffusion coefficients (Kzz) with values of 104 and 108 cm2/s, compared
to the standard 106 cm2/s mainly adopted in this study. We also explored 7
different O/H scenarios, maintaining [C+N+S] at 100 $\times$ solar
metallicity, using a T–P profile computed for 100$\times$ solar metallicity
and an intrinsic temperature (Tint) of 100 K to test the sensitivity of the
model to a higher Tint of 100 K compared to the standard 60 K mainly adopted
in this study. For TOI-270 d, we evaluated one scenario assuming 223$\times$
solar metallicity to benchmark against the metallicity constraint retrieved
from Benneke et al. (2024). Additionally, we investigated 7 different O/H
scenarios, maintaining [C+N+S] at 100$\times$ solar metallicity, assuming a
Bond albedo of 0. Overall, we analyzed 74 (63+2+1+1+7) planetary atmospheric
structures (T–P profiles) and 80 (63+2+7+1+7) planetary atmospheric
photochemical models.

###  2.2 The temperature-pressure profiles for various planetary envelope
accretion scenarios

For each of the elemental parameterizations of planetary envelope accretion
mentioned in Section 2.1, we calculated 74 T–P profiles under radiative-
convective equilibrium using the climate module of the ExoPlanet Atmospheric
Chemistry & Radiative Interaction Simulator (EPACRIS-Climate, Scheucher et
al., in prep), and the results are presented in Figure B2 in Appendix B
(1000$\times$ and 10000$\times$ solar metallicity cases for K2-18 b are
omitted for simplicity).

The model uses the two-stream method from Heng & Marley (2018) to compute
radiative fluxes and incorporates moist adiabatic adjustments following the
methodology of Graham et al. (2021). Water is considered a condensable
component, and its atmospheric concentration is self-consistently adjusted in
line with the moist adiabats. We assumed an intrinsic temperature (Tint) of 60
K (Hu, 2021). For some cases of K2-18 b scenarios, we tested larger Tint of
100 K to test the sensitivity of the model, since Tint can impact deep
atmospheric quenching and its result in the upper atmospheric abundances of
chemical species (Fortney et al., 2020; Tsai et al., 2021).

By default, we set the planetary Bond albedo (Ab) to an Earth-like value of
0.3 for all planets, including K2-18 b, LP791-18 c, and TOI-270 d. For
TOI-270d, we additionally explored seven different O/H ratio scenarios while
maintaining [C+N+S]=100$\times$ solar metallicity, but with a Bond albedo
(i.e., Ab) of 0. In all cases, we assumed an average solar zenith angle of
48.19∘ (Cronin, 2014). The stellar irradiation parameters ($S_{p}$ in W$\cdot$
m-2) were adopted to be 1368 W$\cdot$ m-2 for K2-18 b (Benneke et al.
(2019a)), 3607 W$\cdot$ m-2 for LP 791-18 c (Peterson et al. (2023)), and 4333
W$\cdot$ m-2 for TOI-270 d (Günther et al. (2019)).

###  2.3 Automoatic chemical reaction network generation for H2O-rich
chemistry

A detailed chemical reaction network for H2O-rich atmospheres was constructed
using the Reaction Mechanism Generator (RMG Gao et al. (2016); Johnson et al.
(2022)), a Python-based open-source software. RMG generates chemical networks
using a rate-based iteration algorithm and has been extensively described in
previous literature Gao et al. (2016); Liu et al. (2021). This method’s
application to exoplanet atmospheric studies is detailed in Yang & Hu (2024).
In this work, we sampled temperatures ranging from 300 to 1100 K and pressures
from 10-3 and 102 bar to generate chemical networks tailored for a H2O-rich
scenario of temperate sub-Neptunes, as established by previous radiative
transfer modeling of K2-18 b (Scheucher et al., 2020).

For initial molecular mixing ratio inputs, we used the following: $10^{0}$ for
H2O to account for a H2O-rich scenario. Additional constraints were based on
the recent JWST observations of K2-18 b by Madhusudhan (2023): $10^{-1.89}$
for CH4 and $10^{-2.05}$ for CO2, both determined from two-offsets retrieved
molecular abundances with detection significances of 5$\sigma$ and
3.2$\sigma$, respectively; $10^{-4.46}$ for S and N, derived from no-offset
retrieved molecular abundances with a 2.4$\sigma$ detection significance for
(CH3)2S and based on the upper limit for nitrogen. These values were later
normalized to sum to 1.

The choice of reaction libraries can be found in the supplementary RMG input
file. The details of these libraries can be found in the RMG database RMG
(2023). The completed network comprised 90 species and 1950 reactions,
detailed in the supplementary CHEMKIN file. Of these, 90 species and 1666
reactions were incorporated into the 1-D photochemical kinetic-transport
modeling after the treatment described in Yang & Hu (2024).

To make sure the chemical reaction network generated for the H2O-rich
conditions are also applicable to the $x$acc=1 scenario (i.e., H2:H2O=100:0),
we conducted photochemical modeling of the H2-dominated K2-18 b atmosphere
using the same atomic abundance and the same T–P profile adopted in MODEL 3 of
Wogan et al. (2024). As discussed in Section 3.1.1, the predicted mixing
ratios of major species such as CO2, CH4, H2O, N2, CO, NH3, and SO2 were
almost identical, confirming the robustness of the model results in both the
H2O- and H2-dominated regimes.

The final photochemical network comprised 90 species (54 originally available
in the EPACRIS species, and 36 species newly generated by RMG, not available
in the EPACRIS library, as listed in Table C2 in Appendix C. When compared to
other well-known photochemical networks used for exoplanetary species, within
these 36 newly generated species by RMG, 27 species were not included in
KINETICS (Moses et al., 2011), and 30 species were not included in VULCAN
(Tsai et al., 2017, 2021) (see Table C2 for detail). Among these not-included
species, hydroxymethylene (HCOH) is shown to be a major intermediate species
for the conversion of carbon monoxide into methane and water (Yang et al.,
2023). Additionally, many hydroxyl group species such as CH2CHOH, CH2CH2OH,
HCCOH, CHCHOH, CH3CH2OH, and CH3CHOH participate in the dehydration reactions
that link simple molecules into long and complex chains. For this reason,
although these molecules may be challenging to detect with current
observational techniques, including them in photochemical modeling is
essential, as this can potentially lead to the formation of novel species.

The network included 2009 reactions (343 original EPACRIS reactions = 40
photochemistry reactions + 248 bi-molecular reactions + 34 ter-molecular
reactions + 21 thermo-dissociation reactions, and 1666 reactions newly
generated by RMG. Except for the 40 photochemistry reactions, the other 1969
reactions are forward-reverse reaction pairs. This comprehensive network was
used in 1-D photochemical kinetic-transport atmospheric modeling of various
temperate sub-Neptune atmospheres, as described in Section 2.4.

###  2.4 1-D photochemical kinetic-transport atmospheric modelings

Based on the elemental scenarios outlined in Section 2.1, the T–P profiles
calculated in Section 2.2, and the chemical network tailored for water-rich
systems in Section 2.3, we performed 1D photochemical kinetic-transport
atmospheric modeling of the 80 scenarios using the chemistry module of EPACRIS
(Yang & Hu, 2024). This modeling was conducted to simulate the steady-state
vertical mixing ratios of chemical species in various atmospheric scenarios of
temperate sub-Neptunes.

The eddy diffusion coefficient profile was estimated to be consistently 106
[cm${}^{2}\cdot$s-1] at various altitudes based on Figure 1 in Zhang & Showman
(2018). This value is comparable to that of Earth, which ranges from 4$\times
10^{3}$–106 [cm${}^{2}\cdot$s-1], and Earth has planetary equilibrium
temperature (Teq) of 255 K, similar to K2-18 b. However, some might argue that
Kzz could be much smaller, especially if a short-wavelength absorber causes a
temperature inversion, as it is in Earth’s stratosphere. To test the
sensitivity of the steady-state vertical mixing ratios to changes in the eddy
diffusion coefficients, we conducted additional modelings with coefficients
set at 104 and 108 [cm${}^{2}\cdot$s-1] in selected cases. These tests showed
that the results were not highly sensitive to variations in the eddy diffusion
coefficient, which will be discussed again later. For simulations involving
K2-18 (classified as an M2.8-type star by Montet et al. (2015)) and LP 791-18
(classified as an M6-type star by Crossfield et al. (2019)), we scaled for the
bolometric luminosity using the M5-type stellar spectrum of GJ876 from the
MUSCLES survey III (Loyd et al., 2016). For simulations involving TOI-270
(classified as an M3V-type star by Günther et al. (2019)), the M3V-type
stellar spectrum of GJ644 from the same survey was used for the bolometric
luminosity scaling.

After the models had converged and reached the steady state, we computed the
synthetic transmission spectra of K2-18 b and TOI-270 d based on the molecular
mixing ratio profiles, using the transmission spectra generation module of
EPACRIS (Hu et al., 2013), and compared the resulting transmission spectra
with published JWST observations (Madhusudhan et al., 2023; Benneke et al.,
2024).

##  3 Results and Discussions

###  3.1 Constraining the bulk envelope H2O/H2 ratio

Figure 1: The figure shows the upper atmospheric (P=0.1–2 mbar, corresponding
to the JWST’s primary probing range; Rustamkulov et al. (2023)) CO2-to-CH4
ratios plotted against the deep interior (P=20 bar) H2O-to-H2 ratios. Unless
otherwise noted, the planetary equilibrium temperature (i.e., Teq) is assumed
based on Ab=0.3 and an intrinsic temperature (Tint) of 60 K, with $Z_{\odot}$
denoting solar metallicity. The symbols in the figure represent different
elemental parameterizations for the planetary envelope accretion scenarios.
Diamonds indicate seven O/H scenarios (i.e., H2:H2O from 100:0 to 0:100) with
elemental compositions of carbon, nitrogen, and sulfur ([C+N+S]) of 1$\times
Z_{\odot}$. Circles represent the same O/H ratios but with [C+N+S] of
10$\times Z_{\odot}$, while crosses correspond to 100$\times Z_{\odot}$.
Upward- and downward-pointing triangles indicate the scenario with an H2:H2O
ratio of 50:50 and [C+N+S] of 100$\times Z_{\odot}$, but assuming eddy
diffusion coefficients of 108 and 104 [cm${}^{2}\cdot$s-1], respectively.
Hexagrams also illustrate these O/H ratios with [C+N+S] of 100$\times
Z_{\odot}$, but assuming intrinsic temperature (Tint) of 100 K. X symbols
represent the K2-18 b scenario with 100$\times Z_{\odot}$, based on the T–P
profile from Wogan et al. (2024). Blue symbols indicate the model simulated by
EPACRIS (in this work), and black symbols represent the model by Wogan et al.
(2024). The primary differences between the two models lie in the chemical
networks and the eddy diffusion coefficient profiles used for modeling. An
asterisk indicates 223$\times Z_{\odot}$. Squares indicate JWST observations
with the error bars shown in light blue for K2-18 b (Madhusudhan et al., 2023)
and magenta for TOI-270 d (Benneke et al., 2024). Each color represents the
model simulations for corresponding planets: blue and black for K2-18 b, green
for LP791-18 c, red for TOI-270 d, magenta for TOI-270 d assuming Ab=0. The
color-shaded box regions indicate the implied constraints on the deep-interior
H2O-to-H2 ratios–or envelope O/H ratio–of corresponding planets based on
current studies (light blue for K2-18 b and magenta for TOI-270 d). A much
lighter blue-shaded box region indicates the constraints when accounting for
hotter intrinsic temperature (Tint=100 K) case of K2-18 b with 100$\times
Z_{\odot}$. Grey dotted lines represent standard multipliers of solar
metallicity (1, 10, 100, and 223$\times$), aiding in contextualizing the
current study against conventional benchmarks in solar metallicity.

####  3.1.1 Using the CO2/CH4 ratio as a constraint on the H2O/H2 ratio

Figure 1 displays the upper atmospheric CO2/CH4 ratios plotted against the
deep interior H2O/H2 ratios for various planetary envelope accretion
scenarios, as detailed in Section 2.1. As an example, for the case of H2:H2O =
50:50 and [C+N+S] = 100$\times$ solar metallicity scenario of K2-18 b, as
illustrated in the left panel of Figure 4, we determined the CO2 (red) and CH4
(green) mixing ratios within the pressure range of 0.1–2 mbar, located within
the grey-shaded area, to derive the upper atmospheric CO2/CH4 ratios.
Similarly, we obtained the H2O (blue) and H2 (black) mixing ratios at a
pressure of 200 bar (i.e., $2\times 10^{5}$ mbar) at the bottom to derive the
deep-interior H2O/H2 ratios. This approach allowed us to explore the
relationship between the upper atmospheric CO2/CH4 ratios and the deep-
interior H2O/H2 ratios across various planetary envelope accretion scenarios.
By analyzing these upper atmospheric CO2/CH4 ratios and comparing them to the
CO2/CH4 ratios retrieved from the JWST observation of K2-18 b (Madhusudhan et
al., 2023) and TOI-270 d (Benneke et al., 2024), we can further constrain the
deep-interior H2O/H2 ratio of each planet, respectively.

In general, as expected, we can see a clear consistent linear pattern across
all planetary envelope accretion scenarios with more than 25% water (this
corresponds to 223$\times Z_{\odot}$ grey dotted-line shown in Figure 1). This
is because a H2O-rich envelope creates an oxidative atmosphere favorable for
CO2 formation over CH4, thus increasing the CO2/CH4 ratio as the H2O-content
inside the planetary envelope increases. By leveraging this trend, we can
effectively constrain the deep-interior H2O/H2 ratio using the upper
atmospheric CO2/CH4 ratio.

First, in the case of K2-18 b, as depicted in Figure 1, assuming Ab=0.3 (i.e.,
Teq=255 K), an intrinsic temperature (i.e., Tint) of 60 K, and an eddy
diffusion coefficient of $10^{6}$ [cm2/s], the thicker blue-shaded area
represents the deep-interior H2O/H2 constraints for K2-18 b, which ranges from
$\sim$25 to $\sim$90% water-rich scenarios. This indicates that K2-18 b’s
envelope contains at least $\sim$25% water. This statement contrasts with the
observations by Madhusudhan et al. (2023), which did not detect significant
contributions from H2O and thus did not provide any atmospheric retrievals on
O/H ratios. As discussed later in Section 3.2, in the case of K2-18 b, its low
equilibrium temperature of 255 K causes water to condense out at P$\sim
10^{2}$ mbar, making water detection challenging for JWST observation (see
blue line in the left panel of Figure 4 and the upper panel of Figure 5). This
underscores the importance of accurately addressing water condensation in
atmospheric modeling to extract hidden information.

Adjusting for an Tint=100 K, to account for the sensitivity of the intrinsic
temperature to the upper atmospheric CO2/CH4 ratios due to changes in deep-
interior T–P structure, the lighter blue-shaded area now also marginally
includes the conventional 100$\times$solar metallicity scenario, as denoted as
blue hexagrams in Figure 1. This behavior is primarily attributed to elevated
temperatures in the deep interior of K2-18 b, which favor the formation of CO2
over CH4 (Moses et al., 2011), thus elevating the upper atmospheric CO2/CH4
ratios in H2-rich cases (e.g., 100$\times$solar metallicity) to align with the
lower-end of the JWST observational constraints on CO2/CH4 ratio.

This elevated CO2/CH4 ratio in H2-rich envelope is also consistent with the
modeling result of Wogan et al. (2024), which utilized a T–P profile that is
$\sim$150 K higher at 200 bar compared to the T–P profile computed by EPACRIS
assuming the same Tint of 60 K. However, it is 100 K lower compared to the T–P
profile computed by EPACRIS assuming the Tint of 100 K, which encompasses the
uncertainty. Although the CO2/CH4 ratio predicted by Wogan et al. (2024) is
slightly below the lower end of the JWST observational constraints on CO2/CH4
ratio (see black X symbol in Figure 1), the synthesized transmission spectrum
contained both CO2 and CH4 features. To determine whether the elevated CO2/CH4
ratios in the upper atmosphere of the H2-rich case are predominantly due to
the T–P profile rather than differences in the chemical networks used, we
conducted photochemical modeling of the K2-18 b atmosphere. We used the same
old 100$\times$ solar metallicity from Lodders et al. (2009), where C/O =
0.46, compared to the recently updated value of C/O = 0.55 from Lodders
(2020). We also applied the same T–P profile as in Wogan et al. (2024). The
resulting CO2/CH4 ratio is denoted by a blue X symbol in Figure 1 and exhibits
almost identical value to that modeled by Wogan et al. (2024) (see black X
symbol). This emphasizes that accurate prediction of the interior T–P
structure is crucial for using the upper atmospheric CO2/CH4 ratio to infer
the deep-interior H2O/H2 ratios.

We also tested the sensitivity of the eddy diffusion coefficients to the upper
atmospheric CO2/CH4 ratios for the H2:H2O=50:50 and [C+N+S]=100$\times$solar
metallicity scenario of K2-18 b, denoted by upward- (assuming $10^{8}$
[cm2/s]) and downward-pointing blue triangles ($10^{4}$ [cm2/s]) in Figure 1,
which shows relatively less sensitivity compared to the sensitivity driven by
varying intrinsic temperatures.

In the case of TOI-270 d, as depicted in Figure 1, assuming Ab=0 (i.e.,
Teq=387 K), an intrinsic temperature (i.e., Tint) of 60 K, and an eddy
diffusion coefficient of $10^{6}$ [cm2/s], the magenta-shaded area represents
the deep-interior H2O/H2 constraints for TOI-270 d, which indicates that
TOI-270 d’s envelope contains approximately 15–25% water. These constraints
align well with the quenched-chemistry atmospheric retrievals on O/H ratios
(i.e., horizontal error bars in Figure 1) of the deep atmospheres (1–10 bar)
based on the JWST observations (Benneke et al., 2024). Since Benneke et al.
(2024) also provides a constraint on the TOI-270 d’s metallicity of
approximately $223\times Z_{\odot}$ (note that this corresponds to
H2:H2O=80:20), we ran additional photochemical modeling assuming Ab=0.3 (i.e.,
Teq=354 K), an intrinsic temperature (i.e., Tint) of 60 K, and an eddy
diffusion coefficient of $10^{6}$ [cm2/s]. As shown in Figure 1, the resulting
CO2/CH4 ratio, denoted by a red asterisk, is consistent with the JWST measured
CH4 and CO2, and also falls within the deep-interior H2O/H2 ratio constraint
presented in this work. This demonstrates the robustness of the current
framework for inferring the deep-interior O/H ratio using 1-D photochemical
modeling, particularly when combined with observational data.

Figure 2: The upper atmospheric (P=0.1–2mbar) CO2, CH4, CO, NH3, OCS, and SO2
molecular mixing ratios plotted against the deep interior (P=20 bar) H2O-to-H2
ratios, respectively. All notations follow Figure 1. The black dotted box
indicates the major formation pathways that contribute to SO2 formation,
respectively.

####  3.1.2 Using individual species abundances as further constraints

As described in Section 3.1.1, the upper atmospheric CO2/CH4 ratio can be a
powerful tool for constraining the bulk envelope H2O/H2 ratio. However, this
metric does not retain information about the absolute abundances of carbon,
nitrogen, and sulfur species in the envelope. It is therefore crucial to
investigate whether the abundance of individual gases could provide additional
constraints for characterizing the atmosphere and envelope of temperate sub-
Neptunes. For instance, could measurements of additional gases help
distinguish Tint of 60 K versus 100 K for a planet like K2-18 b? In this
section, we explore the behavior of several individual species across various
planetary envelope scenarios of temperate sub-Neptunes.

Figure 2 shows the upper atmospheric CO2, CH4, CO, NH3, OCS, and SO2 molecular
mixing ratios plotted against the deep interior H2O/H2 ratios for various
planetary envelope scenarios. Generally, a clear pattern emerges between CO2
and CH4. These two molecules exhibit opposite behaviors–CO2 predominates in
water-rich envelopes (i.e., deep interior H2O/H2$\geq$0.1), while CH4 is more
abundant in H2-rich envelopes. Notably, in scenarios with a deep interior
H2O/H2$\sim$1, the presence of both gases is nearly balanced.

Similar to CO2, carbon monoxide (CO) is favored in an oxidizing environment.
However, if there is an excess of oxidizers compared to CO, it will remain
fully oxidized as carbon monoxide. Therefore, it is challenging to exclusively
determine whether the atmosphere is H2-dominated or H2O-dominated based solely
on the CO abundance. Nevertheless, our model indicates that the CO mixing
ratio consistently exceeds 1000 ppm if the C-abundance is greater than
100$\times Z_{\odot}$, as shown in Figure 3.1.2, identifying CO as a strong
indicator of a C-rich planetary envelope. Although the retrieved detection
significance of CO abundance is relatively less credible compared to CO2 and
CH4 due to its sparse infrared absorption line-list density, the largest upper
limits of $10^{-3}$ retrieved from Madhusudhan et al. (2023) suggest that the
carbon abundance should be less than 100$\times Z_{\odot}$. Consequently,
Wogan et al. (2024)’s modeling that assumes 100$\times Z_{\odot}$ for K2-18 b
(black X symbols in Figure 3.1.2) might not be able to explain the JWST
observations of K2-18 b. However, our model, which assumes a carbon abundance
of 10$\times Z_{\odot}$ and a H2:H2O=50:50 ratio can still explain all
retrieved JWST constraints on CO2, CH4, and CO (Madhusudhan et al., 2023). In
the case of TOI-270 d, the retrieved upper limit of CO abundance was
$10^{-1.46}\sim 0.035$, consistent with all scenarios of our model.

Similar to CH4, NH3 indicates reducing envelope conditions (H2-rich) as shown
in Figure 2, and could be a sensitive indicator for constraining the deep-
interior H2O/H2 ratio. For instance, if adopting NH3’s largest upper limits of
$10^{-4.46}\sim 0.000035$ retrieved for K2-18 b from Madhusudhan et al.
(2023), all scenarios with nitrogen abundance of 100$\times Z_{\odot}$ are
already ruled out. However, it is important to note that the entire envelope
could be intrinsically nitrogen-poor, as N-bearing species can effectively
dissolve into the mantle beneath the envelope (Shorttle et al., 2024). This
explanation was also proposed for NH3-depletion (upper limits of $10^{-4.27}$)
observed in TOI-270 d (Benneke et al., 2024).

The upper atmospheric mixing ratios of carbonyl sulfide (OCS) with respect to
the deep interior H2O-to-H2 ratios exhibit very interesting features. As shown
in Figure 2, regardless of the exact OCS abundance being sensitive to Kzz (see
blue upward- and downward-pointing triangles), the upper atmospheric OCS
mixing ratio is generally predicted to be above $\sim 10^{-5}$ (sufficient to
appear as a specific spectral feature at 4.8–4.9 $\mu$m, as shown in Figure 5)
for deep-interiors enriched with more than 10% H2O across various temperate
sub-Neptune scenarios, with Teq ranging from 255 K to 387 K. Moses et al.
(2013) investigated compositional diversity of carbonyl sulfide in hot
Neptune-sized exoplanets using a thermochemical-equilibrium model in the
framework of solar metallicity, but its abundance was less than 1 ppm even at
500$\times Z_{\odot}$ (see Figure 5 in Moses et al. (2013)). However, in our
modeling, the OCS formation mechanism is different since we are investigating
the case of temperate sub-Neptune exoplanets being aH2O-rich interior. As
described in detail later in Section 3.2.1 and 3.3.1, the OCS formation was
attributed to the enriched water content in the deep interior, resulting in an
oxidizing envelope that favors carbon-bearing species in the form of CO and
CO2. Sulfur, primarily from H2S, then reacts with this CO to form OCS (Scheme
5), which is then transported to the upper atmospheric region observable by
the JWST. Additional OCS formation occurs at higher altitudes due to increased
sulfur abundance from continuous H2S photodissociation in the presence of CO
(Scheme 6). The OCS formation is thus a result of the oxidative chemistry due
to the water-rich envelope. This suggests that carbonyl sulfide (OCS) is a
strong indicator of a water-rich envelope for temperate sub-Neptunes,
highlighting OCS as an indicator of temperate sub-Neptune envelopes with over
10% H2O.

As shown in Figure 2, species such as CO2, CH4, NH3, and OCS exhibit intuitive
and consistent behavior with respect to the deep-interior O/H ratio. However,
the upper atmospheric SO2 mixing ratio displays a somewhat indirect pattern
compared to other species, indicating its formation pathways are varied.
Multiple factors–including T–P structure, absolute sulfur abundance, and O/H
ratio–affect SO2 formation, making it more complex. Nonetheless, SO2 formation
pathways can be divided into two representative regimes.

In the regimes with less than approximately $\sim$20% H2O abundance inside the
envelope, SO2 forms though the following reactions at P$\sim$10 mbar:

$\begin{split}\ce{H2S}&\rightarrow\ce{H2}+\ce{S}\\\
\ce{S}+\ce{CO}&\rightarrow\ce{OCS}\\\
\ce{OCS}+\ce{S}&\rightarrow\ce{CO}+\ce{S2}\\\
\ce{S2}&\xrightarrow{\text{h$\nu$}}\ce{S}+\ce{S}\\\
\ce{S}+\ce{CO2}&\rightarrow\ce{SO}+\ce{CO}\\\
\ce{SO}+\ce{SO}&\rightarrow\ce{SO2}+\ce{S}\\\ \end{split}$ (2)

In this regime, CO2 acts as an oxidizer for sulfur due to its thermochemical
instability as a carbon-bearing form (with CH4 being more favorable). Thus,
the amount of SO2 formed is determined by the amount of CO2 (see mixing ratio
patterns at deep interior H2O/H2$\leq$0.4 in both Figures 2). Additional SO2
formation occurs in the upper atmosphere via photochemistry:

$\begin{split}\ce{H2O}&\xrightarrow{\text{h$\nu$}}\ce{H}+\ce{OH}\\\
\ce{S}+\ce{OH}&\rightarrow\ce{SO}+\ce{H}\\\
\ce{SO}+\ce{OH}&\rightarrow\ce{SO2}+\ce{H}\\\ \end{split}$ (3)

This is the proposed photochemical formation pathway for SO2 in the hot
Jupiter-type exoplanet WASP-39 b (Tsai et al., 2023).

The transition region between the $\leq$20% H2O region and the $\geq$20%
region turned out to be very sensitive to the T–P structure of the planet. For
example, when comparing the 100$\times Z_{\odot}$ scenario of K2-18 b assuming
Tint = 60 K and 100 K, there is more than a 6 orders of magnitude difference
between the predicted SO2 mixing ratios. This difference primarily arises from
water condensation, which impacts the stability of H2S, the main precursor for
sulfur-bearing species such as SO2 and OCS. In the scenario where Tint = 100
K, the deep interior at 100 bar is approximately 250 K hotter compared to the
interior assuming Tint = 60 K. At pressures lower than approximately 3 bar,
the entire T–P profile assuming Tint = 100 K is about 5 K higher compared to
the T–P profile assuming Tint = 60 K. This small difference of about 5 K
results in about 15% more H2O abundance at P$\leq\sim$3 bar for the
H2-dominated atmosphere of K2-18 b assuming Tint = 100 K, compared to when
assuming Tint = 60 K. This 15% excess H2O favors the retention of H2S through
the reaction:

$\begin{split}\ce{H2O}+\ce{SH}&\rightarrow\ce{H2S}+\ce{OH}\\\ \end{split}$ (4)

As a result, in the case of H2-dominated K2-18 b assuming Tint = 100 K, H2S
can remain stable up to P$\sim$0.2 mbar, where H2S photodissociates to form
various sulfur-bearing species including SO2. In contrast, with Tint = 60 K,
H2S starts to thermally dissociate into sulfur-bearing species already
starting at P$\sim$30 mbar. This results in significant differences between
the predicted SO2 mixing ratio, as shown in the SO2 panels of Figure 2. This
is also consistent with the OCS case, as evidenced by the rapid decrease of
the OCS mixing ratio within this transition region (see blue crosses and blue
hexagrams near 100$\times Z_{\odot}$ grey dotted line in Figure 2). This T–P
structure-sensitive behavior of SO2 driven by H2O-condensation implies that
the detection of SO2 in temperate sub-Neptunes with an internal H2O envelope
of less than 20% may be inconsistent with a high intrinsic temperature (Tint).

In the H2O-rich regime (more than approximately 20% H2O abundance inside the
envelope), SO2 becomes thermochemically favorable in an oxidizing deep
interior. In these conditions, SO2 becomes abundant with higher water content
and deep-interior temperatures. As shown in Figure 2, SO2 formation correlates
positively with deep-interior temperature and the planetary envelope O/H ratio
(i.e., H2O/H2). For instance, the deep-interior temperature tends to increase
with the planetary equilibrium temperature. Thus, K2-18b has the lowest
equilibrium temperature at 255 K, followed by LP 791-18c (324 K), TOI-270d
(354 K), and TOI-270d assuming Ab = 0 with the highest equilibrium temperature
of 387 K. Higher [C+N+S] abundance also promotes SO2 formation as well.

Although our models spanning various water-rich envelope scenarios don’t
always predict SO2 levels above 10 ppm (which is sufficient to appear as
spectral features in JWST observations), any SO2 level above 10 ppm in our
model indicates a deep-interior with more than 10% H2O inside. For this
reason, along with OCS, our model suggests that any future detection of SO2 in
temperate sub-Neptune atmospheres indicates a water-rich interior of at least
more than 10% H2O.

Although we only used CO2+CH4 in the current study due to the limited or
absent observational constraints on N- or S-bearing species, other species
such as NH3, OCS, or SO2 could serve as additional tools to more precisely
constrain the deep-interior H2O/H2 ratio along with the absolute N- or
S-abundance as described in this section.

####  3.1.3 Constraining the building blocks of temperate sub-Neptunes

Figure 3 presents a comprehensive plot after integrating all the constraints
from the CO2/CH4 ratio and each of the mixing ratios of CO2 and CH4, as
described in Section 3.1.1 and 3.1.2. Notably in this Figure 3, we can
constrain the carbon abundance included in the planetary envelope to within
the range of $10-100\times Z_{\odot}$ for K2-18 b, and approximately
$40-500\times Z_{\odot}$ for TOI-270 d using absolute CO2 and CH4 mixing
ratio. Particularly, the carbon abundance for K2-18 b constrained within the
range of $10-100\times Z_{\odot}$ aligns well with the carbon abundance
implied from the CO abundance shown in Section 3.1.2 and Figure 2.

According to Figure 3, the most probable scenario for TOI-270 d is close to
the H2:H2O=75:25 and [C+N+S]=100$\times Z_{\odot}$ assuming either Ab=0
(magenta cross) or 0.3 (red cross). Although the uncertainty is substantial,
the SO2 mixing ratio constrained from the JWST observation of TOI-270 d by
Benneke et al. (2024) also falls within this O/H ratio range as shown in the
SO2 panel of Figure 3.1.2. In the case of K2-18 b, two scenarios–H2:H2O=50:50
and H2:H2O=25:75, both with [C+N+S]=100$\times Z_{\odot}$ scenarios
(represented by blue crosses) fall within the constraints from the
photochemical mapping (i.e., thick blue shaded-area).

Furthermore, Figure 3 reveals some new insights: TOI-270 d can still fit
within the conventional solar metallicity framework, depicted by a grey solid
line (particularly simulated for various K2-18 b planetary envelope scenarios)
in Figure 3. If we maintain this framework, TOI-270 d would require a
metallicity significantly above 100$\times Z_{\odot}$ value. This is
corroborated by a red asterisk marker on the plot, representing the retrieved
constraint on the metallicity of $\sim$223$\times Z_{\odot}$ from the JWST
observation by Benneke et al. (2024). However, K2-18 b cannot be explained
within this conventional solar metallicity framework. Instead, its
characteristics (i.e., thicker blue-shaded region) suggest efficient ice
accretion within the protoplanetary disk, implying that planetary formation
occurred close to or beyond the ice line (Burn et al., 2024). This underscores
the importance of the O/H ratio as a pivotal variable in classifying
exoplanets, potentially revealing their original locations of planetary
formation. By combining this photochemical mapping method with many upcoming
observational data, we can determine whether the planet accretes H2 into the
envelope with other elements, or efficiently accretes H2O in the form of ice
into its envelope, as shown in Figure 3.

Figure 3: The figure shows a schematic diagram of the feeding zone of planets
in the protoplanetary disk with respect to the ice-line on the top and the
upper atmospheric (P=0.1–2 mbar) CO2+CH4 mixing ratios plotted against the
deep interior (P=20 bar) H2O-to-H2 ratios on the bottom. All notations follow
Figure 1. The only difference is that the constraints on the H2O/H2 ratio for
each planet are now based on the combination of the CO2/CH4 ratio (Section
3.1.1) and individual abundances (Section 3.1.2).

###  3.2 The atmosphere of K2-18 b

####  3.2.1 Overall chemistry in the atmosphere of K2-18 b

The left panel of Figure 4 shows the simulated vertical molecular mixing ratio
profiles of major species for the scenario close to the most probable scenario
for K2-18 b (i.e., H2:H2O=50:50 with [C+N+S]=100$\times Z_{\odot}$) according
to the photochemical mapping detailed in Section 3.1. The first notable thing
to look at is the condensation behavior of water at P$\sim 10^{2}$ mbar. This
is consistent with the computed T–P profiles for various scenarios of K2-18 b
as shown in Figure B2 in Appendix B. As illustrated in Figure B2, water
condensation would begin around P$\sim 10^{2}$ mbar, where the condensation
curves for H2O(Buck, 1981) intersect with the T–P profiles for various O/H
ratios with [C+N+S]=100$\times Z_{\odot}$ scenarios for K2-18 b (see blue
solid lines in Figure B2). Consequently, the JWST observable region of the
upper atmosphere will have a H2-dominated atmosphere with less than $\sim$1%
of H2O, enhancing spectral features due to a decreased mean molecular weight
(Miller-Ricci et al., 2008). The mixing ratios of other molecular species such
as H2, CH4, CO2, N2, H2S, and NH3 in the JWST probe region are primarily
determined by the deep-interior thermal chemistry and are quenched at
pressures greater than $10^{4}$ mbar, before being transported upward. This
quenching behavior is expected since the equilibrium temperatures of temperate
sub-Neptunes are low, leading to longer chemical lifetimes relative to the
vertical mixing timescale. At higher altitudes (i.e., pressures lower than 0.1
mbar), UV-driven photochemistry dominates, photodissociating species including
H2O, CO2, H2S, and NH3.

Another notable feature is the significant presence of carbonyl sulfide (OCS)
in the JWST observable region, as shown in the left panel of Figure 4. OCS
primarily forms in the deep interior through the reaction between sulfur from
H2S and carbon monoxide, which is the favorable carbon-bearing species in
H2O-rich oxidizing envelopes:

$\begin{split}\ce{H2S}&\rightarrow\ce{H2}+\ce{S}\\\
\ce{S}+\ce{CO}&\xrightarrow{\text{M}}\ce{OCS}\\\ \end{split}$ (5)

with M representing any third-body molecule. This summarized scheme for the
deep-interior OCS formation is consistent with the similar vertical mixing
ratio profile shape between OCS and CO at P$\geq 10^{4}$ mbar as shown in the
left panel of Figure 4. OCS is then transported upward via vertical mixing to
P$\sim$10 mbar, where additional OCS forms due to UV-driven photochemistry:

$\begin{split}\ce{H2S}&\xrightarrow{\text{h$\nu$}}\ce{SH}+\ce{H}\\\
\ce{SH}&\xrightarrow{\text{h$\nu$}}\ce{S}+\ce{H}\\\
\ce{S}+\ce{CO}&\xrightarrow{\text{M}}\ce{OCS}\\\ \end{split}$ (6)

with M representing any third-body molecule. This process aligns with the
observed decrease in CO mixing ratios at P$=10^{-4}-10^{-2}$ mbar, above which
CO is replenished by CO2 UV-photodissociation. It should be noted that the OCS
formation pathways here do not directly involve H2O or its photolysis.
Consequently, the OCS abundance is not sensitive to the condensation of water
in the upper atmosphere.

Although sulfur dioxide (SO2) is also predicted to form at higher altitude
(i.e., P$=10^{-4}-10^{-3}$ mbar) through a similar photochemical scheme
described in Tsai et al. (2023), its predicted abundance is insignificant to
appear as notable spectral features observed in the JWST observations of
WASP-39 b Alderson et al. (2023); Rustamkulov et al. (2023); Ahrer et al.
(2023); Feinstein et al. (2023); Powell et al. (2024).

Figure 4: Vertical molecular mixing ratio profiles of major species for the
constrained scenarios for K2-18 b and TOI-270 d in Section 3.1: K2-18 b with
an H2:H2O ratio of 50:50 and [C+N+S] of 100$\times Z_{\odot}$ assuming Ab =
0.3 (left), and TOI-270 d with an H2:H2O ratio of 75:25 and [C+N+S] of
100$\times Z_{\odot}$ assuming Ab = 0 (right). Each color indicates the
corresponding species: H2 in black, H2O in blue, CO2 in red, CH4 in green, H2S
in dark yellow, N2 in dark blue, SO2 in magenta, OCS in light blue, CO in
teal, and NH3 in light brown. The grey-shaded area indicates the JWST primary
probing range (i.e., P=0.1–2 mbar; Rustamkulov et al. (2023))

####  3.2.2 Theoretical Transmission Spectra of the Atmosphere of K2-18 b
generated by EPACRIS

The top panel of Figure 5 compares EPACRIS-generated theoretical transmission
spectra with JWST observations of K2-18 b (Madhusudhan et al., 2023). The
EPACRIS prediction aligns well with the JWST data, particularly in capturing
the CO2 (red) and CH4 (green) features identified by Madhusudhan et al.
(2023). Although NH3 was not detected in the retrieval by Madhusudhan et al.
(2023), EPACRIS-generated transmission spectra suggest its presence at around
3$\mu$m, as well as CO around 4.6$\mu$m, and notably, OCS around 4.8$\mu$m.
This highlights the need for more detailed observations at these spectral
windows to detect these molecules, which may have been overlooked due to low
resolution or insufficient transits. Any potential detection of NH3 would be
particularly significant as it can rule out the possibility of the Hycean
scenario for K2-18 b (Madhusudhan et al., 2021), underscoring the importance
of detailed observations. Additionally, the detection of OCS would be
noteworthy since this molecule has not been detected in exoplanetary
atmospheres so far.

###  3.3 The atmosphere of TOI-270 d

####  3.3.1 Overall chemistry in the atmosphere of TOI-270 d

The right panel of Figure 4 shows the simulated vertical molecular mixing
ratio profiles of major species for the scenario close to the most probable
scenario for TOI-270 d (i.e., H2:H2O=75:25 with [C+N+S]=100$\times Z_{\odot}$
assuming Ab=0) according to the photochemical mapping detailed in Section 3.1.
Different from the K2-18 b scenario, no significant condensation of water is
observed due to the much higher equilibrium temperature of TOI-270 d (Teq= 387
K) compared to that of K2-18 b (Teq= 255 K). As illustrated in Figure B2, all
T–P profiles of various planetary envelope composition scenarios of TOI-270 d
(see red and magenta lines in Figure B2) lie above the water condensation line
(Buck, 1981). Consequently, the entire atmosphere contains a significant
amount of water ($\geq$10%), leading to decreased spectral features due to an
increased mean molecular weight from H2O. The overall chemistry regarding
molecular species such as H2, CH4, CO2, N2, H2S, NH3, and OCS is similar to
that of K2-18 b, as described in Section 3.2.1. In contrast, the upper
atmosphere of TOI-270 d is richer in H2O compared to K2-18 b’s upper
atmosphere where water condensation occurs. This leads to increased SO2
formation due to more H2O being photolyzed by UV radiation, forming H and OH
radicals that gradually oxidize reduced sulfur species (e.g., H2S, S, and S2)
into SO2 (Tsai et al., 2023).

####  3.3.2 Theoretical Transmission Spectra of the Atmosphere of TOI-270 d
generated by EPACRIS

The bottom panel of Figure 5 compares EPACRIS-generated theoretical
transmission spectra with JWST observations of TOI-270 d (Benneke et al.,
2024). The EPACRIS prediction generally aligns well with the JWST data,
particularly in capturing the CO2 (red) and CH4 (green) features identified by
Benneke et al. (2024). Unlike K2-18 b, water is present in the JWST observable
part of TOI-270 d’s atmosphere and contributes to the spectral features.
Although a strong spectral feature around 4.7 $\mu$m cannot be solely
explained by CO-attributed absorption, EPACRIS-generated transmission spectra
suggest that the spectral modulation around 4.8$\mu$m is due to OCS. This,
again, underscores the importance of detailed observations at this spectra
range.

Figure 5: Comparisons between the theoretical transmission spectra generated
by EPACRIS (solid lines) and the JWST observations for (Top) H2:H2O=50:50 and
[C+N+S]=100$\times$solar metallicity scenario of K2-18 b assuming Ab = 0.3
corresponding to the left panel of Figure 4, and (Bottom) H2:H2O=75:25 and
[C+N+S]=100$\times$solar metallicity scenario of TOI-270 d assuming Ab = 0
corresponding to the right panel of Figure 4. The grey symbols with error bars
indicate JWST observations of corresponding exoplanets taken from Madhusudhan
et al. (2023) for K2-18 b (Top), and Benneke et al. (2024) for TOI-270 d
(Bottom). Each color represents a spectrum generated by excluding specific
species: green for no CH4, red for no CO2, light blue for no OCS, blue for no
H2O, teal for no CO, light brown for no NH3, and black for all species
included.

###  3.4 Discussions

In this study, we propose measuring the abundances of CO2 and CH4 (readily
observable within 3–5 $\mu$m by JWST) as the primary metric to determine the
bulk envelope H2O to H2 ratio of temperate sub-Neptunes. Additionally, we
suggest that other species such as CO, NH3, OCS, and SO2 (see Section 3.1.2)
could provide further constraints on the envelope composition despite
uncertainties in the deep-interior temperature. Notably, the inference of the
envelope H2O to H2 ratio does not involve direct measurements of H2O in the
observable part of the atmosphere, which can be depleted due to condensation.

Breaking the degeneracy in scenarios involving a hot interior is an intriguing
question. In the recent study of warm-Neptune WASP-107 b (Teq=750 K), the
combination of CO2 and CH4, which are jointly sensitive to Tint, was used to
decipher the intrinsic temperature of a warm-Neptune exoplanet (Sing et al.,
2024; Welbanks et al., 2024). As mentioned in Section 3.1.1, the model results
are sensitive to Tint when the atmosphere is relatively H2-dominated
($\leq$25% water-rich) as shown in Figure 1 and 2. This sensitivity largely
vanishes when the envelope becomes highly enriched in H2O. Beyond the CO2-CH4
abundances and ratio, our model suggests that the detection of SO2 in a
relatively H2-dominated atmosphere ($\leq$20% internal H2O envelope) of
temperate sub-Neptunes could disfavor a high intrinsic temperature
(Tint$\geq$100 K) scenario.

We have identified various novel formation pathways for SO2 in sub-Neptune
atmospheres. Unlike hot or warm-Jupiters or Neptunes, temperate sub-Neptunes
around M-dwarf stars experience lower radiation on their planetary
atmospheres, making the photochemical formation mechanism of SO2 proposed for
the atmosphere of WASP-39 b ineffective (Tsai et al., 2023). As shown in
Section 3.1.2, SO2 formation in temperate sub-Neptune atmospheres is complex
due to many contributing factors, with a novel mechanism using CO2 as the
oxidizer in weakly H2O-rich atmospheres and direct thermochemical formation in
highly H2O-rich atmospheres. These mechanisms indicate that it is likely to
detect SO2 on exoplanets with a wide range of envelope composition and
temperatures, and the interpretation of each detection will likely require
detailed photochemical modeling.

As shown in Figure B2, the equilibrium temperature differences between K2-18 b
and TOI-270 d distinctly categorize these planets based on whether water-
condensation occurs (K2-18 b) or not (TOI-270 d). The equilibrium temperature
of LP791-18 c falls between these two different regimes, presenting possible
degenerate scenarios. Therefore, by performing photochemical mapping of such
temperate sub-Neptunes (e.g., LP791-18 c), we can cover a broad range of
temperate sub-Neptunes with equilibrium temperatures (Teq) between 250 and 400
K (corresponds to 221 confirmed sub-Neptunes according to NASA’s Exoplanet
Archive (2024)). With the upcoming JWST observations of LP791-18 c, and many
other future observations of temperate sub-Neptunes, the current photochemical
mapping will significantly enhance our understanding of a planet’s interior
O/H composition and potentially its planetary formation mechanisms as well.

Beyond Teq range covered in this study (i.e., Teq$\geq$ 400 K), the
atmospheric chemistry favors CO2 formation over CH4 formation, rendering the
CO2/CH4 ratio less useful. The question remains as to the upper limit for this
equilibrium temperature. For example, if we can extend the applicable Teq to
500 K, there will be 473 corresponding confirmed sub-Neptunes (NASA’s
Exoplanet Archive, 2024) that can be investigated using the current framework.
According to Fortney et al. (2020), the CO2/CH4 ratio was significantly lower
than $10^{-3}$ at Teq = 710 K for 1–100$\times Z_{\odot}$ cases. This implies
that even at Teq = 500 K, thermochemical CO2 formation would still be much
less efficient compared to CH4 formation. Therefore, our current framework
that varies planetary envelope O/H ratios is still applicable to rationalize
any potential CO2 detection along with CH4 in sub-Neptunes with Teq$\sim$500 K
(e.g., Kepler 51 b with Teq=472 K; Libby-Roberts et al. (2020)) or even hotter
(e.g., GJ 3470 b with Teq=597 K; Benneke et al. (2019b)). However, it should
be noted that the planets withTeq of 500–800 K are more likely to have hazy
atmospheres, making them potentially more challenging to characterize via
transmission spectroscopy (Brande et al., 2024).

##  4 Conclusions

In this study, using self-consistent radiative transfer modeling (Scheucher &
Hu, 2024), and a rate-based automatic chemical network generator combined with
a 1D photochemical kinetic-transport atmospheric modeling (Yang & Hu, 2024),
we have extensively investigated various atmospheric scenarios of temperate
sub-Neptunes with Teq=250–400 K. We introduce a new framework that utilizes
the atmospheric CO2/CH4 ratio to gauge the bulk envelope H2O/H2 ratio of
temperate sub-Neptunes, providing new insights into the formation location of
sub-Neptunes relative to the ice-line. Furthermore, our models suggest that
any potential detection of OCS and SO2 serves as strong indicators of at least
a 10% water-rich envelope in temperate sub-Neptunes.

Benchmarking with recent JWST observations of two well-known temperate sub-
Neptunes, K2-18b (Madhusudhan et al., 2023) and TOI-270d (Benneke et al.,
2024), our modeling results suggest the following: The most probable scenario
for TOI-270d’s envelope is H2:H2O $\sim$ 75:25 with [C+N+S] = 100$\times
Z_{\odot}$, implying an original location inside the ice line during planetary
formation. For K2-18b’s envelope, the ratio is H2:H2O $\sim$ 50:50 with
[C+N+S] $\leq$ 100$\times Z_{\odot}$, implying a location beyond the ice line
during planetary formation. The synthesized transmission spectra based on
these models showed good agreement with the JWST observations (Madhusudhan et
al., 2023; Benneke et al., 2024).

While TOI-270 d can be explained by conventional multipliers of solar
metallicity, yielding approximately 230$\times Z_{\odot}$, K2-18 b cannot be
solely explained by the conventional solar metallicity framework. This
requires considering an additional variable: the deep-interior H2O/H2 ratio,
or planetary O/H ratio, which can potentially reveal the planets’ original
locations during their formation. Furthermore, our modeling results indicate
that any potential carbonyl sulfide (OCS) and sulfur dioxide (SO2) features
that appear in the JWST spectra are strong indicators of at least a 10%
H2O-rich envelope in temperate sub-Neptunes.

## Acknowledgements

This research work was carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the National
Aeronautics and Space Administration. This research work was funded by the
Caltech-JPL President’s and Director’s Research and Development Fund. © 2024.
California Institute of Technology. Government sponsorship acknowledged.

##  Appendix A elemental parameterization of planetary envelopes (Section 2.1)

Figure A1 and Table A1 present the various planetary envelope composition
scenarios for sub-Neptunes as discussed in Section 2.1.

Figure A 1: (Top) A schematic diagram of the protoplanetary disk with respect to the ice-line; (Bottom) elemental abundances in various planetary envelope composition scenarios plotted against the Oxygen-to-Hydrogen (O/H) ratio, spanning from hydrogen-rich (i.e., lower O/H) to water-rich (i.e., higher O/H) envelopes. The O/H ratio along the X-axis represents the location within the protoplanetary disk relative to the ice-line, where planets with H2-rich envelopes typically form inside the ice-line, and those with H2O-rich envelopes form beyond the ice-line, as depicted in the top figure. Each open symbol corresponds to different elemental abundance scenarios ($Z_{\odot}$ represents solar metallicity): hexagrams for 10${}^{4}\times Z_{\odot}$, asterisks for 10${}^{3}\times Z_{\odot}$ solar metallicity, crosses for carbon, nitrogen, and sulfur abundances, [C+N+S], of 100$\times Z_{\odot}$, circles for [C+N+S] = 10$\times Z_{\odot}$, and diamonds for [C+N+S] = 1$\times Z_{\odot}$. Colors represent different elemental species: hydrogen (black), helium (cyan), carbon (red), nitrogen (green), oxygen (light blue), and sulfur (light brown). Dotted lines indicate standard multipliers of solar metallicity for each element. Table A 1: Elemental composition profiles utilized to investigate a range of planetary atmospheric scenarios in this study. We explored atmospheres spanning from hydrogen-rich (H2) to water-rich (H2O) compositions and varying from 1$\times$ to 100$\times$ solar metallicity. For each scenario, we systematically varied the H2 to H2O ratio, beginning at 100:0 and progressing through intermediate stages of 90:10, 75:25, 50:50, 25:75, and 10:90, until reaching a completely water-rich environment at a 0:100 ratio. This range of scenarios, detailed in the main text and depicted in Figure A1, illustrates variations in planetary formation locations relative to the ice line. Additionally, some cases were modeled at higher metallicities, including 223$\times$, 1000$\times$, and 10000$\times$ solar metallicity. H2:H2O accretion ratio |  H2-rich | $\Longleftrightarrow$ |  H2O-rich
---|---|---|---
100:0 | 90:10 | 75:25 | 50:50 | 25:75 | 10:90 | 0:100
1$\times$ | H | 0.921775 | 0.881567 | 0.826877 | 0.749393 | 0.685186 | 0.651685 | 0.631113
He | 0.077379 | 0.074004 | 0.069413 | 0.062909 | 0.057519 | 0.054706 | 0.052979
C | 0.000272 | 0.000272 | 0.000272 | 0.000272 | 0.000272 | 0.000272 | 0.000272
N | 0.000065 | 0.000065 | 0.000065 | 0.000065 | 0.000065 | 0.000065 | 0.000065
O | 0.000495 | 0.044078 | 0.103360 | 0.187348 | 0.256945 | 0.293258 | 0.315557
S | 0.000013 | 0.000013 | 0.000013 | 0.000013 | 0.000013 | 0.000013 | 0.000013
10$\times$ | H | 0.914815 | 0.878810 | 0.824291 | 0.747049 | 0.683043 | 0.649647 | 0.629140
He | 0.076795 | 0.073773 | 0.069196 | 0.062712 | 0.057339 | 0.054535 | 0.052814
C | 0.002700 | 0.002700 | 0.002700 | 0.002700 | 0.002700 | 0.002700 | 0.002700
N | 0.000648 | 0.000648 | 0.000648 | 0.000648 | 0.000648 | 0.000648 | 0.000648
O | 0.004913 | 0.043941 | 0.103036 | 0.186762 | 0.256141 | 0.292341 | 0.314570
S | 0.000129 | 0.000129 | 0.000129 | 0.000129 | 0.000129 | 0.000129 | 0.000129
100$\times$ | H | 0.850591 | 0.853369 | 0.800428 | 0.725422 | 0.663269 | 0.630840 | 0.610926
He | 0.071404 | 0.071637 | 0.067193 | 0.060896 | 0.055679 | 0.052956 | 0.051285
C | 0.025103 | 0.025103 | 0.025103 | 0.025103 | 0.025103 | 0.025103 | 0.025103
N | 0.006022 | 0.006022 | 0.006022 | 0.006022 | 0.006022 | 0.006022 | 0.006022
O | 0.045679 | 0.042668 | 0.100053 | 0.181356 | 0.248726 | 0.283878 | 0.305463
S | 0.001201 | 0.001201 | 0.001201 | 0.001201 | 0.001201 | 0.001201 | 0.001201
223$\times$ | H | 0.748399 |  |  |  |  |  |
He | 0.062838 |  |  |  |  |  |
C | 0.060745 |  |  |  |  |  |
N | 0.014572 |  |  |  |  |  |
O | 0.110539 |  |  |  |  |  |
S | 0.002907 |  |  |  |  |  |
1000$\times$ | H | 0.499741 |  |  |  |  |  |
He | 0.041960 |  |  |  |  |  |
C | 0.147484 |  |  |  |  |  |
N | 0.035379 |  |  |  |  |  |
O | 0.268377 |  |  |  |  |  |
S | 0.007059 |  |  |  |  |  |
10000$\times$ | H | 0.097516 |  |  |  |  |  |
He | 0.008188 |  |  |  |  |  |
C | 0.287791 |  |  |  |  |  |
N | 0.069036 |  |  |  |  |  |
O | 0.523694 |  |  |  |  |  |
S | 0.013775 |  |  |  |  |  |
|  |  |  |  |  |  |  |

##  Appendix B The temperature-pressure (T–P) profiles for various planetary
envelope scenarios (Section 2.2)

Figure B2 represents the temperature-pressure T–P profiles for different
planetary envelope scenarios, calculated as described in Section 2.2.

Figure B 2: The figure shows 72 temperature-pressure T–P profiles for
different planetary envelope scenarios, calculated as described in Section 2.2
(1000$\times$ and 10000$\times$ solar metallicity cases for K2-18 b are
omitted for simplicity). $Z_{\odot}$ represents solar metallicity. Unless
otherwise specified, the planetary equilibrium temperature (i.e., Teq) assumes
a Ab of 0.3 and an intrinsic temperature (Tint) of 60 K by default. Each color
corresponds to a specific planetary envelope scenario: blue for K2-18 b, cyan
for K2-18 b assuming Tint=100 K, green for LP791-18 c, red for TOI-270 d,
black for the 223$\times Z_{\odot}$ scenario of TOI-270 d, and magenta for
TOI-270 d assuming Ab=0. The solid lines (except cyan and black) represent
seven different O/H scenarios (i.e., H2:H2O from 100:0 to 0:100) with the
elemental composition of carbon, nitrogen, and sulfur ([C+N+S]) of 100$\times
Z_{\odot}$, dashed lines for 10$\times Z_{\odot}$, and dotted lines for
1$\times Z_{\odot}$, as detailed in Section 2.1. The grey solid-dashed line
indicates the condensation curves for H2O (solid and liquid states) from Buck
(1981). As illustrated in the figure, only for cases of K2-18 b (i.e., blue
and cyan lines), water condenses to form clouds below the photosphere (i.e.,
the pressure range at $\sim$ 1 mbar where the JWST conducts its observations).
In contrast, other planets, LP791-18 c (green lines) and TOI-270 d (red,
black, and magenta lines), are too hot for water condensation to occur.

##  Appendix C RMG-generated species that are not included in the original
EPACRIS species library (Section 2.3)

Table C2 represents 36 newly included molecular species in the chemical
network tailored for H2O-dominated atmospheres by RMG described in Section
2.3.

Table C 2: 36 newly included molecular species in the chemical network tailored for H2O-dominated atmospheres by RMG Species | SMILESa | Species | SMILESa
---|---|---|---
CH2OHb,c | [CH2]O |  HCOH | [C-]=[OH+]
CH2CHO | [CH2]C=O |  H2CC | [C]=C
CH2CHOH | C=CO |  CH2CH2OH | [CH2]CO
CHCHO | [CH]=C[O] |  HCCOH | C#CO
CHCHOH | [CH]=CO |  CH3CH2OH | CCO
CH3CHOH | C[CH]O |  OCHCO | O=[C]C=O
HOCH2O | [O]CO |  OCHO | [O]C=O
CH3C(O)O | CC([O])=O |  HSO2 | O=[SH]=O
HOS | O[S] |  Sac | [S]d
HSSb,c | S=[SH] |  HSSHc | SS
CH2SH | [CH2]S |  HCCS | [S]C#C
H2SS | S=[SH2] |  NCN | [N]=C=[N]
CH2N2 | [CH2]N=[N] |  H2CNb,c | C=[N]
cNCN | C1=NN=1 |  CH2NCN | [CH2]N=C=[N]
CH2NHb,c | C=N |  NCOH | OC#N
NNHb,c | [N]=N |  CH3NHc | C[NH]
CH2NNH | [CH2]N=N |  CH2NNH2 | NN=C
CH2NH2b,c | [CH2]N |  CHNH | [CH]=N

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${\mathbf{g}}({\mathbf{x}}_{i})=z^{(j)}\varphi_{\operatorname{r}}(2z^{(j)}{\mathbf{w}}^{(j){\scriptscriptstyle{\mathsf{T}}}}{\mathbf{x}}_{i}-1)=z^{(j)}={\mathbf{y}}_{i}.\qed$

###  E.2 Proofs for Section 4.2

See 9

###### Proof.

Let ${\mathbf{G}}$ denote all solutions to (VP) on the sampled parity dataset,
so $\left\|{\mathbf{g}}-\chi\right\|_{L^{2}{(\boldsymbol{\nu}_{n})}}=0$ for
all ${\mathbf{g}}\in{\mathbf{G}}$. By Proposition 1, we can write each
${\mathbf{g}}$ as
${\mathbf{g}}(x)=g_{\boldsymbol{\mu}}(x)+{\mathbf{v}}^{\scriptscriptstyle{\mathsf{T}}}x+\mathbf{c}$,
where $\boldsymbol{\mu}\in\mathcal{M}$, ${\mathbf{v}}\in\mathbb{R}^{d}$, and
$\mathbf{c}\in\mathbb{R}$. Furthermore, we can assume that
$g_{\boldsymbol{\mu}}(0)=0$ by absorbing the value of
$g_{\boldsymbol{\mu}}(0)$ into $\mathbf{c}$ (at the cost of losing the
evenness of $\boldsymbol{\mu}$, but evenness is not needed in the sequel).
Lemma 1, Theorem 20, and Theorem 4 together imply that every
${\mathbf{g}}\in{\mathbf{G}}$ satisfies
$\left\|{\mathbf{g}}\right\|_{\mathcal{R}}\leq Cd$ for some absolute constant
$C>0$. Let $\mathcal{E}$ be the event that
$\max\\{|\mathbf{c}|,\|{\mathbf{v}}\|_{2}\\}\leq 8Cd$ (for all
${\mathbf{g}}\in{\mathbf{G}}$), and let $\mathcal{E}^{\operatorname{c}}$ be
its complement; event $\mathcal{E}$ occurs with probability at least
$1-\delta/2$ by Lemma 27, for another absolute constant $C^{\prime}>0$.

Since, for each ${\mathbf{g}}\in{\mathbf{G}}$, we have
${\mathbf{g}}({\mathbf{x}}_{i})=\chi({\mathbf{x}}_{i})$ for every example
$({\mathbf{x}}_{i},\chi({\mathbf{x}}_{i}))$ in the sampled parity dataset, it
follows that
$\left\|\operatorname{clip}\circ\,{\mathbf{g}}-\chi\right\|_{L^{2}{(\boldsymbol{\nu}_{n})}}=\left\|{\mathbf{g}}-\chi\right\|_{L^{2}{(\boldsymbol{\nu}_{n})}}=0$
for all such ${\mathbf{g}}\in{\mathbf{G}}$. For any $t>0$,

$\displaystyle\operatorname{\mathbb{P}}\left[\sup_{{\mathbf{g}}\in{\mathbf{G}}}\left\|\operatorname{clip}\circ\,{\mathbf{g}}-\chi\right\|_{L^{2}{(\nu)}}^{2}\geq
t\right]$
$\displaystyle\leq\operatorname{\mathbb{P}}\left[\sup_{{\mathbf{g}}\in{\mathbf{G}}}\left\|\operatorname{clip}\circ\,{\mathbf{g}}-\chi\right\|_{L^{2}{(\nu)}}^{2}\geq
t\mid\mathcal{E}\right]+\operatorname{\mathbb{P}}\left[\mathcal{E}^{\operatorname{c}}\right]$
$\displaystyle=\operatorname{\mathbb{P}}\left[\sup_{{\mathbf{g}}\in{\mathbf{G}}}\left\|\operatorname{clip}\circ\,{\mathbf{g}}-\chi\right\|_{L^{2}{(\nu)}}^{2}-\left\|\operatorname{clip}\circ\,{\mathbf{g}}-\chi\right\|_{L^{2}{(\boldsymbol{\nu}_{n})}}^{2}\geq
t\mid\mathcal{E}\right]+\operatorname{\mathbb{P}}\left[\mathcal{E}^{\operatorname{c}}\right]$
$\displaystyle\leq\operatorname{\mathbb{P}}\left[\sup_{g\in\mathcal{G}_{0}}\left\|\operatorname{clip}\circ\,g-\chi\right\|_{L^{2}{(\nu)}}^{2}-\left\|\operatorname{clip}\circ\,g-\chi\right\|_{L^{2}{(\boldsymbol{\nu}_{n})}}^{2}\geq
t\right]+\delta/2,$

where

$\mathcal{G}_{0}:=\left\\{x\mapsto
g(x)+v^{\scriptscriptstyle{\mathsf{T}}}x+c:\left\|g\right\|_{\mathcal{R}}\leq
Cd,\ \max\\{\|v\|_{2},|c|\\}\leq 8Cd\right\\}.$

Define

$t_{0}:=4\underbrace{\operatorname{\mathbb{E}}\left[\sup_{g\in\mathcal{G}_{0}}\frac{1}{n}\sum_{i=1}^{n}{\boldsymbol{\epsilon}}_{i}g({\mathbf{x}}_{i})\right]}_{\operatorname{Rad}_{n}(\mathcal{G}_{0})}\,+\,4\sqrt{\frac{\log(2/\delta)}{n}}.$

Above, $\operatorname{Rad}_{n}(\mathcal{G}_{0})$ denotes the Rademacher
complexity of $\mathcal{G}_{0}$, where
${\boldsymbol{\epsilon}}_{1},\dotsc,{\boldsymbol{\epsilon}}_{n}\sim_{\operatorname{iid}}\operatorname{Unif}(\\{\pm
1\\})$, independent of ${\mathbf{x}}_{1},\dotsc,{\mathbf{x}}_{n}$. Since, for
any $y\in\\{\pm 1\\}$, the mapping
$z\mapsto(y-\operatorname{clip}(z))^{2}=(1-y\operatorname{clip}(z))^{2}$ is
$4$-Lipschitz and has range $[-4,4]$, it follows by standard Rademacher
complexity arguments [see, e.g., Meir and Zhang, 2003, Theorem 8] that

$\operatorname{\mathbb{P}}\left[\sup_{g\in\mathcal{G}_{0}}\left\|\operatorname{clip}\circ\,g-\chi\right\|_{L^{2}{(\nu)}}^{2}-\left\|\operatorname{clip}\circ\,g-\chi\right\|_{L^{2}{(\boldsymbol{\nu}_{n})}}^{2}\geq
t_{0}\right]\leq\delta/2.$

So it remains to show that $t_{0}\leq\epsilon$ under the assumption $n\geq
C_{0}((d^{3}+\log(1/\delta))/\epsilon^{2})$ for suitably large absolute
constant $C_{0}>0$. The second term in the definition of $t_{0}$ is at most
$\epsilon/2$ provided that $C_{0}$ is chosen large enough. To bound the first
term ($\operatorname{Rad}_{n}(\mathcal{G}_{0})$), we use the fact that

$\operatorname{Rad}_{n}(\mathcal{G}_{0})=\operatorname{Rad}_{n}(\mathcal{G}_{1})+\operatorname{Rad}_{n}(\mathcal{G}_{2})$

where $\mathcal{G}_{1}:=\\{g:\left\|g\right\|_{\mathcal{R}}\leq Cd\\}$ and
$\mathcal{G}_{2}:=\\{x\mapsto
v^{\scriptscriptstyle{\mathsf{T}}}x+c:\max\\{\|v\|_{2},|c|\\}\leq 8Cd\\}$.
Theorem 10 of Parhi and Nowak [2021a] implies

$\operatorname{Rad}_{n}(\mathcal{G}_{1})\leq\frac{2\cdot(Cd)\cdot\sqrt{d}}{\sqrt{n}}=O\left(\sqrt{\frac{d^{3}}{n}}\right),$

while Theorem 3 of Kakade et al. [2008] implies

$\operatorname{Rad}_{n}(\mathcal{G}_{2})\leq\sqrt{d+1}\cdot\sqrt{(8Cd)^{2}}\cdot\sqrt{\frac{2}{n}}=O\left(\sqrt{\frac{d^{3}}{n}}\right).$

By choosing $C_{0}$ large enough, it follows that
$\operatorname{Rad}_{n}(\mathcal{G}_{0})\leq\epsilon/8$. Hence, we have shown
that $t_{0}\leq\epsilon$ as required. ∎

##  Appendix F Proofs for Section 5

###  F.1 Proof of Theorem 10

######  Theorem 30 (Detailed version of Theorem 10).

Suppose $f\colon\varOmega\to[-1,1]$ is given by
$f(x)=\phi(v^{\scriptscriptstyle{\mathsf{T}}}x)$ for some unit vector
$v\in\mathbb{S}^{d-1}$ and some $\phi\colon[-\sqrt{d},\sqrt{d}]\to[-1,1]$ that
is $L$-Lipschitz and $\rho$-periodic for
$\rho\in[\left\|v\right\|_{\infty},1]$. Let
$\sigma_{\rho,v}:=\sqrt{2\rho\|v\|_{1}-1}$, and fix any $\epsilon\in(0,1)$.
There exists a function $g\colon\varOmega\to\mathbb{R}$ such that the
following properties hold:

  1. 1.

$|f(x)-g(x)|\leq\epsilon$ for all $x\in\\{\pm 1\\}^{d}$;

  2. 2.

$g$ is represented by a neural network of width at most

$O\left(\frac{dL(\sigma_{\rho,v}\sqrt{\log(1/\epsilon)}+\rho\log(1/\epsilon))}{\epsilon^{2}}\right);$

  3. 3.

$g$ satisfies

$\left\|g\right\|_{\mathcal{R}}=O\left(\frac{L^{2}(\sigma_{\rho,v}\sqrt{\log(1/\epsilon)}+\rho\log(1/\epsilon))(\sigma_{\rho,v}+\sqrt{\log(d/\epsilon)})}{\epsilon}\right).$

###### Proof.

We first describe the (randomized) construction of our approximating neural
network ${\mathbf{g}}\colon\mathbb{R}^{d}\to\mathbb{R}$. For
$w\in\mathbb{Z}^{d}$, define $h_{w}\colon\mathbb{R}^{d}\to\mathbb{R}$ by
$h_{w}(x):=\phi(v^{\scriptscriptstyle{\mathsf{T}}}x+\rho
w^{\scriptscriptstyle{\mathsf{T}}}x)$. Let
${\mathbf{w}}\in\mathbb{Z}^{d}\setminus\\{-(1/\rho)v\\}$ be a random vector
with distribution to be specified later in the proof. Let
${\mathbf{w}}^{(1)},\dotsc,{\mathbf{w}}^{(k)}$ be i.i.d. copies of
${\mathbf{w}}$ for a positive integer $k>(9(d+1)\ln(2))/\epsilon$, and let
${\mathbf{h}}^{(j)}:=h_{{\mathbf{w}}^{(j)}}$ for each $j$. Observe that each
${\mathbf{h}}^{(j)}$ can be written as
${\mathbf{h}}^{(j)}(x)=\phi^{(j)}(x^{\scriptscriptstyle{\mathsf{T}}}{\mathbf{u}}^{(j)})$
for

${\mathbf{u}}^{(j)}:=\frac{1}{\|v+\rho{\mathbf{w}}^{(j)}\|_{2}}(v+\rho{\mathbf{w}}^{(j)})\in\mathbb{S}^{d-1}\quad\text{and}\quad\phi^{(j)}(z):=\phi(\|v+\rho{\mathbf{w}}^{(j)}\|_{2}z),$

where $\phi^{(j)}\colon\mathbb{R}\to[-1,1]$ is $L_{j}$-Lipschitz for
$L_{j}:=L\|v+\rho{\mathbf{w}}^{(j)}\|_{2}$ (using the $L$-Lipschitzness of
$\phi$). Let $\tau>0$ be a value (depending on $\rho$, $v$, and $\epsilon$)
also to be specified later. By Lemma 31 (with
$t:=\tau/\|v+\rho{\mathbf{w}}^{(j)}\|_{2}$ and $\delta:=\epsilon/3$), there
exist $\tilde{\mathbf{h}}^{(1)},\dotsc,\tilde{\mathbf{h}}^{(k)}$ such that:

  * •

(H1) $\tilde{\mathbf{h}}^{(j)}\colon\mathbb{R}^{d}\to\mathbb{R}$ is
represented by a neural network of width at most $O(\tau L/\epsilon)$;

  * •

(H2) $\|\tilde{\mathbf{h}}^{(j)}\|_{\mathcal{R}}=O(\tau
L^{2}\|v+\rho{\mathbf{w}}^{(j)}\|_{2}/\epsilon)$;

  * •

(H3) $|\tilde{\mathbf{h}}^{(j)}(x)-{\mathbf{h}}^{(j)}(x)|\leq\epsilon/3$ for
all $x\in\\{\pm 1\\}^{d}$ such that
$|x^{\scriptscriptstyle{\mathsf{T}}}{\mathbf{u}}^{(j)}|\leq\tau/\|v+\rho{\mathbf{w}}^{(j)}\|_{2}$;

  * •

(H4) $|\tilde{\mathbf{h}}^{(j)}(x)-{\mathbf{h}}^{(j)}(x)|\leq 1$ for all
$x\in\mathbb{R}^{d}$.

Our approximating neural network
${\mathbf{g}}\colon\mathbb{R}^{d}\to\mathbb{R}$ is defined by

${\mathbf{g}}(x):=\frac{1}{k}\sum_{j=1}^{k}\tilde{\mathbf{h}}^{(j)}(x).$

By construction and using properties H1 and H2 (above), the following
properties of ${\mathbf{g}}$ are immediate:

  * •

(G1) ${\mathbf{g}}$ is represented by a neural network of width at most
$O(k\tau L/\epsilon)$;

  * •

(G2)
$\max\\{\|{\mathbf{g}}\|_{\mathcal{R}},\|{\mathbf{g}}\|_{\mathscr{V}_{2}}\\}=O(\tau
L^{2}\max_{j\in[k]}\|v+\rho{\mathbf{w}}^{(j)}\|_{2}/\epsilon)$.

Note that these properties are given in terms of $\tau$, which has yet to be
specified, as well as $\max_{j\in[k]}\|v+\rho{\mathbf{w}}^{(j)}\|_{2}$, which
is a random variable. So, in the remainder of the proof, we choose a
particular distribution for ${\mathbf{w}}$ (and hence also for
${\mathbf{w}}^{(1)},\dotsc,{\mathbf{w}}^{(k)}$) and a value of $\tau$ that,
together, will ultimately allow us to establish the existence of an
approximating neural network with the desired properties via the probabilistic
method.

We first specify the probability distribution of ${\mathbf{w}}$ and establish
some of its properties. We let
${\mathbf{w}}=({\mathbf{w}}_{1},\dotsc,{\mathbf{w}}_{d})$ be a vector of
independent random variables ${\mathbf{w}}_{1},\dotsc,{\mathbf{w}}_{d}$ with
$p_{i}:=\operatorname{\mathbb{P}}[{\mathbf{w}}_{i}=-2\operatorname{sign}\left(v_{i}\right)]=\left|v_{i}\right|/(2\rho)$
and $\operatorname{\mathbb{P}}[{\mathbf{w}}_{i}=0]=1-p_{i}$. Note that
$p_{i}\in[0,1/2]$ for all $i$ since we have assumed
$\rho\geq\left\|v\right\|_{\infty}$, so the distribution of ${\mathbf{w}}$ is
well-defined. Furthermore, observe that ${\mathbf{w}}\neq-(1/\rho)v$ almost
surely (since $v\neq 0$ and $\rho\geq\|v\|_{\infty}$ by assumption),
$\operatorname{\mathbb{E}}[v+\rho{\mathbf{w}}]=0$, and

$\operatorname{\mathbb{E}}\left[\left\|v+\rho{\mathbf{w}}\right\|_{2}^{2}\right]=\sum_{i=1}^{d}\mathop{\operatorname{Var}}(\rho{\mathbf{w}}_{i})=\sum_{i=1}^{d}4\rho^{2}\cdot\frac{\left|v_{i}\right|}{2\rho}\cdot\left(1-\frac{\left|v_{i}\right|}{2\rho}\right)=2\rho\left\|v\right\|_{1}-\left\|v\right\|_{2}^{2}=\sigma_{\rho,v}^{2}.$

Moreover, $\|v+\rho{\mathbf{w}}\|_{2}$ is a function of independent random
variables ${\mathbf{w}}_{1},\dotsc,{\mathbf{w}}_{d}$ that satisfies the
$(2|v_{1}|,\dotsc,2|v_{d}|)$-bounded differences property. By McDiarmid’s
inequality (Lemma 15) and Jensen’s inequality,

$\operatorname{\mathbb{P}}\left[\|v+\rho{\mathbf{w}}\|_{2}\geq\sigma_{\rho,v}+\sqrt{2\ln(1/\delta)}\right]\leq\delta\quad\text{for
all $\delta\in(0,1)$}.$ (2)

Finally, for any fixed $x\in\\{\pm 1\\}^{d}$,
$x^{\scriptscriptstyle{\mathsf{T}}}(v+\rho{\mathbf{w}})=\sum_{i=1}^{d}x_{i}(v_{i}+\rho{\mathbf{w}}_{i})$
is a sum of $d$ independent, mean-zero random variables, with variance
$\mathop{\operatorname{Var}}\left(x^{\scriptscriptstyle{\mathsf{T}}}(v+\rho{\mathbf{w}})\right)=\sigma_{\rho,v}^{2}$
and $|x_{i}(v_{i}+\rho{\mathbf{w}}_{i})|\leq 2\rho$ almost surely for each
$i$. By Bernstein’s inequality (Lemma 14),

$\operatorname{\mathbb{P}}\left[\left|x^{\scriptscriptstyle{\mathsf{T}}}(v+\rho{\mathbf{w}})\right|\geq\sigma_{\rho,v}\sqrt{2\ln(2/\delta)}+2\rho\ln(2/\delta)/3\right]\leq\delta\quad\text{for
all $x\in\\{\pm 1\\}^{d}$, $\delta\in(0,1)$}.$ (3)

We now show that ${\mathbf{g}}$ has the desired properties with positive
probability. Since $w^{\scriptscriptstyle{\mathsf{T}}}x$ is an integer for any
$w\in\mathbb{Z}^{d}$ and $x\in\\{\pm 1\\}^{d}$, and since
$v+\rho{\mathbf{w}}^{(j)}\neq 0$ almost surely, the $\rho$-periodicity of
$\phi$ implies that $g_{{\mathbf{w}}^{(j)}}(x)=f(x)$ for all $x\in\\{\pm
1\\}^{d}$ and all $j\in[k]$. Therefore, the intermediate (random) function
${\mathbf{g}}_{1}\colon\mathbb{R}^{d}\to[-1,1]$ defined by
${\mathbf{g}}_{1}(x):=\frac{1}{k}\sum_{j=1}^{k}{\mathbf{h}}^{(j)}(x)$
satisfies ${\mathbf{g}}_{1}(x)=f(x)$ for all $x\in\\{\pm 1\\}^{d}$. For each
$x\in\\{\pm 1\\}^{d}$, let

${\mathbf{r}}(x):=|\\{j\in[k]:|x^{\scriptscriptstyle{\mathsf{T}}}(v+\rho{\mathbf{w}}^{(j)})|\geq\tau\\}|=|\\{j\in[k]:|x^{\scriptscriptstyle{\mathsf{T}}}{\mathbf{u}}^{(j)}|\geq\tau/\|v+\rho{\mathbf{w}}^{(j)}\|_{2}\\}|.$

Using the approximation properties of $\tilde{\mathbf{h}}^{(j)}$ (i.e., H3 and
H4 from above), we have for each $x\in\\{\pm 1\\}^{d}$,

$\left|{\mathbf{g}}(x)-{\mathbf{g}}_{1}(x)\right|=\frac{1}{k}\left|\sum_{j=1}^{k}(\tilde{\mathbf{h}}^{(j)}(x)-{\mathbf{h}}^{(j)}(x))\right|\leq\left(1-\frac{{\mathbf{r}}(x)}{k}\right)\cdot\frac{\epsilon}{3}+\frac{{\mathbf{r}}(x)}{k}\cdot
1.$

This final expression is at most $\epsilon$ if ${\mathbf{r}}(x)\leq
2k\epsilon/3$. We choose $\tau$ such that for any $x\in\\{\pm 1\\}^{d}$, we
have
$\operatorname{\mathbb{P}}[|x^{\scriptscriptstyle{\mathsf{T}}}(v+\rho{\mathbf{w}})|>\tau]\leq\epsilon/3$.
By (3), it suffices to choose

$\tau:=\sigma_{\rho,v}\sqrt{2\ln(6/\epsilon)}+2\rho\ln(6/\epsilon)/3.$

By a multiplicative Chernoff bound (Lemma 13) and a union bound over all
$x\in\\{\pm 1\\}^{d}$,

$\operatorname{\mathbb{P}}\left[\exists x\in\\{\pm 1\\}^{d}\ \text{s.t.}\
{\mathbf{r}}(x)>2k\epsilon/3\right]\leq 2^{d}\cdot
e^{-k\epsilon/9}<\frac{1}{2},$

where the final inequality uses the choice of $k>(9(d+1)\ln 2)/\epsilon$.
Therefore, with probability more than $1/2$, we have ${\mathbf{r}}(x)\leq
2k\epsilon/3$ for all $x\in\\{\pm 1\\}^{d}$, and hence

$|{\mathbf{g}}(x)-f(x)|=|{\mathbf{g}}(x)-{\mathbf{g}}_{1}(x)|\leq\epsilon\quad\text{for
all $x\in\\{\pm 1\\}^{d}$}.$ (4)

Finally, by (2) an a union bound over all $j\in[k]$, we have that with
probability more than $1/2$,

$\max_{j\in[k]}\|v+\rho{\mathbf{w}}^{(j)}\|_{2}\leq\sigma_{\rho,v}+\sqrt{2\ln(2k)}.$
(5)

So, there is a positive probability that both (4) and (5) hold simultaneously,
and in this event, it can be checked (via G1 and G2 above) that the function
${\mathbf{g}}$ satisfies the desired properties in the theorem. ∎

######  Lemma 31.

Suppose $f(x)=\phi(v^{\scriptscriptstyle{\mathsf{T}}}x)$ is an $L$-Lipschitz
function for $v\in\mathbb{S}^{d-1}$, $\phi:\mathbb{R}\to[-1,1]$, and $L\geq
1$. For any $t\in[1,\sqrt{d}-1]$ and $\delta\in(0,1)$, there exists a neural
network $g$ of width $O(\frac{tL}{\delta})$ such that:

  1. 1.

$\left\|g\right\|_{\mathcal{R}}=O(\frac{tL^{2}}{\delta})$;

  2. 2.

$\left|f(x)-g(x)\right|\leq\delta$ for all $x$ with
$\left|v^{\scriptscriptstyle{\mathsf{T}}}x\right|\leq t$;

  3. 3.

$\left|f(x)-g(x)\right|\leq 1$ for all $x\in\mathbb{R}^{d}$;

  4. 4.

$g(x)=0$ for all $x$ with
$\left|v^{\scriptscriptstyle{\mathsf{T}}}x\right|\geq t+\frac{1}{L}$; and

  5. 5.

$g$ is a ridge function that in direction $v$.

###### Proof.

We first introduce an $L$-Lipschitz function $\phi_{t}$ (visualized in Figure
1) that perfectly fits $\phi$ on the interval $[-t,t]$ and is zero in
$(\infty,-t-\frac{1}{L}]\cup[t+\frac{1}{L},\infty]$:

$\phi_{t}(z):=\begin{cases}\phi(z)&\text{if $z\in[-t,t]$};\\\
\operatorname{sign}\left(\phi(t)\right)\max\left\\{\left|\phi(t)\right|-L(z-t)),0\right\\}&\text{if
$z\geq t$};\\\
\operatorname{sign}\left(\phi(-t)\right)\max\left\\{\left|\phi(-t)\right|-L(-z+t)),0\right\\}&\text{if
$z\leq-t$}.\end{cases}$

Figure 1: A visualization of how the truncated $\phi_{t}$ (gray) is generated
from $\phi$ (blue), $t$, and $L$.

Then, there exists a piecewise-linear function $\psi_{t}$ that

  * •

point-wise approximates $\phi_{t}$ to accuracy $\delta$;

  * •

has $\psi_{t}(z)=\phi_{t}(z)$ for all $z\not\in[-t,t]$;

  * •

has $\frac{2tL}{\delta}$ evenly-spaced knots on the interval $[-t,t]$ where
$\psi_{t}$ exactly fits $\phi_{t}$; and

  * •

is $L$-Lipschitz.

As a result $\psi_{t}$ can be written as a neural network with
$\psi_{t}(z)=\sum_{j=1}^{m}a^{(j)}\varphi_{\operatorname{r}}{z-b^{(j)}}$ where
$m=\frac{2Lt}{\delta}$, $b^{(j)}\in[-t-\frac{1}{L},t+\frac{1}{L}]$, and
$\left|a^{(j)}\right|\leq 2L$.

By taking $g(x):=\psi_{t}(v^{\scriptscriptstyle{\mathsf{T}}}x)$, we have a
neural network that satisfies conditions 2, 3, 4, and 5. The bound on
$\left\|g\right\|_{\mathcal{R}}$ is immediate from the fact that $g$ can be
expressed as a neural network with $O(\frac{tL}{\delta})$ neurons with unit
weights, biases in $[-\sqrt{d},\sqrt{d}]$, and bounded coefficients $a^{(j)}$.
∎

###  F.2 Proof of Theorem 11

See 11

###### Proof.

We prove the claim by a reduction to Theorem 3. That is, we show that an
interpolant with better $\mathcal{R}$-norm than the bound stipulates can be
used to construct a neural network that contradicts Theorem 3.

To do so, we consider a lower dimension $d^{\prime}=4\lfloor d/4q\rfloor-4$
and create a mapping from points $z\in\\{\pm 1\\}^{d^{\prime}}$ to
$x_{z}\in\\{\pm 1\\}^{d}$. We define $a\in[0,4q-1]$ such that $2a\equiv
d\pmod{4q}$. For any $z$, we define $x_{z}$ as follows:

$x_{z}=(\underbrace{z_{1},\dots,z_{1}}_{q},\dots,\underbrace{z_{d^{\prime}},\dots,z_{d^{\prime}}}_{q},\underbrace{1,\dots,1}_{a},\underbrace{-1,\dots,-1}_{d-d^{\prime}q-a}).$

Observe that

$\vec{1}^{\scriptscriptstyle{\mathsf{T}}}x_{z}=q\vec{1}^{\scriptscriptstyle{\mathsf{T}}}z+2a-d+d^{\prime}q\equiv
q\vec{1}^{\scriptscriptstyle{\mathsf{T}}}z\pmod{4q}.$

Due to the periodicity of cosine and the fact that $d^{\prime}$ is a multiple
of 4,

$\cos(\frac{2\pi}{\rho}v^{\scriptscriptstyle{\mathsf{T}}}x_{z})=\cos(\frac{\pi}{2}\vec{1}^{\scriptscriptstyle{\mathsf{T}}}z)=\chi(z).$

Consider some $g(x)=\phi(w^{\scriptscriptstyle{\mathsf{T}}}x)$ with
$\|g-\cos(\frac{2\pi}{\rho}v^{\scriptscriptstyle{\mathsf{T}}}\cdot)\|_{\infty}\leq\frac{1}{2}$.
Define $w^{\prime}\in\mathbb{R}^{d^{\prime}}$ such that
$w^{\prime}_{i}:=\sum_{j=1}^{q}w_{(i-1)q+j}$. Observe that
$\left\|w^{\prime}\right\|_{2}\leq\sqrt{q}$ and that
$w^{\scriptscriptstyle{\mathsf{T}}}x_{z}=w^{\prime{\scriptscriptstyle{\mathsf{T}}}}z+c_{w}$,
where $c_{w}$ depends only on the remaining elements of $w$. Define
$\tilde{g}(z)=\phi(w^{\prime{\scriptscriptstyle{\mathsf{T}}}}z+c_{w})$. Then,

$|\tilde{g}(z)-\chi(z)|=|\phi(w^{\scriptscriptstyle{\mathsf{T}}}x_{z})-\cos(\frac{2\pi}{\rho}v^{\scriptscriptstyle{\mathsf{T}}}x_{z})|\leq\frac{1}{2}$

for all $z\in\\{\pm 1\\}^{d^{\prime}}$. Since translation can only decrease
the $\mathcal{R}$-norm (by exhausting some neurons to effectively behave
linearly in the domain) namely,
$\left\|\tilde{g}\right\|_{\mathcal{R}}\leq\left\|w^{\prime}\right\|_{2}\left\|\phi^{\prime}\right\|_{\operatorname{TV}}=\left\|w^{\prime}\right\|_{2}\left\|g\right\|_{\mathcal{R}}$,
Theorem 3 implies that $\left\|g\right\|_{\mathcal{R}}=\Omega({d^{\prime
3/2}}/{\sqrt{q}})$. The theorem statement follows by plugging in $q$ and
$d^{\prime}$. ∎